A mathematical model of the slow force response to stretch in rat ventricular myocytes.
ABSTRACT We developed a model of the rat ventricular myocyte at room temperature to predict the relative effects of different mechanisms on the cause of the slow increase in force in response to a step change in muscle length. We performed simulations in the presence of stretch-dependent increases in flux through the Na(+)-H(+) exchanger (NHE) and Cl(-)-HCO(3)(-) exchanger (AE), stretch-activated channels (SAC), and the stretch-dependent nitric oxide (NO) induced increased open probability of the ryanodine receptors to estimate the capacity of each mechanism to produce the slow force response (SFR). Inclusion of stretch-dependent NHE & AE, SACs, and stretch-dependent NO effects caused an increase in tension following 15 min of stretch of 0.87%, 32%, and 0%, respectively. Comparing [Ca(2+)](i) dynamics before and after stretch in the presence of combinations of the three stretch-dependent elements, which produced significant SFR values (>20%), showed that the inclusion of stretch-dependent NO effects produced [Ca(2+)](i) transients, which were not consistent with experimental results. Further simulations showed that in the presence of SACs and the absence of stretch-dependent NHE & AE inhibition of NHE attenuated the SFR, such that reduced SFR in the presence of NHE blockers does not indicate a stretch dependence of NHE. Rather, a functioning NHE is responsible for a portion of the SFR. Based on our simulations we estimate that in rat cardiac myocytes at room temperature SACs play a significant role in producing the SFR, potentially in the presence of stretch-dependent NHE & AE and that NO effects, if any, must involve more mechanisms than just increasing the open probability of ryanodine receptors.
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ABSTRACT: Stretch induces immediate and delayed inotropic effects in mammalian myocardium via distinct mechanosensitive pathways, but these effects are poorly characterized in human cardiac muscle. We tested the effects of stretch on immediate and delayed force response in failing human myocardium. Experiments were performed in muscle strips from 52 failing human hearts (37 degrees C, 1 Hz, bicarbonate buffer). Muscles were stretched from 88% of optimal length to 98% of optimal length. The resulting immediate and delayed (ie, slow force response [SFR]) increases in twitch force were assessed without and after blockade of the sarcoplasmic reticulum (SR; cyclopiazonic acid and ryanodine), stretch-activated ion channels (SACs; gadolinium, streptomycin), L-type Ca2+-channels (diltiazem), angiotensin II type-1 (AT1) receptors (candesartan), endothelin (ET) receptors (PD145065 or BQ123), Na+/H+ exchange (NHE1; HOE642), or reverse-mode Na+/Ca+ exchange (NCX; KB-R7493). We also tested the effects of stretch on SR Ca2+ load (rapid cooling contractures [RCCs]) and intracellular pH (in BCECF-loaded trabeculae). Stretch induced an immediate (<10 beats), followed by a slow (5 to 10 minutes), force response. Twitch force increased to 232+/-6% of prestretch value during the immediate phase, followed by a further increase to 279+/-8% during the SFR. RCC amplitude significantly increased, but pHi did not change during SFR. Inhibition of SACs, L-type Ca2+ channels, AT1 receptors, or ET receptors did not affect the stretch-dependent immediate or SFR. In contrast, the SFR was reduced by NHE1 inhibition and almost completely abolished by reverse-mode NCX inhibition or blockade of sarcoplasmic reticulum function. The data demonstrate the existence of a functionally relevant, SR-Ca2+-dependent SFR in failing human myocardium, which partly depends on NHE1 and reverse-mode NCX activation.Circulation Research 06/2004; 94(10):1392-8. · 11.86 Impact Factor
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ABSTRACT: 1. Liquid ion-exchanger Cl- -sensitive micro-electrodes were used to make continuous measurements of the intracellular Cl activity, aCli, of quiscent sheep cardiac Purkinje fibres in vitro. 2. aCli was higher than that expected from a passive distribution, (which would have been about 5 mM). It was 3--4 times hiable; EC1 was about 35 mV positive to Em. It was over twice as high in the nominal absence of bicarbonate/CO2 (when the buffer-system was HEPES/O2) but was not always so stable, and ECl was about 20 mV positive to Em. 3. Experiments designed to assess the maximum possible error likely to occur in the measurement of aCli showed that this could not be large and that the estimates of ECl were accurate to within 8 mV. 4. The ability of Cl to move down both concentration and potential gradients was established by demonstrating a loss of aCli in Cl-free solutions and a gain when Em was depolarized positive to ECl in high-K solutions. In both cases, the changes were complete within about 100--160 min. 5. The decline of aCli in Cl-free solutions (glucuronate-substituted) was not significantly affected by changes of [Ca]o from 0 to 12 mM or by the depolarizations of Em of up to 60 mV that sometimes occurred in low or zero [Ca]o. 6. Only 2--3 mM-aClo was sufficient to impede substantially the ready loss of aCli in HEPES-buffered solutions. 7. In high-K solutions (45 mM), Cl appeared to be passively distributed since, at equilibrium, Em and ECl differed by less than 2 mV. 8. In HEPES-buffered Tyrode, ECl of quiescent papillary muscle of the guinea-pig was, on average, 39 mV positive to Em. 9. It is concluded that liquid ion-exchanger Cl- -sensitive micro-electrodes are suitable for studying the Cl regulation of sheep Prukinje fibres, and probably of other cardiac tissues. The measurements of resting aCli are quite accurate when using either HEPES or bicarbonate-buffered Tyrode. The results are discussed in relation to estimates of the apparent membrane Cl permeability under various conditions and the possible existence of an inwardly directed 'Cl pump'.The Journal of Physiology 11/1979; 295:83-109. · 4.38 Impact Factor
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ABSTRACT: Intracellular [Na+] ([Na+]i) is centrally involved in regulation of cardiac Ca2+ and contractility via Na+-Ca2+ exchange (NCX) and Na+-H+ exchange (NHX). Previous work has indicated that [Na+]i is higher in rat than rabbit ventricular myocytes. This has major functional consequences, but the reason for the higher [Na+]i in rat is unknown. Here, resting [Na+]i was measured using the fluorescent indicator SBFI, with both traditional calibration and a novel null-point method (which circumvents many limitations of prior methods). In rabbit, resting [Na+]i was 4.5 +/- 0.4 mM (traditional calibration) and 4.4 mM (null-point). Resting [Na+]i in rat was significantly higher using both the traditional calibration (11.1 +/- 0.7 mM) and the null-point approach (11.2 mM). The rate of Na+ transport by the Na+ pump was measured as a function of [Na+]i in intact cells. Rat cells exhibited a higher V(max) than rabbit (7.7 +/- 1.1 vs. 4.0 +/- 0.5 mM x min(-1)) and a higher K(m) (10.2 +/- 1.2 vs. 7.5 +/- 1.1 mM). This results in little difference in pump activity for a given [Na+]i below 10 mM, but at measured resting [Na+]i levels the pump-mediated Na+ efflux is much higher in rat. Thus, Na+ pump rate cannot explain the higher [Na+]i in rat. Resting Na+ influx rate was two to four times higher in rat, and this accounts for the higher resting [Na+]i. Using tetrodotoxin, HOE-642 and Ni2+ to block Na+ channels, NHX and NCX, respectively, we found that all three pathways may contribute to the higher resting Na+ influx in rat (albeit differentially). We conclude that resting [Na+]i is higher in rat than in rabbit, that this is caused by higher resting Na+ influx in rat and that a higher Na+,K+-ATPase pumping rate in rat is a consequence of the higher [Na+]i.The Journal of Physiology 03/2002; 539(Pt 1):133-43. · 4.38 Impact Factor
A Mathematical Model of the Slow Force Response to Stretch in Rat
Steven A. Niederer*yand Nicolas P. Smith*y
*Bioengineering Institute, University of Auckland, Auckland, New Zealand; andyComputing Laboratory, University of Oxford,
the presence of stretch-dependent increases in flux through the Na1-H1exchanger (NHE) and Cl?-HCO?
receptors to estimate the capacity of each mechanism to produce the slow force response (SFR). Inclusion of stretch-dependent
0%, respectively. Comparing [Ca21]idynamics before and after stretch in the presence of combinations of the three stretch-
dependent elements, which produced significant SFR values (.20%), showed that the inclusion of stretch-dependent NO effects
of SACs and the absence of stretch-dependent NHE & AE inhibition of NHE attenuated the SFR, such that reduced SFR in the
the SFR. Based on our simulations we estimate that in rat cardiac myocytes at room temperature SACs play a significant role in
producing the SFR, potentially in the presence of stretch-dependent NHE & AE and that NO effects, if any, must involve more
mechanisms than just increasing the open probability of ryanodine receptors.
