Vol. 284, Issue 4, H1217-H1229, April 2003
Cross-bridge kinetics modeled from myoplasmic
[Ca2+] and LV pressure at 17°C and after 37°C and
17°C ischemia
Samhita S.
Rhodes1,
Kristina M.
Ropella1,
Said H.
Audi1,5,
Amadou K. S.
Camara2,
Leo G.
Kevin2,
Paul
S.
Pagel1,2,6, and
David F.
Stowe1,2,3,4,6
1 Department of Biomedical Engineering, Marquette
University, Milwaukee 53233; Departments of
2 Anesthesiology and 3 Physiology,
4 Cardiovascular Research Center, and
5 Department of Pulmonary Medicine and Critical Care,
Medical College of Wisconsin, Milwaukee 53226; and
6 Veterans Affairs Medical Center, Milwaukee, Wisconsin
53295
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ABSTRACT |
We modeled changes in contractile element
kinetics derived from the cyclic relationship between myoplasmic
[Ca2+], measured by indo 1 fluorescence, and left
ventricular pressure (LVP). We estimated model rate constants of the
Ca2+ affinity for troponin C (TnC) on actin (A) filament
(TnCA) and actin and myosin (M) cross-bridge
(A · M) cycling in intact guinea pig hearts
during baseline 37°C perfusion and evaluated changes at 1)
20 min 17°C pressure, 2) 30-min reperfusion (RP) after
30-min 37°C global ischemia during 37°C RP, and
3) 30-min RP after 240-min 17°C global ischemia
during 37°C RP. At 17°C perfusion versus 37°C perfusion, the
model predicted: A · M binding was less
sensitive; A · M dissociation was slower;
Ca2+ was less likely to bind to TnCA with
A · M present; and Ca2+ and TnCA
binding was less sensitive in the absence of
A · M. Model results were consistent with a
cold-induced fall in heart rate from 260 beats/min (37°C) to 33 beats/min (17°C), increased diastolic LVP, and increased phasic
Ca2+. On RP after 37°C ischemia vs. 37°C
perfusion, the model predicted the following:
A · M binding was less sensitive;
A · M dissociation was slower; and
Ca2+ was less likely to bind to TnCA in the absence of
A · M. Model results were consistent with reduced
myofilament responsiveness to [Ca2+] and diastolic
contracture on 37°C RP. In contrast, after cold ischemia
versus 37°C perfusion, A · M association and
dissociation rates, and Ca2+ and TnCA association rates,
returned to preischemic values, whereas the dissociation rate
of Ca2+ from A · M was ninefold
faster. This cardiac muscle kinetic model predicted a better-restored
relationship between Ca2+ and cross-bridge function on RP
after an eightfold longer period of 17°C than 37°C ischemia.
indo 1; ischemia-reperfusion injury; hypothermia; isolated
hearts; four-state model
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INTRODUCTION |
THE MECHANISM OF
Ca2+-contraction coupling in cardiac muscle is attributed
in large part to changing myoplasmic Ca2+ concentration
([Ca2+]m) and myofilament sensitivity to
Ca2+. Modeling the cyclic relationship between left
ventricular (LV) pressure (LVP) and [Ca2+]m
in intact beating hearts may help to elucidate the impact of regulatory
proteins and kinetics of actin and myosin cross-bridge interaction on
Ca2+-contraction coupling. It is well known that
the relationship between [Ca2+]m and force is
complicated by the various cellular mechanisms that alter
Ca2+ handling. Previous model (14, 23, 27, 32, 38,
40, 53-55) either measured or used [Ca2+] and
force recordings from isolated myocardial fibers or skinned cardiac
muscle strips collected from various species. These preparations were
typically studied at room temperature and stimulated to contract at
slow rates. Although an isolated muscle strip is simpler to validate,
the intact heart model allows a more physiological assessment of
Ca2+-contraction coupling and cross-bridge kinetics at body
temperature and spontaneous heart rate.
Our first goal was to determine whether a four-state mathematical model
described previously (4, 10, 43) (Fig.
1) reproduces the dominant features of
Ca2+-contraction coupling from the relationship between
[Ca2+]m and developed LVP in guinea pig
isolated, spontaneously beating, and normothermic hearts. The model is
based on interactions between actin and myosin for cross-bridge
formation and includes binding of Ca2+ to troponin C (TnC)
on the actin myofilament (TnCA) (32, 40, 53, 55) while
assuming rapid switching of tropomyosin between permissive and
nonpermissive states (4, 10, 12, 38, 43). In contrast to
previous studies that used aequorin bioluminescence, we measured
[Ca2+]m using the fluorescent probe indo 1. These two techniques result in different Ca2+ transient
shapes, which may impact on model predictions and interpretations.
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Fig. 1.
Block diagram of biochemical model relating the
input/output relationship between myoplasmic [Ca2+]
([Ca2+]m) and left ventricular pressure (LVP)
adapted from Baran et al. (4) and Shimizu et al.
(43). The four-state model is governed by five
differential equations (see APPENDIX) along with model
initial conditions. TnCA, troponin C (TnC) protein molecule on the
actin (A) myofilament; M, myosin head; +, weak bonds;
· , strong bonds. The sequence of events from
phasic [Ca2+]m to contraction are as follows:
Ca2+ binds to TnCA, tropomyosin shifts so M and A can bind
forming an actinomyosin cross bridge, Ca2+ dissociates from
TnCA with cross-bridge attached, and finally the cross-bridge breaks.