We developed a model of the rat ventricular myocyte at room temperature to predict the relative effects of different
function increased diastolic filling results in an increase in
cardiac output to match venous return. If the increased
diastolic volume is maintained, the myocardium responds
increase in developed tension is attributed to the Frank
Starling mechanism and occurs independently of changes in
by the combination of length-dependent increases in both
isometric tension and Ca21sensitivity (1). The slow force
response (SFR) to stretch, which follows this initial increase,
occurs overa periodof 10–15 min and resultsin a further 20–
50% increase in actively generated tension (2). Although the
myocyte (4) preparations the underlying mechanism(s) and
signaling pathway(s) remain debated (5). Recently three
potential hypotheseshave emergedas thelikelycandidates to
cause the SFR: 1), stretch activated flux through the Na1-H1
exchanger (NHE) (6–8), 2), increased conductance of cations
through stretch-activated channels (SAC) (6), and 3), nitric
oxide (NO) signaling (9).
The SFR is often (7,8,10), although not always (4),
observed in the presence of increased [Na1]i. Recently the
increase in [Na1]ihas been attributed to an increase in NHE
activity due to an increase in the rate of exchanger cycling
through a stretch-activated pathway (7,8,10–12). The re-
sulting increase in [Na1]i, in turn, enhances the [Ca21]i
influx on the Na1-Ca21exchanger (NCX) increasing the
peak of the [Ca21]itransient and hence developed tension
SACs are reported as modulators of the action potential
(13) and [Ca21]itransient (14,15) morphology in response to
stretch. Zeng et al. (16) reported that SACs in rat cardiac
myocytes conduct both K1and Na1, although not Ca21.
Hence, nonspecific cation SACs provide a potential path for
Na1into the cell following stretch, resulting in an increase in
[Na1]iand the corresponding enhanced [Ca21]iinflux on
NCX, as outlined above for stretch-activated NHE. Calaghan
and White (6) found SACs to have a significant effect on
the SFR in rat myocytes and papillary muscles. However,
von Lewinski and co-workers found no effect of SACs on
the slow force response in both human (8) and rabbit (11)
(17,18). Recently, Villa-Petroff et al. (9) found an increase in
the open probability of the ryanodine receptor (RyRs) in the
presence of a stretch-induced increase in NO. Villa-Petroff
et al. (9) hypothesized that the increased open probability
of the RyRs would cause the SFR. In this hypothesis it was
postulated that NO regulation would increase the quantity
This NO mediated mechanism clearly requires a func-
tioning SR, specifically one that contains Ca21and releases
Ca21during excitation. Inhibition of Ca21release and
Submitted August 19, 2006, and accepted for publication January 18, 2007.
Address reprint requests to Nicolas P. Smith, Oxford University, Comput-
ing Laboratory, Wolfson Bldg., Parks Rd., Oxford, OX1 3QD, UK. E-mail:
? 2007 by the Biophysical Society
4030 Biophysical Journal Volume 92June 2007 4030–4044
uptake from the SR causes a significant decrease in SFR in
failing human myocardium (11) but not in rat (1,6) or rab-
bit (19). Calaghan and White (6) were also unable to detect
any effect of NO on the SFR in rat isolated myocytes and
papillary preparations when NO synthesis was inhibited by
A wide range of experimental data is available to quantify
the electrophysiology and contraction of cardiac myocytes,
models. Models provide a means to test both the individual
SFR. As recent experimental studies (6) and reviews (17)
have postulated that a combination of stretch-dependent ele-
ments may cause the SFR.
Previous modeling studies have identified transmembrane
Na1flux as the most likely cause of the SFR to stretch (20)
and quantified the effects of SACs and tension-dependent
the model was based on the Luo Rudy guinea pig cell model
(22) and compared with either rabbit (20) or rat (21) SFR
at room temperature to make direct comparisons between
simulations and experiments, within a framework, which
maintains a consistent species focus both in model simula-
tions and in comparisons with experimental results. The
framework developed is capable of simulating the major
to test new hypotheses in the future. The electrophysiology is
quantified by the Pandit et al. (23) rat myocyte model. The
Ca21dynamics were replaced with the recently proposed
coupled L-type Ca21channel-RyR (LCC-RyR) model,
developed by Hinch et al. (24), which has been fitted using
rat myocyte data. Active tension is calculated using the
framework proposed by Niederer et al. (25), also parameter-
ized from largely rat literature. The electrophysiology, Ca21
dynamics, and active contraction models form a base compu-
tational representation of a cardiac cell.
To represent the stretch-dependent mechanisms, we incor-
porate the additional modeling elements of stretch-dependent
pH regulation, SACs, and stretch-dependent increased NO
production and the resulting increased Ca21release from the
RyRs. We then estimate the relative effects of individual and
combinations of stretch-dependent elements on the SFR and
[Ca21]itransient. The simulation results are then compared
with experimental data to estimate the feasibility of and
The base cardiac cell model quantifies the Ca21dynamics, active tension,
and the membrane potential. The stretch-dependent pHiregulation, SAC
conductance, and NO production modeling elements are then added. Finally
the Na1-K1pump is adapted to better represent rat experimental data and
ensure the correct [Na1]ifrequency response.
The model was coded in CellML (26) and all model development and
simulations were performed using the freely available Cellular Open Re-
source (COR) (27).
Basic cell model
As noted above, the base cell model is developed by coupling the Pandit
et al. (23) electrophysiology, the Hinch et al. (24) Ca21dynamics, and the
Niedereret al. (25)contraction model.All three modelshavebeendeveloped
to represent rat cardiac myocytes at room temperature. In this study we have
used the endocardial parameter set from the Pandit et al. (23) electrophys-
iology model study. The Pandit et al. (23) cell model relies on common pool
Ca21dynamics, which are incapable of representing graded Ca21release
(28) an intrinsic property of the cardiac myocyte (29). To achieve a graded
release of SR Ca21we replace the Ca21dynamics in the Pandit et al. (23)
framework with the computationally efficient and biophysical LCC-RyR
model from Hinch et al. (24). These electrophysiology elements have been
coupled to active tension using the model of Niederer et al. (25), which
accounts for the effects of Ca21-activated contraction.
The units of all the models were converted to mm, mS, ms, mV and mA.
The changes in ion concentrations in the model are calculated from a flux of
ions divided by the cell volume. During an action potential the changes in
[Ca21]iare considerably greater than [Na1]ior [K1]iand hence are more
Hinch et al. (24) Ca21model values as opposed to the Pandit et al. (23)
electrophysiology model values. The [Ca21]ibinding to troponin C kinetics
experimental measurements. To ensure that the Hinch et al. (24) model
number of regulatory units was increased to 75,000 from 50,000. The SR
leak current conductance was then reduced to 5 3 10?6ms?1to ensure an
equilibrium SR Ca21concentration of 700 mM. The model currents and
capacitance were defined in per mm2using 1.534 3 10?2mm2as the
capacitive area (30) to be compatible with tissue simulations. The Na1-K1
pump [Na1]idependence and maximum flux were also adapted to match
recent experimental results from rat myocytes (31). Validation of the
complete model including the additional length-dependent components is
Cell/muscle length is a parameter in both the model of contraction and the
stretch-dependent model elements. In all cases length is defined by the
in experiments used to quantity the stretch-dependent elements are recorded
as a percentage of the length at which maximum active tension is generated,
which is problematic to quantify at a sarcomere scale. To accommodate this,
we define all of the stretches relative to the resting cell length or l ¼ 1. The
significance of this assumption is tested below.
pHiin the cardiac myocyte is maintained through a combination of pH
regulated cotransporters and exchangers. The NHE and Na1? HCO?
cotransporter (NBC) act as acid extruders and Cl?? HCO?
and Cl?-OH?exchanger (or equivalently the Cl?-H1cotransporter) (CHE)
act as base extruders (32). In this study pH regulation will be modeled using
the six-state CHE, AE, and NBC transporter models developed by Crampin
and Smith (33). A NHE model was developed that took a greater account of
Na1dependencies due to the range of [Na1]ivalues encountered during the
Modeling the Slow Force Response4031
Biophysical Journal 92(11) 4030–4044
SFR and the proposed importance of NHE in causing the SFR. The NHE
model consists of a transporter and a regulatory component, as proposed
previously by Crampin and Smith (33). The transporter component was
modeled using a simplified two-state exchanger model, where a-values
correspond to the forward and backward rates between the two states with
1 and 2 signifying the rates when Na1or H1are bound to the exchanger,
data and making use of the thermodynamic constraint that a1
The regulatory component modeled the intracellular allosteric ‘‘proton
modifier site’’ (36) and was represented by a simple Hill scaler.
A complete [Na1]iand pHidependent data set is not currently available
for NHE so the model was fit using Na1dependent data normalized by the
maximum measured flux, from sheep Purkinje fibers (35) and pHidependent
data from rat (34). The model does not take account of extracellular ion
dependencies as these are fixed in all simulations to pHo¼ 7.4 and [Na1]o¼
140 mM. Transporter kinetics are described by Eq. 1,
0:01415ms?1;KNa¼ 21:49mM;KH¼ 1:778310?10ðpKH¼ 9:75Þ;Ki¼
4:187310?5ðpKi¼ 6:38Þ;nh ¼ 2; and a?
The model parameters were fit using rat pHi-dependent flux data (34) with
[Na1]iset to 10 mM (see Fig. 1 A) and normalized [Na1]iflux data from
sheep Purkinje fibers with pHi¼ 6.4 (35) (see Fig. 1 B).
Change in pHifor a given proton flux (JH) (where JHcomprises the sum
of the JNHE, JCHE, JAE, and JNBC) was modeled using Eq. 2, where biis the
intrinsic buffering capacity.
2are defined in Eq. 1.
where the intrinsic buffering capacity (Eq. 3) is contributed to via the two
independent buffers b1and b2:
where Cb1¼ 31.2 mM, pKa1¼ 6.4, Cb2¼ 6.85 mM, and pKa2¼ 7.48. The
buffering capacity equation parameters were refit from the original values
recorded by Swietach et al. (37) to account for a 50% decrease in buffering
capacityat 22?C, basedon the 40% decrease observedfor a 10?C decrease in
temperature (38). The transporter models outlined in Eq. 1 are also fitted
from experimental data measured at 37?C. Che’en et al. (38) found that NHE
and NBC flux have a Q10of 2. Assuming that the Q10value is similar for
CHE and AE (39) in rat cardiac myocytes all transport fluxes were scaled to
35% of the values at 37?C to represent model fluxes at 22?C.
A simple background flux with a linear voltage dependence (Eq. 4) was
added with a conductance of 4 3 10?6mS to ensure a resting pHiconsistent
with a resting pHiof 7.26 in HEPES solution reported by Yamamoto et al.
(34). Adding 5% CO2caused a small drop in pHito 7.21, which is within the
experimentally measured range of no change (40) and 0.15 pH units (34).
The addition of the H1current is not based on experimental evidence,
however, similar currents have been included in earlier modeling studies
(41). The implications of this current and its justification are discussed in
more detail in the model critique below.
bi¼ lnð10Þ 10?pHi1Cb1
IH;b¼ gH;bðVm? EHÞ:
of charge must be accounted for by including intracellular Cl?(Cl?
model. As both AE and CHE bring Cl?into the cell a small Cl?leak current
membrane potential was ??80 mV and the reversal potential of Cl?? ?55
mV (42), with an extracellular Cl?concentration of 126 mM (43).
i) in the
ICl;b¼ gCl;bðVm? EClÞ:
NHE & AE stretch dependence
The stretch dependence of NHE was demonstrated by Alvarez et al. (7) in
HEPES solution (eliminating NBC flux), by showing that stretching rat
papillary muscles induced an alkolisis of 0.09 6 0.01 pH units, which could
be removed by the NHE blocker amiloride. At the same time a rise in [Na1]i
of 5.5–6 mM and an increase in peak active tension was recorded, which
Alvarez et al. (7) attributed to the combined effects of increased NHE
activity and increased Ca21influx on NCX. Increased NHE flux in response
to stretch has also been proposed in papillary (6,10,12,44–46), trabeculae
(7,8), ventricular myocytes (6), and ventricular strip (11) preparations in
human (11), ferret (44), rabbit (8), rat (6,7), and cat (45,10) preparations,
suggesting that the NHE response to stretch is an intrinsic property of the
myocyte and independent of species. In bicarbonate buffered solutions the
cellular pH has been shown to remain constant followingstretch (46). This is
cardiac myocytes for experimental measurements (points) (34) and fitted
model simulations with pHo¼ 7.4, [Na1]o¼ 140 mM, and [Na1]i¼ 10 mM
(line). (B) Comparison between fitted model, with [Na1]o¼ 140 mM, pHi¼
6.4, and pHo¼ 7.4 (solid line) and normalized flux across the NHE recorded
in sheep Purkinje fibers (solid squares) as a function of [Na1]i(35).
(A) Steady-state flux across NHE as a function of pHiin rat
4032 Niederer and Smith
Biophysical Journal 92(11) 4030–4044
hypothesized to be a result of an increase in flux through AE, which com-
pensates the increased flux through NHE (46).
The stretch responses of both NHE & AE have been shown to be the
result of a protein kinase C dependent pathway (46). In this study we assume
that stretch-dependent phosphorylation of AE and NHE causes an increase
in flux by altering the dissociation constant of protons binding to the
intracellular pHiregulatory site (Kiin Eq. 1 for NHE). We also assumed that
the effects of stretch are instantaneous as the slow force responseof stretch is
expected to take place over a longer timescale than phosphorylation and
there is little information available to quantify any time dependence.
pHiis determined by the equilibrium of acid carrier flux within the
‘‘permissive zone’’, as defined by Leem et al. (32). Although the fluxes are
small, changes in either the acid influx or efflux will cause a shift in the pHi.