Model parameters are rate constants (see Table 1);
K1 and K3 are measures of
Ca2+-binding affinity of myofilaments before cross-bridge
formation, and K2 and K4
are measures of Ca2+-binding affinity of myofilaments in
the presence of an attached cross bridge. Ka and
Kd are measures of cross-bridge association and
dissociation rates, respectively, in the presence of bound
Ca2+, and Kd' is a measure of the
cross-bridge dissociation rate in the absence of bound
Ca2+. Model rate constants and their units are indicated by
their sites of action. Note that A and M cannot form cross bridges in
the absence of Ca2+; however, because this is a loose
coupling model, once a cross bridge has been formed it no longer needs
associated Ca2+ to remain attached.
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We (1, 2, 11, 44, 52) have reported that hypothermia and
ischemia-reperfusion injury result in dissociation between Ca2+ and contraction and relaxation. Our second goal
was to determine whether this four-state model is also capable of
describing imposed dissociations in Ca2+-contraction
coupling induced by hypothermic perfusion and warm and cold
ischemia-reperfusion injury in the isolated heart
preparation. We used the model to predict LVP in response to
[Ca2+]m during 17°C perfusion, after
short-term 37°C ischemia, and after long-term 17°C
ischemia. We postulated that changes in the modeled rate
constants will help to interpret altered contractile function after
cold versus warm ischemia due to changes in myofilament Ca2+ handling and cross-bridge kinetics.
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METHODS |
Langendorff heart preparation.
The investigation conformed to the National Institutes of Health
Guide for the Care and Use of Laboratory Animals (NIH
Publication No. 85-23, Revised 1996). Prior approval was obtained from
the Medical College of Wisconsin and Marquette University Animal
Studies Committees. The preparation of guinea pig hearts at our
laboratory has been described in detail (1, 15, 47, 52).
Briefly, 30 mg of ketamine and 1,000 units of heparin were injected
intraperitoneally into albino English short-haired guinea pigs
(n = 16) 15 min before the animals were
decapitated. The animals were euthanized after they became
unresponsive to noxious stimulation. After thoracotomy, the inferior
and superior vena cavae were cut, and the aorta was cannulated distal
to the aortic valve. Each heart was immediately perfused via the aortic
root with cold oxygenated modified Krebs-Ringer (KR) solution
(equilibrated with 97% O2-3% CO2) at an
aortic root perfusion pressure of 55 mmHg, and the heart was then
rapidly excised. The KR perfusate (pH 7.39 ± 0.01, PO2 620 ± 10 mmHg) was filtered (5 µm
pore size) in-line and had the following calculated composition
(nonionized, in mM): 137 Na+, 5 K+, 1.2 Mg2+, 2.5 Ca2+, 134 Cl
, 15.5 HCO
, 1.2 H2PO
, 11.5 glucose, 2 pyruvate, 16 mannitol, 0.05 EDTA, and 0.1 probenecid and 5 U/l insulin. Perfusate and bath temperatures were maintained at
37.2 ± 0.1°C with the use of a thermostatically controlled water circulator.
LVP was measured isovolumetrically with a transducer (fluid-filled
catheter system) connected to a thin saline-filled latex balloon
inserted into the LV through the mitral valve from a cut in the left
atrium. The balloon volume was adjusted initially to a diastolic LVP of
0 mmHg so that any subsequent increase in diastolic LVP reflected an
increase in LV wall stiffness. Pairs of bipolar electrodes were placed
in the right atrial appendage, right ventricular apex, and LV base to
monitor spontaneous heart rate and atrial-ventricular conduction time.
Coronary flow (aortic inflow) was measured at constant temperature and
perfusion pressure (55 mmHg) with an ultrasonic flowmeter (model T106X,
Transonic Systems; Ithaca, NY) placed directly into the aortic inflow line.
Measurement of myoplasmic free Ca2+
in intact hearts.
Experiments were carried out in a light-blocking Faraday cage. The
heart was partially immobilized by hanging it from the aortic cannula,
the pulmonary artery catheter, and the LV balloon catheter. The heart
was immersed continuously in the bath at 37°C. The distal end of a
trifurcated fiber silica fiber-optic cable (optical surface area 3.85 mm2) was placed gently against the LV epicardial surface
through a hole in the bath. A rubber O-ring was placed over the fiber optic tip to seal the hole and netting was applied around the heart for
optimal contact with the fiber-optic tip. This maneuver did not affect
LVP. Background autofluorescence was determined for each heart after
initial perfusion and equilibration at 37°C.
Indo 1-AM (Sigma; St. Louis, MO), a Ca2+ indicator, was
freshly dissolved in 1 ml of DMSO containing 16% (wt/vol) pluronic
I-127 (Sigma) and diluted to 165 ml with modified KR solution. Each heart was then loaded with indo 1-AM for 30 min with the recirculated KR solution at a final indo 1-AM concentration of 6 µM. Loading was
stopped when the fluorescence intensity at 385 nm increased by
~10-fold. Residual interstitial indo 1-AM was washed out by perfusing
the heart with standard perfusate for another 20 min. Probenecid (100 µM) was present in the perfusate to retard cell leakage of indo 1. We
(47) have reported that loading and washout of indo 1 reduces LVP ~25%; this effect is due to the vehicle and to
myoplasmic Ca2+ buffering by indo 1.
Fluorescence emissions at 385 and 456 nm (F385 and
F456) were recorded with the use of a modified luminescence
spectrophotometer (SLM Aminco-Bowman II, Spectronic Instruments;
Urbana, IL). The LV region of the heart was excited with light from a
xenon arc lamp, and the light was filtered through a 360-nm
monochromator with a bandwidth of 16 nm. The beam was focused onto the
ingoing fibers of the optic bundle. The arc lamp shutter was opened
only for 2.5-s recording intervals to prevent photobleaching. Emission fluorescence was collected by fibers of the remaining two limbs of the
cable and filtered by square interference filters (Corion; Franklin,
MA) at 385 nm (390 ± 5 nm) and 456 nm (460 ± 5 nm). Time-control studies (n = 9) have shown that, although
both F385 and F456 decline over time, the
F385-to-F456 ratio remained stable, indicating
no change in effective measured [Ca2+]m, and
developed LVP after indo 1 loading was not significantly altered after
6 h, including 4 h of hypothermia (47).