The changes in pHirecorded following stretch were used to determine the
changes in NHE & AE. The changes in NHE flux following stretch were
constrained by the observed 0.1 increase in pHi in the absence of
extracellular CO2for a 10% stretch (7). Here we model the change in pH
using the hypothesis proposed by Alverez et al. (7), that acid efflux through
NHE is increased following stretch, while acting against a constant acid
influx on CHE and the background proton flux (Eq. 4). Fig. 2 A shows how a
30% decrease in the dissociation constant of protons to the NHE regulatory
site model causes an increase in pHiof 0.1 pH units and a 0.00054 mMms?1
increase in Na1influx (see DJNHEFig. 2 A) at steady state.The model for the
change in AE flux following stretch was fit by the experimental observations
of no change in pHiin 5%CO2/ 95% O2solution, in the presence of a 30%
decrease in the dissociationconstant of protons to the model NHE regulatory
site, as noted above. Fig. 2 B shows how a 28% increase in the dissociation
constant to the AE regulatory site balances the increase in NHE flux fol-
lowing a 10% stretch, resulting in no change in pHiand a 0.0036 mM ms?1
increase in Na1influx (see DJNHEFig. 2 B) at steady state. Experimental
results record between no change (12) and a 0.03 6 0.01 (46) drop in pH in
5% CO2/95% O2solution, in the absence of NHE for an 8% and 6% stretch,
respectively. Here the simulated increase in AE flux, following stretch,
resulted in a 0.01 pH unit drop in pH, which is within the experimentally
measured results. The stretch dependence of the dissociation constants are
quantified in the model by Eqs. 6 and 7 for NHE & AE, respectively:
gNHE¼ 1 ? bNHEðl ? 1Þ
gAE¼ 11bAEðl ? 1Þ;
where gNHE and gAE are the length-dependent scalers of the binding
affinities for NHE & AE, respectively, and bNHE¼ 2.804 and bAE¼ 2.5 are
the strain coefficients of NHE & AE, respectively.
Stretch-activated channels have been reported in isolated myocytes (16),
multicellular (47), and whole heart preparations (48). Models for stretch-
activated channels have been proposed by Zeng et al. (16) for rat myocytes
and Kohl and Sachs (49) and Healy and McCulloch (50) for guinea pig
myocytes. Recently Li et al. (51) have reported that the stretch-dependent
currents in rat myocytes are the result of a stretch-activated K1specific
current (IKo) and a nonspecific stretch activated channel (Ins). Healy and
of the IKp current (52) and modeled Ins with a linear current voltage
relationship (16). In this study we apply the same method, fitting the
conductance and length dependence of Insand IKofor the rat myocyte using
I-V curves recorded at 22?C during a 10% regional strain by Li et al. (51).
Subsequently, we separatedInsintoNa1andK1componentsas proposedby
Li et al. (51) (see Eq. 9). The relative conductance of Na1and K1through
SACs (r ¼ gNa=gK) can be estimated given the reversal potential of Ins(Ens)
of ?10 mV and the reversal potential of Na1and K1(53), which are ?65
and ??85 mV, respectively, using Eq 8:
r ¼Ens? EK
giving r ¼ 1. The resulting cation SAC current is defined by:
gSL;ns¼ bnsðl ? 1Þ
Ins;Na¼ r gnsgSL;nsðV ? ENaÞ
Ins;K¼ gnsgSL;nsðV ? EKÞ
The stretch-dependent K1specific current (Eq. 10) is defined by:
gSL;Ko¼ bKOðl ? 1Þ10:7
45ðV ? EKÞ;
where l is the extension ratio, gnsand gKoare equal to 4.1 3 10?7mS and
1.2 3 10?6mS, respectively, and bns¼ 10 and bKO¼ 3 are the strain
coefficients for nonspecific SACs and K1SACs, respectively. The strain
dependenceofthe conductance, gSL;KoandgSL;ns; was fit witha linearfunction
tothe experimentallyrecordedI-V curve, recorded attwo strainpoints,0 and
10% (51). The background K1current was substituted for IKo, because at
bufferedsolution (no AE or NBC). Solid anddash-dottedlines show the acid
efflux and influx as functions of pHibefore stretch, respectively. Following
10% stretch, the flux through NHE increases shifting the acid efflux curve to
the right from the solid line to the dashed line. Resulting in a 0.1 pH unit
increase in pHiand an increase in flux through NHE of DJNHE¼ 0.00054
mM ms?1. (B) Change in acid flux following a 10% stretch in 5% CO2
bicarbonate buffered solution.Solid lines and dashed lines are the acid efflux
and influx as functions of pHi, respectively. Following a 10% stretch, acid
efflux shifts right and acid influx shifts left as indicated by the arrows. An
enlargement of the changeinintersectionoftheacidinfluxandeffluxisshown
mM ms?1and no change in pHi.
(A) Change in acid flux following 10% stretch in HEPES
Modeling the Slow Force Response4033
Biophysical Journal 92(11) 4030–4044
resting membrane potential IKowas approximately equal to the background
K1current and IKowas not included in the original Pandit et al. (23) cell
model. The resulting model produces a current of Ins¼ ?3.795 pA mm?2
and IKo¼ 0.0416 pA mm?2with 10% strain and Vm¼ ?80 mV.
Nitric oxide dependence of ryanodine receptors
Increased NO production has been recorded following stretch (9). Villa
Petroff et al. (9) proposed that NO acts as a secondary messenger of stretch
by selectively increasing the open probability of the RyRs (9). Although NO
has been found to act on other SR Ca21handling mechanisms, namely
L-type Ca21channels and SERCA and through cGMP pathways (54), these
were all ruled out by Villa Petroff et al. (9). Villa-Petroff et al. (9) found that
a 7% and 12% stretch caused an approximate twofold and fourfold increase
in Ca21spark frequency. In the model proposed in this study, Hinch et al.
(24) represent Ca21release from the RyRs as a three-state model (see Fig.
3). We assume that the increased Ca21spark frequency is a result of an
increased probability of a RyR shifting from the closed (C) to the open (O)
state in the quiescent model, because only a small proportion of RyRs are
in the inactivated (I) state when the cell is in a quiescent state. Solving for
the quiescent steady-state RyR model, such that [Ca21]ds? [Ca21]i, gives
Simplifying gives, O ? b1/b?as b1? b?and m1? m?. The fraction
of RyRs in the open state in the quiescent cell was increased by decreasing
theRyR rateof closing(b?)becauseNO has beenreportedto predominantly
affect active tension relaxation (55) although similar results are achieved in
the simulations described belowby altering the openingrate of RyRs (results
not shown). The decrease in b?is modeled by reducing the RyR proportion
of time closed in the active mode by a strain dependent scaler gNO(see Eq.
13), with a strain coefficient of bRyR¼ ?6.47, such that a strain of 7% and
12% induce an increase in the proportion of RyRs in the open state by a
factor of 1.83 and 4.48, respectively, effectively close to the Villa Petroff (9)
Sodium potassium pump
The Na1-K1pump (NaK) model in the Pandit cell model (23) is taken from
a guinea pig model (22) and scaled to balance the flux of Na1and K1to
achieve a resting [Na1]iconcentration of 10.7 mM. The model is based on
guinea pig data and has a near linear (Hill coefficient 1.5) dependence on
[Na1]i. Recent experimental results from Despa et al., (31) found that the
NaK pump is highly dependent on [Na1]iin rat, with a Hill coefficient of
4.0, although the binding affinity of [Na1]iremains the same. Considering
this, the Hill coefficient of NaK was increased to 4.0, although higher than
previous values (3.0) (56), the Hill coefficient is taken directly from a fit to
recent experimental data and so was not changed. The maximum flux was
increased to 9.5 mA/mm2to fit the model [Na1]i-frequency response to data
from Despa et al. (31) (see Fig. 4).
The rat electrophysiology model described here is an
ensemble of models. Many of the model components used
here have been previously verified. The ensemble model’s
stability over prolonged periods is essential for modeling the
slow force response, which takes place over ?15 min. In the
quiescent state there arenochangesin[Na1]i,[K1]i,[Ca21]i,
[Na1]i, [K1]i, [Ca21]i, and Vmchanged by ,0.1%.
To ensure that the model components were coupled ap-
propriately the maximum tension and action potential du-
ration were calculated as functions of frequency. The final
model exhibits nontrivial frequency responses, which are
qualitatively similar to those of rat, with both measures of
action potential duration and peak active tension increasing
with frequency. Fig. 5, A and B, show tension traces at 2,
1.25, and 0.2 Hz and the peak tension as a function of pacing
frequency, respectively. As both figures demonstrate, the
model. The model has three states: closed (C), open (O), and inactivated (I).
The opening rate (b1) between C and O and both activation (m1) and
inactivation (m?) rates between I and C are [Ca21]dsdependent. The closing
rate (b?) between O and C is constant.
Schematic of RyR model from the Hinch et al. (24) Ca21
(points) taken from Despa et al. (60). Maximum flux through NaK was fitted
to achieve model [Na1]iresponse (dashed line).
Plot of [Na1]ias a function of frequency. Experimental data
4034Niederer and Smith
Biophysical Journal 92(11) 4030–4044
peak tension increases with pacing frequency as observed
75%, and 90%, respectively. The model exhibits increasing
values for all measures of action potential duration, qualita-
tively matching additional rat experimental data (58).
Slow force response
SFR simulations were performed after the cell reached a
steady state at 1 Hz pacing frequency, in 5% CO2and l ¼ 1.
Steady state was defined by constant [Na1]iand [K1]iover a
period of 20 min. The cell was then paced for 5 min at resting
length (l ¼ 1) and then stretched by 10% (l ¼ 1.1) for 15
min. Fig. 6 A shows an example simulation, where SACs are
present. The force response to stretch was separated into a
Frank Starling effect and SFR. The Frank Starling effect was
defined as the percentage change in tension between before
the stretch and 10 s after the stretch was applied ((T2? T1)/T1
from Fig. 6 A). The SFR was taken as the percentage change
in tension between 10 s and 15 min after the stretch was
where T2and T3are defined in Fig. 6 A.
Applying a step in extension ratio in the presence of
different model permutations of the three stretch-dependent
elements computationally isolates which individual mecha-
nism or combination of mechanisms are capable of repro-
ducing the experimentally observed SFR. A factorial set of
experiments was performed using the model as defined in
Table 1. The resulting impact of each individual factor or
combinations of factors on the SFR was calculated from the
results in Table 1 (59). Fig. 6 B shows the impact of the
(inthis caseSACs).The SACsare theonlystretch-dependent
element that was capable of reproducing a SFR consistent
with experimental observations, with stretch-dependent pH
having only 37% of the impact of SACs and NO-induced
Fig. 6 B).
Calcium response to stretch
most significant factor in causing the SFR. Although this
suggests that SACs play a role in generating the SFR, it does
not rule out NO-induced opening of the RyRs or increased
flux through NHE & AE as complementary mechanisms
because the SFR was still significant when either pH or both
pH and NO stretch-dependent elements were included in the
presence of SACs (see Table 1). As a further test of which
mechanisms are involved in SFR we compared simulated
[Ca21]itransients before, 10 s after, and 15 min after a 10%
stretch, with experimental results from Kentish and Wrzoesk
(1). The factorial analysis identified combinations 3, 4, and 8
tension transient with no strain (l ¼ 1),
at 0.2 Hz (dash-dotted line), 1.25 Hz
(dashed line), and 2 Hz (solid line). (B)
Increase in peak tension with pacing
frequency between 0.1 and 2 Hz with
no strain (l ¼ 1). (C) Model simula-
tions of action potential, at 0.2 Hz
(dash-dotted line) and 2 Hz (solid line).
(D) The time taken for 90% (APD90,
dotted line), 75% (APD75, dash-dotted
line), 50% (APD50, dashed line), and
25% (APD25, solid line) repolarization
with pacing frequency between 0.1 and
(A) Model simulations of
Modeling the Slow Force Response4035
Biophysical Journal 92(11) 4030–4044
simulations in the presence of SACs and stretch-dependent
during the course of the stretch compared to an 11% increase
observed by Kentish and Wrzoesk (1). It is important to note
that this quantitative comparison is made with the only single
furo-2 trace that was available for comparison.
Fig. 8 compares the [Ca21]itransient of these stretch-
dependent element combinations with results from Kentish
and Wrzoesk (1).
The inclusion of the SACs modeling element resulted in
[Ca21]itransients that qualitatively matched experimental
results. However, in the presence of NO effects both the peak
[Ca21]i, time to peak [Ca21]i,and RT50times were notably
different from experimental results. The stretch dependence
which is interesting to contrast with previous experimental
studies that found that inhibition of NHE attenuated the
increase in [Ca21]iassociated with SFR (6).
NHE & AE
Surprising, in the context of the comparisons between
experimental data and the results in Fig. 7, is that the stretch
dependence of NHE & AE caused only nominal changes in
the SFR and the [Ca21]itransient. Increased NHE activity is
consistently attributed with causing the SFR in experimental
results, primarily due to the removal or reduction of the SFR
when NHE is blocked by amiloride or cariporide. To test the
effects of blocking NHE on the SFR we compared simulated
SFR values calculated with normal and inhibited NHE and
including SAC or SAC and pH stretch-dependent elements.
In NHE inhibited simulations NHE was inhibited for the
duration of the experiment. In both normal and NHE
inhibited simulations the cell was paced at 1 Hz at resting
length (l ¼ 1) in the presence of 5% CO2until it reached a
steady state. SFR values were then calculated after 15 min of
stretch as described above.
Fig. 9 compares simulated control and NHE inhibited SFR
values in the presence of SACs and SACs and NHE & AE
stretch-dependent elements. We observe a 32% and 56%
decrease in SFR after NHE inhibition in models containing
SAC and SAC, and NHE & AE stretch-dependent elements,
In this study we have added background Cl?and H1fluxes,
increased the number of LCC-RyR units, and added stretch
dependencies to NHE & AE, and stretch-activated channels.
The sensitivity of the SFR model results to each of these
modeling element was quantified by the change in peak
[Ca21]i, [Na1]i,and the SFR in the presence of SACs and pH
stretch-dependent elements, after a 10% perturbation in key
parameters, as shown in Table 2. The sensitivity of model
results to the strain dependence of the stretch-dependent
elements was also tested in simulations containing each
relevant stretch-dependent element individually. Results are
shown in Table 3. Simulations were performed at described
above with all measurements taking place 15 min after a 10%
strain (l ¼ 1.1) was imposed.
ence of SACs. Cell is paced at 1 Hz at rest (l ¼ 1) after 5 min a 10% stretch
(l ¼ 1.1) is applied for 15 min. T1, T2, and T3are the active tension before,
10 s after, and 15 min after stretch, respectively.(B) Impact of individual and
combinations of SACs (SAC), stretch-dependent NHE & AE (pH), and
increased open probability of RyRs due to stretch released NO (NO) on the
SFR, as determined from a factorial experiment design. The impact of each
factor or combination was normalized against the maximum impact (in this
(A) Example tension trace from SFR simulation, in the pres-
TABLE 1 SFR factorial experiment design
Experiment PH SACNO SFR (%)
Each experiment includes a combination of the stretch-dependent SAC, NO,
and pH modeling elements as indicated by a 1 (present) or 0 (absent) and
the resulting SFR as defined by Eq. 14.
4036Niederer and Smith
Biophysical Journal 92(11) 4030–4044
The source of the SFR is currently debated. Experimental
studies have reported conflicting results on the cause of SFR,
finding SACs, stretch-dependent NHE & AE, and stretch-
dependent, NO-induced changes in RyR dynamics as poten-
tial sources of the SFR. We have employed mathematical
modeling to initially characterize discrete experimental data
sets for the three hypothesized mechanisms and estimate to
what extent individual or combinations of these mechanisms
are necessary to reproduce the SFR. We proposed three tests
to characterize the capability of these elements to replicate
tension and [Ca21]i transient experimental results after
stretch. The first test measured the capacity of individual
and combinations of the three mechanisms to replicate
experimental SFR tension results, under physiological con-
ditions. The second test compared simulated changes in the
[Ca21]itransient after stretch with experimental results. The
third test compared changes in the simulated SFR after the
inhibition of NHE with experimental results.