Experimental protocol and data collection.
Both normothermic and hypothermic experimental protocols (Fig.
2) began with a 30-min stabilization
period, during which initial background data was obtained at 37°C in
each heart. Control data were obtained 30 min after the dye washout and
before global ischemia. Eight hearts were subjected to 30 min
of global ischemia at 37°C, followed by a 2-h, 37°C
reperfusion period (group A). We reported that this resulted
in an infarct size of 52 ± 3% of total ventricular weight
(2). Eight other hearts were cooled to 17°C and
subjected to 240 min of global ischemia at 17°C, followed by
a 2-h 37°C reperfusion period (group B). This resulted in
an infarct size of 36 ± 3% (11). Thus there were
five experimental conditions: 1) 37°C control,
2) 17°C control, 3) 17°C perfusion,
4) 37°C ischemia-reperfusion, and 5)
17°C ischemia-reperfusion. Perfusate and bath were maintained at 37°C by a heated water circulator and at 17°C by a
parallel-refrigerated water circulator. The order of experiments was
randomized. At the end of each experiment, 100 µM MnCl2
was infused for 10 min to quench the myoplasmic indo 1 Ca2+
signal so that the resulting signal consisted only of the nonmyoplasmic fraction. This allowed for back correction of the nonmyoplasmic component from the total cell Ca2+ transients, as described
in the APPENDIX. All analog signals were digitized
(PowerLab/8 SP, ADInstruments; Castle Hill, Australia) and recorded at
125 Hz (Chart and Scope version 3.63, ADInstruments) on Power Macintosh
G4 computers (Apple; Cupertino, CA) for later analysis using MATLAB
(MathWorks; Natick, MA) and Excel software (Microsoft; Redmond, WA).
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Fig. 2.
Experimental protocols for normothermic (A; group
A) and hypothermic (B; group B)
ischemia (I)-reperfusion (R) model. Reported results are for
the five conditions shown in italics. These conditions are
1) 37°C control, 2) 17°C control,
3) 17°C perfusion, 4) 37°C
ischemia-reperfusion, and 5) 17°C
ischemia-reperfusion. Data for conditions 4 and
5 are at 30 min into warm reperfusion.
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We obtained simultaneous recordings of
[Ca2+]m and LVP (Fig.
3) at designated time points during the
protocol. Total exposure to the 350-nm excitation wavelength light was
62.5 s. Customized software was developed in MATLAB for off-line
signal processing of the recorded data. LVP and fluorescence data were
digitally low-pass filtered using a fourth-order bidirectional
Butterworth filter at 20 Hz. After the data were filtered, the
F385-to-F456 ratio was converted to
[Ca2+]m as detailed in the
APPENDIX. Ca2+ transients shown were corrected
for the nonmyoplasmic fraction; only [Ca2+]m
data are shown.
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Fig. 3.
A: group A; B: group B.
Plot of simultaneously recorded [Ca2+]m and
isovolumic LVP (LVPexp) under several conditions. Diastolic
[Ca2+] events were detected using a threshold-blanking
period algorithm and the points were used as time markers for creating
the averaged [Ca2+] and LVP transient. Note the longer
time scale for 17°C perfusion.
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Computing averaged
[Ca2+]m and LVP transients.
Times of occurrence of peak diastolic [Ca2+]m
were obtained over the 2.5-s recordings with the use of a simple
threshold and blanking period event-detection algorithm triggered on
the filtered [Ca2+]m signal. The amplitude
threshold was set at 65% of the minimum signal amplitude, and local
minima were detected in a 144-ms window after threshold crossing. These
local minima corresponded to the diastolic Ca2+ for each
cardiac cycle. The Ca2+ and simultaneously obtained LVP
signals between each consecutively detected Ca2+ diastolic
point were aligned and averaged on a point-by-point basis to form the
averaged Ca2+ and LVP transient signals. Cycle length,
heart rate, and time-dependent indexes of LVP and
[Ca2+]m were recorded for use in data
analysis and modeling.
Description and assumptions of mathematical model.
The mathematical model applied to our data is based on the interaction
between TnCA and myosin, in the presence of Ca2+, for
cross-bridge formation and force development (4, 10, 43).
The model (Fig. 1) consists of four stages and is governed by five
differential equations, as described in the APPENDIX. The rate constants are described in Table 1.
LVP, as predicted by the model (LVPmod), is proportional to
the number of cross-bridges formed:
[Ca · TnCA · M] + [TnCA · M], where A · M is the actin and myosin cross-bridge (4). Chemomechanical coupling in
cardiac muscle includes a positive feedback mechanism, termed
cooperativity (32), which is responsible for the rise in
force. There are various hypotheses for the mechanism of cooperativity,
including the effect of Ca2+ binding to TnCA on myofilament
Ca2+ sensitivity, effect of cross-bridge formation on the
rate of formation of neighboring cross-bridges, and effect of
Ca2+ binding to TnCA on neighboring tropomyosin units
(32, 38, 40). In accordance with Baran et al.
(4) and Shimizu et al. (43), we accounted for
cooperativity by allowing K1 and
Ka to vary according to the following functions,
as described in Table 1
<IT>+</IT>[TnCA<IT>·</IT>M](<IT>t</IT>)}<SUP>0.5</SUP><IT>+&bgr;</IT><SUB>1</SUB>](/content/vol284/issue4/fulltext/H1217/img003.gif)
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<IT>+</IT>[TnCA<IT>·</IT>M](<IT>t</IT>)}<SUP>2</SUP><IT>+&bgr;</IT><SUB>a</SUB>](/content/vol284/issue4/fulltext/H1217/img004.gif)
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where 
and 
are adjustable second-order rate constants and
t refers to time. The -parameter represents the slopes of
the K1 and Ka curves and
is a measure of the sensitivity of the cooperative mechanism; an
increase in represents a stronger positive biochemical feedback.