Stretch-dependent elements involved in the SFR
In the first of the three tests to quantify the effects of the three
previously proposed stretch-dependent mechanisms on the
SFR we performed a factorial analysis. A factorial analysis
provides an indicator of the relative impact of each, or
combinations, of factors on a system metric (59). In this case
the system was the cell model, the factors were the stretch-
dependent elements, and the metric is the SFR as defined
above by Eq. 14. Table 1 summarizes the results of the
individual experiments performed in the factorial analysis
and Fig. 6 B summarizes the effects of each factor and factor
combination on the SFR or system metric. In the process of
performing a factorial analysis the SFR values are calculated
in the presence of the different permutations of the three
The first test identified SACs as a significant contributor to
the SFR, both in individual experiments and the factorial
analysis. Acting independently SACs caused a 32% SFR
compared with 0.87% and 0.00% for stretch-dependent
NHE & AE, and RyR dynamics, respectively (see Table 1),
compared with experimental values of 20–50% (2). The
factorial analysis found stretch-dependent NHE & AE has
changes in RyR dynamics have a negative individual and
synergistic impact on SFR.
The second of the three tests compared simulated [Ca21]i
(1). All the models tested contained SACs and different
combinations of stretch-dependent NHE & AE and stretch-
dependent, NO-induced changes in RyR dynamics. All three
stretch-dependent element combinations predicted an in-
crease in peak [Ca21]icomparable with experimental results
(see Fig. 8, A and B). However, the inclusion of NO effects
caused a significant drop in peak [Ca21]i10 s after the stretch
was applied, which was not seen by Kentish and Wrzoesk
(1). Simulations in the presence of NO caused an increase in
[Ca21]i between model simulations
in the presence of stretch-dependent
NHE & AE andSACsandexperimental
results from Kentish and Wrzosek (1).
(A) 340:380ratio (a measureof [Ca21]i)
transients prestretch (solid line and
open circles) and immediately after
stretch (solid line and solid circles).
(B) Simulation [Ca21]itransients pre-
stretch (solid line and open circles) and
10 s after stretch (solid line and solid
circles); inset shows enlargement of
crossover of transients at the peak. (C)
340:380 ratio transients immediately
after stretch (solid line and solid circles)
and 15 min after stretch (solid line and
transients 10 s after stretch (solid line
and solid circles) and 15 min after
stretch (solid line and open diamonds)
inset shows enlargement of transients at
Example comparison of
Modeling the Slow Force Response4037
Biophysical Journal 92(11) 4030–4044
the time to peak [Ca21]iimmediately after stretch; this is
contrary to both experimental results and simulations in the
absence of NO, which recorded a nominal decrease (see Fig.
8, C and D). Simulations in the absence of stretch-dependent
NO effects compared well with experimental results, show-
ing a monotonic decrease in RT50immediately and 15 min
after stretch. Whereas in the presence of NO effects there was
an increase in RT50immediately following stretch (see Fig.
8, E and F). All three simulations and experimental results
recorded an initial increase in RT90followed by a plateau
during the SFR (see Fig. 8, G and H). Hence, of the three
stretch-dependent element combinations tested, the model,
which included NO-induced changes in RyR dynamics
produced stretch-induced changes in the [Ca21]itransient
that were not consistent with experimental results.
The third test aimed to elucidate the importance of any
stretch dependence of NHE & AE, which has been widely
attributed as the cause of the SFR, as noted above. We tested
the effects of blocking NHE on the SFR and the effects of
model structure on the rise in [Na1]iafter stretch. It is well
blockers (7,8,11). Hence, we predicted that, in the absence of
ulation indices of [Ca21]i transients
(plots b, d, f, and h) and experimental
indices of 340:380 ratio (a measure of
[Ca21]i) transients (1) (plots a, c, e, and
g) beforestretch, 10s aftera 10%strain,
and 15 min after a 10% strain. Model
simulations contain four combinations
SACs (SAC, dashed line and open
squares), SACs and stretch-dependent
NHE & AE (SAC & pH, dash-dotted
line and solid squares), SACs and
stretch-dependent NO effects (SAC &
NO dotted line and open diamonds),
and SACs and stretch-dependent NO
and NHE & AE effects (SAC & NO &
pH, solid line and solid diamonds). (A
and B) Peak measure of [Ca21]i. (C and
D) Time to peak. (E and F) Time for the
transient to fall to half of its maximum
amplitude (RT50). (G and H) Time for
the transient to fall to 10% of its maxi-
mum amplitude (RT90).
4038Niederer and Smith
Biophysical Journal 92(11) 4030–4044
stretch-dependent NHE, the simulated SFR would remain
unchanged when NHE was blocked. However, as Fig. 9
NHE regardless of theinclusion of stretch-dependent NHE &
AE in the presence of SACs; such that, in model simulations,
nonessential contributor to the SFR.
Comparing model findings with experimental data
Model simulations tested three hypothesized sources of the
producing the SFR, whereas, stretch-dependent, NO-induced
changes in RyR dynamics caused [Ca21]itransients after
stretch, which were not consistent with experimental mea-
surements. However, the role of SACs in the SFR is still
RyRs are controversial, whereas stretch-dependent NHE
regulation is consistently attributed to being the source of the
SFR. We now aim to rationalize the model results on each of
the three proposed mechanisms in the context of the ap-
parently inconsistent experimental data.
One consistent finding in SFR studies is an effect of
[Na1]i. This has been attributed to increased [Ca21]iinflux
on NCX (10). In concurrence with earlier modeling (20) and
experimental (6) studies we find that the buildup of [Na1]iin
response to stretch is the most likely path through which
stretch modulated elements affect force. The route through
which extracellular Na1enters the cell is proposed to be
through NHE or nonspecific SACs, as noted above. How-
ever, model simulations containing solely stretch-activated
NHE & AE were incapable of causing a significant rise in
[Na1]iand hence, tension, after stretch. The reason for this
can be demonstrated graphically. Fig. 10 shows the total Na1
efflux (thick solid line) and influx (thin solid line) at rest (l ¼
1) as a function of [Na1]iwith all other state variables fixed
to their values in the quiescent state. The intersection of the
two curves ([Na1]i¼ 8.2 mM) approximates resting [Na1]i
before stretch (10 mM (60)). After stretch the influx curve
shifts to the right, as indicated by the arrows, for simulations
including the NHE & AE (dash-dotted line), SACs (dashed
line), and both (dotted line) stretch-dependent elements. It is
clear from this graph that the increase in Na1influx required
to shift [Na1]iis significant, given that the efflux of [Na1]i,
through NaK, has a fourth-order dependence on [Na1]i.
Regardless of the inclusion of SACs the increase in Na1
influx through NHE (DJNHEin Fig. 10) is ?0.0035 mM
ms?1, which is insufficient to cause more than an ?0.7 mM
change in [Na1]i. This compares to the DJSACof 0.023 mM
ms?1, which is ;83 larger and capable of causing an ?4.5
mM increase in [Na1]i in the quiescent state. Hence, it
appears unlikely that the increase of [Na1]iflux through
stretch-dependent stimulation of NHE is sufficient to shift
[Na1]isignificantly in rat myocytes.
However, blocking NHE consistently attenuates the SFR
(6,8). In experiments where NHE is blocked, the blocking
agent is added 15–25 min (6,8) before stretch and the muscle
is then left to reach steady state. At rest NHE makes up 25%
(29% in the model) of the basal Na1flux in rat myocytes
(60), hence blocking NHE results in a drop in resting [Na1]i
before stretch. In simulations performed in the absence of
stretch-dependent NHE & AE, in the presence of SACs, and
with NHE blocked, the final [Na1]ivalue after 15 min of
stretch is less than if NHE were functional. Considering that
NCX flux is a cubic function of [Na1]iand, hence, as [Na1]i
increases, the sensitivity of NCX to [Na1]ialso increases,
then for the same change in [Na1]iand starting at a lower
TABLE 2 Model parameter sensitivity
ParameterValue SFR (%)Peak [Ca21]i(%)
2 3 10?5
4.4 3 10?6
Percent change in peak [Ca21]iand [Na1]iand change in SFR with 10%
increase in each of the added or altered parameters in model simulation
containing SACs and pH stretch-dependent elements.