The -parameter represents the static value of
K1 and Ka at 0 LVP; an
increase in indicates an increase in the basal value of either
parameter.
We had several assumptions for this model. First,
Ca2+ can dissociate from TnCA before the transition of the
cross bridges from the strong to the weak conformation or before the
myosin head detaches from the actin molecule, also known as "loose
coupling" (32). Second, the model does not distinguish
low-affinity sites for Ca2+ binding to TnCA from
high-affinity sites. Thus each Ca2+ that attaches itself to
TnCA is assumed to cause a conformational change in TnCA, which in turn
initiates cross-bridge attachment and force generation
(32). Third, the transition between weak and strong
cross-bridge conformations is rapid and not rate limiting. Therefore,
once actin and myosin cross-bridge attachment occurs, it is in a strong
force generating state. It was proposed that the myosin heads first go
through a weak state when the angle between the myosin head and the
actin molecule is at 90° and then cycle into the strong force
generating state when this angle is at 45° (18). Fourth,
changes in sarcomere length have little effect on the relationship
between myoplasmic Ca2+ and LVP, as we measured
isovolumetric LVP. Kentish and Wrzosek (28) reported that
lengthening of the rat trabecula muscle caused an increase in twitch
force but had no effect on the magnitude of the Ca2+
transient, suggesting an increase in myofilament Ca2+
sensitivity. However, Shimizu et al. (43) found that
unlike in isolated muscle, in the in vivo heart there were no apparent length-dependent changes in myofilament Ca2+ binding or
cross-bridge cycling as predicted by the four-state model.
Model implementation.
The model was executed with the use of customized software developed in
MATLAB and evaluated solely from data we obtained in guinea pig
isolated hearts, where [Ca2+]m was measured
with the use of the fluorescent dye indo 1 and isovolumetric LVP
(LVPexp) was measured using a transducer connected to a
saline-filled latex balloon placed in the LV. The governing differential equations were solved numerically using fourth-order Runge-Kutta with a 0.4-ms step size and the appropriate initial conditions (10, 43), as shown in the APPENDIX.
Experimentally measured and averaged Ca2+ transients were
the inputs to our model. Model rate constants were optimized using
commercially available algorithms based on constrained quasi-Newton
methods that guarantee linear convergence in MATLAB and were estimated
to minimize the root-mean-square error between LVPmod and
LVPexp at the sampled time points
Several constraints were imposed on the model rate constants
during optimization: 1) to ensure a positive feedback,
1 and a must be >0, 2)
K'd must be greater than
Kd because cross bridges dissociate more readily
in the absence of attached Ca2+;
K'd accounts for the
physiological difference between contraction and relaxation kinetics
(4), and 3) the maximum rate of
Ca2+ binding to TnCA with attached cross bridges must be
greater than the maximum rate of Ca2+ binding to TnCA with
no attached cross bridges (K2 > K1); this concept incorporates the idea of a
positive feedback mechanism to explain the delay in rise in LVP during
contraction (25).
Statistical analysis.
All experimental measurements and model rate constants are expressed as
means ± SE. All experimental observations and model rate
constants computed during 17°C perfusion and after 37°C and 17°C
ischemia-reperfusion injury were compared with the 37°C
baseline control by one-way analysis of variance for repeated measures, followed by Tukey's comparison of means post hoc test (MINITAB statistical software version 13.3, Minitab; State College, PA). Differences among means were considered statistically significant at
P < 0.05 (two-tailed).
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RESULTS |
Figure 3 shows representative tracings of simultaneously obtained
LVPexp and [Ca2+]m under control,
17°C perfusion, and during 37°C reperfusion after warm and cold
ischemia. Note the different time scales for cold perfusion
data compared with warm perfusion and reperfusion data after warm and
cold ischemia. Detected diastolic Ca2+ points are
shown for control data. There was beat-to-beat variability in
Ca2+ transient morphology after warm or cold
ischemia-reperfusion injury compared with before
ischemia. From Tables 2 and 3,
there were no significant differences in experimental observations or model rate constants at 37°C between the two temperature control groups, i.e., before a change in temperature and
ischemia-reperfusion.
Hypothermic perfusion.
Figure 4 displays a typical plot of the
coupling between [Ca2+]m and
LVPexp during 37°C perfusion and during 17°C perfusion for one heart. LVPmod is plotted as a function of
[Ca2+]m and is represented by the solid line
for both data. The model described the Ca2+-contraction
coupling with an error of 3.1 ± 0.4% between LVPexp and LVPmod at 37°C and an error of 3.4 ± 0.6% at
17°C perfusion. As seen in Figs. 3 and 4, and listed in Table 2,
hypothermia markedly reduced heart rate by eightfold while markedly
increasing diastolic LVP and systolic and diastolic
[Ca2+]m. Cold perfusion resulted in a marked
slowing in contraction and relaxation rates as noted from the reduced
values of the maximal and minimal rates of pressure increase and
decrease over time, respectively, compared with control.
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Fig. 4.
Plot of sample tracings of LVP vs.
[Ca2+]m averaged over all beats in the 2.5-s
recordings during normothermic (37°C) and hypothermic (17°C)
perfusion. Experimental data are represented by open symbols and the
model fit is represented by a smooth line.