TABLE 3Stretch dependence sensitivity
Parameter ValueSFR (%) Peak [Ca21]i(%)
2.75 and 3
3.3 and 11
Percent change in peak [Ca21]iand [Na1]iand change in SFR with 10%
increase in length dependence of the stretch-dependent elements solely in
the presence of the relevant element.
SACs (SAC) and SACs and stretch-dependent NHE & AE (pH) in the
presence (control) and absence (NHE inhibition) of NHE.
SFR following 10% stretch for 15 min for models containing
Modeling the Slow Force Response 4039
Biophysical Journal 92(11) 4030–4044
initial [Na1]iwill cause a smaller increase in Ca21influx
through NCX and hence attenuate the SFR. This can be seen
in Fig. 9, where in the absence of NHE & AE stretch
dependencies inhibiting NHE causes a 30% drop in SFR in
the presence of SACs. Hence, the decrease in SFR observed
in the presence of NHE blockers suggests that the function-
ing NHE provides a significant contribution to the SFR alone
rather than any specific dependencies.
Another means of testing the capacity of NHE stimulation
to cause the SFR and the accompanying changes in [Na1]iis
to consider the change in proton concentration that occurs
during the SFR. The change in pH, following stretch,
recorded in the absence of bicarbonate was ?0.1 pH units,
which equates to an ;0.02-mM increase in free proton
concentration and a 2-mM increase in the concentration of
protons bound to buffers in the model. Whereas [Na1]i
increases by 5.5–6 mM (7) in rat and 3 mM (8) in rabbit
cardiac myocytes after stretch. As NHE operates in a 1:1 (61)
ratio it would be surprising if NHE was the sole source of the
additional Na1, as the change in protons (2 mM) is far less
than the change in [Na1]iin rat, although, similar to the
increase in [Na1]iobserved in rabbit.
The second pathway for Na1into the cell is on nonspecific
cation SACs. SACs are consistently reported in cardiac myo-
cytes (13) and have been reported to carry Na1(13,16,
51,62,63) through the nonspecific cation SAC subset. How-
ever, the significance of SACs in the SFR is not unanimous
myocardium. In rabbit ventricular strips von Lewinski (8)
found the action potential amplitude, resting membrane po-
tential, and action potential duration were unchanged follow-
ing a 10% stretch. These observations appear inconsistent
with the existence of functioning SACs, which would be
expected to modulate the action potential in some form. In
Gd31to block SACs, however, Gd31also blocks NCX (64)
(6). This compares with rat myocytes, where SACs were
mycin (6) and the action potential adapts significantly after
Furthermore,theprimary pathfor[Na1]iinto thecell, after
stretch, could be species dependent. A notable difference
between rats and other species used in SFR experiments is
the high resting [Na1]ivalue (66). The change in [Na1]i
(7) than in rabbit (8) and cat (46) (?2–3 mM) preparations.
Key [Na1]itransporters, namely NCX (67), NHE (34), and
NaK (60) are also different in rat and low [Na1]ispecies and
any stretch-induced increase in [Na1]iinflux will result in an
increase in NaK and decrease in NHE and NCX Na1flux,
causing a net decrease in Na1influx from the three
transporters. Hence, a role for stretch-induced NHE stimula-
of this study.
The third hypothesis tested was the stretch-dependent
release of NO and the resulting increase in the open prob-
ability of RyR. As discussed above, model results are not
consistent with the role of NO proposed by Villa Petroff et al.
(9) in the SFR, nor are the model simulations consistent with
NO effects on the RyRs in combination with either SACs, or
NHE & AE stretch dependencies. The interpretation of the
simulations is further confounded by experimental studies
that do not provide a clear role for NO in response to stretch.
Although Villa-Petroff et al. (9) found the SFR to be NO
dependent, this experiment has not been able to be replicated
by Calaghan and White (6) and is inconsistent with the
simulation results presented here. Two of the key assump-
tions of this hypothesis, namely: 1), the need for a func-
tioning SR (as defined above) to observe a SFR; and 2), NO
activates the RyRs, are both contested. von Lewinski (11)
found that blocking Ca21uptake and release from the SR
with cyclopiazonic acid and ryanodine attenuated the SFR.
However, in rat the SFR is still present when the SR Ca21
release and uptake is inhibited by thapsigargin and ryanodine
(6) or cyclopiazonic acid and ryanodine (1). Similar results
have also been shown in the rabbit (19). SR Ca21content is
also observed to rise during the SFR but this is not required
for the SFR to occur (19). The effect of NO on RyRs is also
debated; separate experimental studies have found NO to
both inactivate (68) and activate (69) cardiac RyRs. Further
still, increased open probability of RyRs has been shown to
and/or pH stretch-dependent elements, following stretch, in the quiescent
cell. Fluxes are calculated as a function of [Na1]iwith [K1]i, [Ca21]i,
[Ca21]SR, and Vmclamped at their quiescent values of 144.19, 6.508 3
10?5, 0.6817 mM, and ?79.90 mV, respectively. Na1efflux through NaK
(bold solid line), Na1influx prestretch (solid line), and Na1influx following
10% stretch in the presence of stretch-dependent NHE & AE (pH, dot-
dashed line), stretch-dependent NHE & AE, and SAC (SAC & pH dotted
line), and SACs (SAC dashed line). DJNHEindicates the change in Na1flux
caused by the addition of stretch-dependent NHE. DJSACindicates the
change in Na1flux caused by the addition off SACs.
Changes in Na1influx and efflux in the presence of SAC
4040Niederer and Smith
Biophysical Journal 92(11) 4030–4044
cause a transient increase in systolic [Ca21]itransients (70),
as a result of SR feedback mechanisms, which are captured
by the model (see Fig. 11). This observation directly con-
flicts with the hypothesis of Villa Petroff et al. (9) and is
highlighted by the inability of increased RyR open proba-
bility to cause a significant SFR (see Table 1). Hence, the
role of NO, if any, in the SFR appears unlikely to solely
involve the sensitization of RyRs. Casadei and Sears (17)
postulated that the increase in Ca21influx on NCX, due to
[Na1]iaccumulation, in conjunction with an increase in RyR
open probability may allow a sustained increase in systolic
[Ca21]i. In model simulations the addition of NO stretch-
dependent effects caused a decrease in SFR after stretch
regardless of the presence of other stretch-dependent ele-
ments. However, whereas the physiological effects of nitro-
sylation occur over a period of several minutes (9), they were
assumed in the model to act instantaneously (due to limited
time transient data). Hence, it is possible that a gradual
modulation of RyR sensitivity may affect both the capacity
of NO to cause the SFR and potential transient effects of NO
on the SFR. We suggest that further experiments are required
to confirm a role for stretch released NO activation of RyRs
in the SFR to stretch. We suggest that such experiments
could have the additional dual goals of determining the
significance, if any, of gradual RyR modulation and
potentially expanding the proposed hypothesis to include
other Ca21or Na1handling mechanisms.