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Figure 5 shows the model-predicted
relationship between the cooperative parameters
K1 and LVPmod, and
Ka and LVPmod, during perfusion at
37°C and 17°C. Note that slopes of the K1
and Ka curves are markedly reduced at 17°C
perfusion compared with 37°C perfusion; this indicates reduced
cooperativity for both K1 and Ka. Changes in these curves are also reflected
in altered values for the - and -parameters associated with
K1 and Ka.
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Fig. 5.
Plot of the cooperative rate constants (see Table 1)
K1 (A) and Ka
(B) as a function of the total number of estimated cross
bridges ([Ca · TnCA · M] + [TnCA · M]) during controls (groups
A and B), 17°C perfusion (group B),
normothermic reperfusion after warm ischemia (group
A), and normothermic reperfusion after cold ischemia
(group B). Note the similarity in estimated values of
K1 and Ka for control in
groups A and B and reperfusion in group
B. In contrast, note the flat curves of K1
and Ka for the 17°C perfusion in group
B and after warm short-term ischemia in group
A, which indicate a loss of sensitivity in the two cooperative
mechanisms.
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Table 3 lists model rate constants
obtained from fitting LVPexp data collected during
perfusion at 37°C and 17°C to LVPmod. The model
predicted decreased affinity of Ca2+ for TnCA, depressed
sensitivity in cross-bridge formation, and depressed basal rate of
cross-bridge formation, but no significant change in the basal rate of
Ca2+ binding to TnCA at 17°C compared with 37°C
perfusion. In addition, K2,
K3, Kd, and
K'd were all significantly slower
during 17°C perfusion than during 37°C perfusion. This indicated
that cold perfusion not only resulted in a slower dissociation of cross bridges, both in the presence or absence of bound Ca2+, but
also reduced the affinity of Ca2+ for TnCA in the presence
of formed cross bridges.
Figure 6 shows the typical
[Ca2+]m and LVPexp transients,
model-predicted concentrations of contractile elements, and best-fitted LVPmod over one cardiac cycle during 37°C (Fig. 6,
left, thick line) and 17°C perfusion (Fig. 6,
right, thick line). Note the 8- to 10-fold greater cycle
length of cold perfusion compared with warm perfusion. As reported
previously by Baran et al. (4), we also found that the
concentration of bound cross bridges without associated
Ca2+ ([TnCA · M]) was very small
compared with the concentration of bound cross-bridges with associated
Ca2+
([Ca · TnCA · M]). Note
also the multiphasic nature of the [Ca · TnCA]
transient both during cold and warm perfusion. This is due to
[Ca · TnCA] being influenced by two
simultaneous, unbalanced, yet opposing forces of cooperativity; namely,
K1 acts to slowly increase
[Ca · TnCA] and Ka acts
to rapidly decrease [Ca · TnCA]. Note that warm
perfusion exhibited three distinct plateaus in the
[Ca · TnCA] transient, whereas cold perfusion
exhibited three peaks that decrease in magnitude, possibly resulting
from excess Ca2+ loading.
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Fig. 6.
Plot of model-predicted changes in the various
contractile elements over one cardiac cycle. Note the eightfold longer
time scale for cold perfusion (right) than for warm
perfusion (left).
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Reperfusion after normothermic ischemia and hypothermic
ischemia.
Figure 7 shows the cyclic relationship
between [Ca2+]m and LVPexp during
1) 37°C control, 2) 30 min after 37°C global
ischemia, and 3) 30 min after 4-h 17°C global
ischemia. LVPmod as a function of
[Ca2+]m is shown in solid lines for all data
sets. The model successfully described the cyclic
Ca2+-contraction relationship with errors of 3.1 ± 0.4%, 2.4 ± 0.3%, and 3.2 ± 0.3% between
LVPexp and LVPmod for all three conditions, respectively. Normothermic perfusion for 30 min after 37°C
ischemia-reperfusion injury did not change heart rate or phasic
[Ca2+]m, but it depressed systolic LVP and
elevated diastolic LVP markers of contractile dysfunction compared with
37°C and 17°C baseline values. In addition, rates of contraction
and relaxation were slowed after short-term warm
ischemia-reperfusion injury.
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Fig. 7.
Plot of sample tracings of LVP vs.
[Ca2+]m averaged over all beats in the 2.5-s
recordings during normothermic (37°C) and hypothermic (17°C)
perfusion. Experimental data are represented by open symbols and the
model fit is represented by a smooth line.
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Figure 5 shows that both K1 and
Ka curves were markedly flattened 30 min after
30-min 37°C global ischemia compared with control and that
the values of K1 and Ka
changed very little with changes in the number of cross bridges with or
without bound Ca2+, i.e.,
[Ca · TnCA · M] + [TnCA · M]. Thus the model predicted a loss of
the positive feedback mechanism governing myofilament Ca2+
sensitivity and kinetics of cross-bridge cycling after
normothermic ischemia-reperfusion injury. Changes in the values
of the - and -parameters associated with
K1 and Ka (Table 3)
indicated decreased affinity of Ca2+ for TnCA, depressed
cross-bridge formation, increased basal rate of Ca2+
binding to TnCA, and decreased basal rate of cross-bridge formation. In
addition, Kd and
K'd were significantly reduced from
control by 68% and 42%, respectively, indicating slower cross-bridge
dissociation in the presence and absence of TnCA-bound
Ca2+.
Warm reperfusion for 30 min after 4-h 17°C ischemia did not
change the heart rate or diastolic LVP from before ischemia at 37°C; however, systolic LVP remained depressed and systolic and diastolic [Ca2+]m remained elevated compared
with 17°C baseline values. In addition, rates of contraction and
relaxation were slower after prolonged cold ischemia. From Fig.
5, it can be seen that K1 and
Ka curves after 4-h 17°C global
ischemia were virtually indistinguishable from trends observed
in the baseline 37°C perfusion curves. The values of
1, 1, and a were unchanged
from control; however, the basal rate of cross-bridge formation
decreased significantly. The only other significant change noted was
the increase of K4 from baseline control.