The model proposed here is necessarily an approximation,
and thus it is important to consider the embedded limitations
and assumptions. The limitations of the model and motiva-
tions for the model adaptations are now discussed. These
include: 1), the structure of the [Na1]iregulation and stretch
dependence of NHE model, 2), the addition of pH regulation
required the development of a simplified NHE model and the
inclusion of Cl?and H1leak fluxes, 3), adding in stretch
dependencies required the fitting of a rat parameter set for the
Healy and McCulloch SAC model (50), representing the
stretch dependence of RyR dynamics and NHE & AE fluxes,
4), the definition of strain, and 5), the challenges inherent in
One potential reason for the lack of evidence for NHE
stretch dependence shown by model results may be due to
assumptions about the model structure, which we now
explicitly test. In the model development it was assumed
that NHE flux was reduced by increased [Na1]i, whereas
some experimental results find nominal inhibition at physi-
ological concentrations of [Na1]i(35). Even though there is
no experimental evidence, it is possible that CHE is also
stretch dependent, which would increase the flux through
NHE required to cause a 0.1 pH unit change in pH following
stretch in HEPES solution, as noted above. It has also been
proposed that the NaK pump has a third- (56) and not fourth-
order dependence on [Na1]i, which would alter the change in
[Na1]iin response to an increased [Na1]iinflux on NHE. All
three assumptions about the model structure attenuate the
in the presence of stretch-dependent NHE & AE and the
absence of SACs, [Na1]iincreased by 0.8 mM and caused an
SFR of 0.87% following 15 min of stretch. If NHE inhibition
by [Na1]iis removed from this model by fixing [Na1]iin Eq.
1 to the diastolic [Na1]ivalue at 1 Hz (11.33 mM) then the
increase in [Na1]iis 0.84 mM. If CHE were also stretch
dependent we would observe a greater increase in NHE flux
but still no change in pHi. To model this we clamped pHiand
increased the flux of NHE following stretch twofold; under
these conditions [Na1]iincreases by 2.52 mM causing a SFR
of12%. Thirdly, changingthe[Na1]idependence oftheNaK
pump model to a third-order dependence and increasing the
relationship on [Na1]i, caused a 0.92-mM increase in [Na1]i.
Thus, the rise in [Na1]iafter stretch is only sensitive to the
increase in NHE flux after stretch but doubling the increase
conclusion that stretch-dependent NHE & AE alone cannot
replicate experimental SFR values.
To achieve resting pHivalues in HEPES and bicarbonate
buffered solutions in quiescent cells a background proton
flux was added. Initially, combining the models of NHE,
NBC, AE, and CHE resulted in a resting pHiwas 7.25 and
7.38 in bicarbonate and HEPES buffered solution, respec-
tively. Although, it was possible that metabolic pH produc-
tion could reduce these high pHivalues. However, taking
basal metabolism as 6.3 mWg?1(71), converting to nmol
ms?1using the scaling factors from Gibbs and Loiselle (72)
and assuming ?1:1 ratio between O2consumption and CO2
production, results in a persistent CO2flux of 12.6 nmol
ms?1. Including this persistent flux in the model results in a
decrease in pHi in HEPES solution of ,0.01 pH units.
Increasing the CO2flux by a factor of 100–1.26 mM ms?1
transient. Initially the [Ca21]itransient is at a steady state at a 1 Hz pacing
frequency. After 5 s a 50% increase in RyR open probability was applied by
decreasing the rate of closing of the RyRs by 50%.
The effects of increased RyR sensitization on the [Ca21]i
Modeling the Slow Force Response 4041
Biophysical Journal 92(11) 4030–4044
caused pHito drop to 7.31. As such we concluded that the
deviation in model pHifrom experimental results was not
due to the absence of basal metabolic H1production. During
paced simulations the accumulation of lactic acid or other
metabolic byproducts could also contribute to changes in
pHi. However, Bountra et al. (73) found that pHichanged by
?0.01 pH units per Hz in bicarbonate buffered solutions,
which is less than the mean 6 SE of the resting pH ex-
perimental measurements to which we are fitting. As such,
no form of metabolic acid production was included in our
model. A small voltage-dependent background leak flux was
added, as in earlier modeling studies (41), with a conduc-
tance of 4 3 10?6mS, which lowered resting pHito 7.21 and
7.26 in bicarbonate and HEPES buffered solution, respec-
tively. The background flux is parameterized to ensure viable
resting pHivalues and is not based on any experimental data.
It is possible that the use of a mix of models (rat and guinea
pig) creates a discrepancy in fluxes at resting pHi, which we
have filled with the leak current. However, this cannot be
determined until a complete data set is available for rat
proton equivalent exchangers. Table 2 shows that [Na1]i,
[Ca21]i, and SFR are insensitive to the conductance of the
background leak and as such demonstrate that the final
conclusions of this study are independent of the leak current.
To compare a broad range of stretch-dependent mecha-
nisms requires a standard definition of stretch. Ideally sar-
comere length would be recorded in a standard preparation,
however, the experimental studies that provide the multiple
data sets required to parameterize the model use a variety of
definitions. To accommodate the range of strain measure-
ments and preparation types we assumed that the strain was a
percentage of the reference length. To quantify the signifi-
[Na1]i, and peak [Ca21]i to perturbations in the length
dependence of the three stretch-dependent elements. Table 3
shows that only the length dependence of the SAC channels
halving the length dependence of SACs results in a SFR of
8.28%, considerably greater than either NO or pH dependent
elements acting alone or in combination. Also, increasing the
conclusions are affected by the definition of strain.
Although we have made every effort to parameterize and
validate the model against a wide range of experimental data,
it is important not to underestimate the complexity of bio-
logical systems. Specifically, we are cognizant, that just be-
cause the quantitative mechanism embedded in a model
explains experimental data, it does not mean that this is a
unique and/or correct explanation. This study uses modeling
to make quantitative comparisons between different hypoth-
esized causes of the SFR based on a set of rationalized
assumptions. We have identified the SACs as the most
plausible of the three stretch-dependent elements tested. This
does not mean that SACs are definitely the cause of the SFR.
However, based on the current experimental data available
in rat ventricular myocytes.
functions including many stretch-dependent elements com-
monly excluded form electrophysiology cell models. The
model provides a framework to test new hypotheses of the
and act as a starting point for developing coupled electro-
mechanics models of rat cardiac myocytes. The code was
written in CellML (26) and is freely available at www.cellml.
org. The efficient computational form of the model means that
it is suitable for single-cell and multicell simulations. The
electrophysiology equations are posed to be compatible with
deformation mechanics to facilitate the use of the model in
coupled electromechanics simulations. This provides a means
to link between cellular and whole organ spatial scales and
further the goals of the IUPS physiome project (74).
In this study we developed a detailed, biophysical and
dynamically stable model of the rat cardiac myocyte at room
temperature. The model is unique in its use of species and
temperature-specific data to determine the model parameters
and steady-state action potential duration, [Na1]i, and force
frequency responses. The model is also capable of replicat-
ing the nontrivial changes in [Ca21]itransients in response to
stretch and changes in RyR dynamics, as highlighted in the
inset of Figs. 7 B and 11, respectively.
We have compared the capacity of individual and combi-
nations of stretch-dependent NHE & AE, SACs, and stretch-
the SFR, theeffects ofthese elements on the[Ca21]itransient
after stretch, and the changes in SFR after NHE inhibition.
Simulations containing stretch-dependent, NO-induced RyR
increased open probability were not capable of replicating
both the SFR and the changes in [Ca21]idynamics observed
after stretch. The increased flux through NHE & AE after
stretch is capable of contributing to the SFR, however, NHE
alone plays an important role in generating the SFR,
independent of any stretch dependencies. Hence, our results
predict that SACs play an important role in producing the
& AE but that the effects of NO, if any, are likely to include
more than an increased open probability of RyRs.
The authors thank Professor Richard Vaughan-Jones and Dr. Pawel
Swietach for helpful discussions.
4042Niederer and Smith
Biophysical Journal 92(11) 4030–4044
S.N. thanks the New Zealand Vice Chancellors Committee (NZVCC) and
Zealand through grant No. 04-UOA-177 and the National Institutes of Health
through the NIH multiscale grant No. RO1-EB005825-01.
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