Figure 6 also shows the typical [Ca2+]m
transients, model-predicted concentrations of various contractile
proteins, and finally LVPexp and LVPmod over
one cardiac cycle during 37°C perfusion (Fig. 6, left,
thick line), 30 min after 30 min warm ischemia (Fig. 6,
left, thin line), and 30 min after 240 min cold
ischemia (Fig. 6, left, dotted line). Note again the
multiphasic nature of the [Ca · TnCA] transient
during warm perfusion. By contrast, warm ischemia-reperfusion
injury resulted in a [Ca · TnCA] transient that
exhibits only two phases, with the first peak considerably larger than
the second one. Cold ischemia-reperfusion injury showed three
phases similar to control but these were less plateaulike. The
magnitude of the first phase after warm or cold ischemia was higher than control; this could be due to ischemia-induced
Ca2+ loading.
| |
DISCUSSION |
The main findings of this study were 1) the four-state
model with cooperativity is capable of interpreting actinomyosin
cross-bridge kinetics and myofilament Ca2+ handling in
guinea pig isolated hearts from the instantaneous relationship between
indo 1-measured [Ca2+]m and LVP under
normothermic, nonischemic conditions, 2) the model
predicts a marked decrease in association and dissociation rates
between actin and myosin heads and reduced myofilament Ca2+
binding during hypothermic perfusion with a markedly slowed heart rate,
and 3) the model predicts much better preservation of
cooperativity in both cross-bridge kinetics and myofilament
Ca2+ handling after long-term cold
ischemia-reperfusion injury than after short-term warm
ischemia-reperfusion injury despite much higher
[Ca2+]m on reperfusion after cold ischemia.
Mathematical modeling of cross-bridge kinetics from experimental
data.
Cardiac muscle cells contract and relax rhythmically and synchronously
due to the cyclic influx and efflux of Ca2+ ions into the
intracellular space (4). Changes in myocardial contractile
force depend on changes in [Ca2+]m, changes
in the responsiveness of the myofibrils for a given [Ca2+]m, or a combination of both (19,
39, 51). The contractile performance of cardiac muscle is
dependent on the amplitude of the Ca2+ transient (upstream
mechanism), the affinity of TnCA for Ca2+ (central
mechanism), and the response of the actin and myosin myofilaments to
occupancy of the Ca2+-binding sites on TnCA (downstream
mechanism) (5, 39, 51). The upstream mechanism is limited
by a plateau in the maximal contractile effort for increasing
[Ca2+]m (44). It is difficult to
distinguish among combinations of the three mechanisms that together
are responsible for altering the relationship between Ca2+
and the contractile effort.
The mathematical model used in this study was adapted from models first
developed in single myocardial fibers and intact hearts of various
species (4, 38, 40, 43, 55). The rate constants previously
developed and reported (4, 10) used the bioluminescence probe aequorin as the indicator for Ca2+. We measured
[Ca2+]m by indo 1 fluorescence rather than by
aequorin bioluminescence. There is no general agreement as to which of
these indicators provides the most accurate estimate of free ionized
Ca2+. The rate constants we report in Table 3 are clearly
different from those reported by Baran et al. (4), likely
because of the difference in the shape of Ca2+ transient
during diastole. Baran et al. (4) also noted that the
aequorin measurement is characterized by a faster decay of [Ca2+]m during diastole than that observed
for the Ca2+-sensitive fluorescent probes, including indo 1 (29). A comparison between the predicted rate constants of
Baran et al. (4) and our rate constants from Table 3
during control conditions show relatively higher
K2 and K4 and lower
K3, Kd and
K'd. These differences may be
attributed in our technique to the slower decline in
[Ca2+]m measured during diastole that
contributes to faster myofilament Ca2+ cycling and slower
dissociation of cross bridges.
Effects of hypothermic perfusion on model kinetics.
It is well documented that mild hypothermia (22°-30°C) has a
positive inotropic effect. We have shown that this rise in contractile force is accompanied by a rise in [Ca2+]m in
the isolated heart (11). The mechanism of this increased contractility is not clearly understood (30, 42). The
hypotheses for this effect are prolonged duration of excitation and
contraction, slowed cross-bridge cycling rates, and tissue alkalosis.
Hypothermia may also cause increased myofilament Ca2+
loading via inhibited sarcolemmal Ca2+ pump, inhibited
Na+/Ca2+ exchange, and slowed
Ca2+-induced Ca2+ release and Ca2+
reuptake by the sarcoplasmic reticulum (17, 31, 35, 42, 48). Previous studies (20, 24) present
contradictory results on the effects of hypothermia on myofilament
Ca2+ sensitivity in isolated myofibrils. Our results from
paced isolated hearts indicate that, whereas there is an increase in
myofilament Ca2+ sensitivity at 27°C, at 17°C
contractility and relaxation are both impaired despite a marked
increase in phasic [Ca2+]m (44).
The model predicted that cooling the heart would cause a marked slowing
of the myofibrillar mechanism controlling the development of LVP or
force. Hypothermic perfusion caused a large decrease in the intrinsic
heart rate. The marked reduction in Ca2+ binding to TnCA,
in the presence or absence of attached cross bridges, points to reduced
myofilament Ca2+ sensitivity at 17°C perfusion, as we
reported previously at controlled heart rates (44). This
model predicted that reduced myofilament Ca2+ sensitivity
may underlie the unchanged phasic LVP despite the large increase in
phasic [Ca2+]m during 17°C perfusion
compared with 37°C perfusion. In addition to slower myofilament
Ca2+ binding and cross-bridge association, the rate of
cross-bridge dissociation either in the presence or absence of bound
Ca2+ was significantly slower, and this contributed to the
experimentally measured increase in diastolic pressure. This predicted
decrease in cross-bridge dissociation rate may be a result of a
hypothermia-induced reduction in ATP hydrolysis (48); with
fewer cross bridges dissociating per unit time, the contractile
apparatus becomes stiffer.
Effects of normothermic ischemia-reperfusion injury on
model kinetics.
Reperfusion after ischemia can result in reversible injury,
such as stunning, or irreversible myocardial damage, such as
infarction. Ischemia-reperfusion injury is well known to cause
a complex biochemical dissociation in the coupling of Ca2+
cycling to the generation of pressure and work (14, 52). Ischemia-reperfusion injury may occur in patients with coronary artery disease or after surgical procedures like coronary angioplasty, angiography, coronary artery bypass grafting, and cardiac valve replacement. Ischemia-reperfusion injury may result in dilated cardiomyopathy, myocardial infarction, lethal arrhythmias, valvular dysfunction, necrosis, and apoptosis (26). During
cardiac ischemia, there is a gradual increase in diastolic
pressure due to decreased compliance associated with the decline in ATP
(49).
A possible mechanism of ischemia and rigor development is
prolonged cross-bridge-dependent activation of the actin filament, even
at low [Ca2+]m and low ATP concentration. In
one model of myocardial ischemia-reperfusion injury
(14), decreased ATP hydrolysis and increased intracellular pH, indicators of ischemia-reperfusion injury were applied to a
previously developed mathematical model (Oxsoft HEART) to test how
elevated myoplasmic Ca2+ might lead to dysrhythmias.
However, until now, the link between actin and myosin cross-bridge
attachment and altered myofilament Ca2+ binding
associated with ischemia-reperfusion injury has not been modeled using
actual beat-to-beat changes in Ca2+ and LVP at different temperatures.
Our prior study (52) showed diastolic contracture and
reduced myofilament responsiveness to [Ca2+]m
after warm ischemia-reperfusion injury. The present results confirm that reperfusion after warm ischemia depresses systolic LVP and elevates diastolic LVP, whereas values of systolic and diastolic [Ca2+]m return to control values.
The model predicted that cross-bridge cycling rates are also depressed
because estimated values of Ka and
Kd were smaller than preischemic values.
A decrease in the estimated value of Kd suggests
continued actin and myosin affinity during a relative scarcity of
[Ca2+]m. Thus the model predicts that after
warm, short-term ischemia-reperfusion injury the LV remains in
partial contracture even during diastole, as confirmed experimentally.
K1 and Ka curves were
flat on reperfusion indicating a reduced cooperativity mechanism, a
depressed affinity of Ca2+ for TnCA, and depressed
cross-bridge formation.
Effects of hypothermic ischemia-reperfusion injury on model
kinetics.
Hypothermia is the most cardioprotective mechanism against
ischemia. Moderate hypothermia prolongs the time to stunning or permanent damage. Mild and moderate hypothermia is widely used to
protect hearts during open heart surgery. As temperature is lowered,
ischemic time before damage occurs is lengthened; however, hypothermia of increasing duration also has deleterious effects on
Ca2+ handling and contractility (13, 15, 16,
44-47, 50). We have previously explored the altered
association between cyclic Ca2+ fluctuations and
contraction and relaxation in intact hearts during mild (27°C) and
moderate (17°C) hypothermia (47) and after
ischemia-reperfusion injury (1, 2, 15, 52). Protection afforded by hypothermia during ischemia includes better tissue perfusion, improved metabolic function, fewer reperfusion dysrhythmias, and reduced infarct size on reperfusion (15, 16).
Our experimental data showed that after 4 h of cold
ischemia and 30-min warm reperfusion, diastolic LVP did not
increase, although systolic LVP was depressed and
[Ca2+]m loading occurred during the 4-h cold
ischemia. The model predicted that on reperfusion after cold
ischemia, unlike after warm ischemia, cooperative
elements in the affinity of Ca2+ for TnCA
(K1) and cross-bridge formation
(Ka) are restored to preischemic trends.
Also, Kd and
K'd were both indistinguishable from preischemic values; this may explain the absence of
diastolic contracture after cold ischemia. Thus we predict that
the mechanisms responsible for Ca2+-contraction coupling
are better preserved after cold ischemia. Hypothermia-induced
cardioprotection results in preservation of mitochondrial function,
which in turn results in better ATP synthesis than warm
ischemia (36). Because Kd
and K'd are both affected by ATP
hydrolysis, the dissociation rates of the cross bridges were comparable
to the warm preischemia value; this could be attributed to a
relative preservation of ATP. The model also predicted an increase in
the rate of dissociation of Ca2+ from the formed cross
bridges that was ninefold faster after long-term cold ischemia
versus the 37°C baseline control. This dramatic increase in the
dissociation rate may contribute to the observed excess in systolic and
diastolic [Ca2+]m. However, this rise in
diastolic Ca2+ did not translate into a proportional rise
in diastolic LVP; this may be attributed to a stabilization in
myofilament Ca2+ responsiveness and cross-bridge kinetics.
Justification of model and techniques.
Our results indicated that this four-state model adapted from the work
of Baran et al. (4) and Shimizu et al. (43)
was able to link observed changes in Ca2+-contraction
coupling in the intact heart to predictable changes in myofilament
Ca2+ association and cross-bridge formation under
physiological and pathological conditions. However, there are
differences between our experimental techniques, data, and predicted
model parameters and those of previous investigations that must be considered.
Most previous models were developed and validated using data recorded
from isolated cardiac muscle fibers from frogs, cows, ferrets, and rats
(3, 23, 32, 38, 40, 55). However, isolated hearts are more
sensitive to changes in [Ca2+]m than isolated
muscle preparations (54). Therefore, it was essential to
use the intact heart to model cyclic changes in the [Ca2+]m-LVP relationship. Interspecies
differences in Ca2+ handling have been reported previously
between the rat and ferret isolated muscle preparations
(54) and between rat and guinea pig isolated trabeculae
(37). The action potential and Ca2+ handling
characteristics of guinea pig myocardial cells resemble human
myocardial cells more than do any of the rodent species (33, 34,
37). Therefore, it was an important goal to apply this kinetic
model of cross-bridge cycling and Ca2+ handling
specifically to the guinea pig isolated heart.
Differences in preparations and Ca2+ measurements may give
different information in cross-bridge kinetic models. Most models of
contraction and relaxation kinetics have relied on
[Ca2+]m measured by the bioluminescent
indicator aequorin injected into cells (4, 10, 23, 43,
55). But aequorin signals cannot establish diastolic
[Ca2+]m, and they suffer from high-frequency
noise; therefore, their Ca2+ transients must be averaged
over several cardiac cycles to improve the signal-to-noise ratio
(39). Compared with aequorin, Ca2+-sensitive
fluorescent dyes exhibit a linear response over a wider range of
Ca2+, making them more attractive for measuring
[Ca2+]m (6). Fluorescent dyes
like indo 1 and fura 2 are more sensitive Ca2+ indicators
than aequorin, especially for diastolic
[Ca2+]m (4, 6). Perhaps because
of this the shapes of the Ca2+ transients produced by the
bioluminescent and fluorescent indicators are different, especially
during the fall of the [Ca2+]m transient
(29). Moreover, the indo 1 Ca2+ technique is
ratiometric; this ensures a stable Ca2+ signal over several
hours (44).
We selected indo 1 as the fluorescent indicator for Ca2+.
Unlike aequorin, which is iontophoretically restricted to the myocyte compartment, indo 1 in its AM form is believed to pass through all
membranes and thus enters nonmyoplasmic compartments, such as the
mitochondria and nucleus, as well as endothelial and vascular cells.
This nonmyoplasmic fraction requires a correction to obtain an accurate
measure of [Ca2+]m. Because we compared our
rate constants from hypothermic perfusion and after
ischemia-reperfusion injury to those determined during the
baseline warm perfusion conditions, our interpretations on cross-bridge
kinetics are not influenced by our Ca2+ measurement technique.
In summary, we have successfully used a previously described
mathematical model to reliably predict the relationship between fluctuating [Ca2+]m and LVP over the cardiac
cycle in the guinea pig isolated heart under both normothermic and
hypothermic ischemic conditions. The changes in estimated
values of the rate constants from pre- to postischemia appear
to concur with known or proposed changes in cross-bridge kinetics and
myofilament Ca2+ handling in the presence of hypothermia
and after ischemic-reperfusion injury. This mathematical
characterization should facilitate future studies by which we can
predict effects of inotropic drug interventions, ischemic or
pharmacological preconditioning, or heart failure, on the phasic
relationship between [Ca2+]m and LVP.
| |
APPENDIX |
Calculating
[Ca2+]m.
The Ca2+ transient obtained from the fluorescence ratio of
F385 to F456 is nonlinearly related to
[Ca2+]. Calibration curves were derived according to the
protocols of Brandes et al. (8, 9), which used
modifications of a standard equation for fluorescence indicators
(22).
Total intracellular Ca2+ concentration
([Ca2+]tot) was calculated as
![[Ca<SUP>2<IT>+</IT></SUP>]<SUB>tot</SUB> (nM)<IT>=</IT>S<SUB>456</SUB><IT>·K</IT><SUB>d</SUB>[(R<SUB>tot</SUB><IT>−</IT>R<SUB>min</SUB>)<IT>/</IT>(R<SUB>max</SUB><IT>−</IT>R<SUB>tot</SUB>)]](/content/vol284/issue4/fulltext/H1217/img006.gif)
|
(A1)
|
where Rtot is the instantaneous measured
F385tot/F456tot, S456 is
F456 (for 0 [Ca2+])/F456 (for
saturating [Ca2+]) = 2.4, Rmax is
F385 (for saturating [Ca2+])/F456
(for saturating [Ca2+]) = 5.986, Rmin is
F385 (for 0 [Ca2+])/F456 (for 0 [Ca2+]) = 0.059, and the Kd
of indo 1 is 149 nM at 37°C and 254 nM at 17°C (44).
Nonmyoplasmic (primarily mitochondrial) Ca2+
concentration ([Ca2+]mito) was calculated
similarly
![[Ca<SUP>2<IT>+</IT></SUP>]<SUB>mito</SUB><IT>=</IT>S<SUB>456</SUB><IT>·K</IT><SUB>d</SUB>[(<IT>R</IT><SUB>mito</SUB><IT>−R</IT><SUB>min</SUB>)<IT>/</IT>(R<SUB>max</SUB><IT>−</IT>R<SUB>mito</SUB>)]](/content/vol284/issue4/fulltext/H1217/img007.gif)
|
(A2)
|
where Rmito was calculated as the ratio of the
nonmyoplasmic fluorescence, F385mito and
F456mito, respectively. Nonmyoplasmic fluorescence was
measured at the end of each experiment after perfusing hearts with 100 µM MnCl2 for 10 min to quench fluorescence derived from
the myoplasmic compartment (7, 21, 41, 52). F385mito and F456mito were calculated at each
time point by multiplying the residual mitochondrial fluorescence
fractions (f385 and f456) by total
end-diastolic fluorescence so that
|
(A3)
|
Similar to Eqs. A1 and A2,
[Ca2+]m was calculated as
![[Ca<SUP>2<IT>+</IT></SUP>]<SUB>m</SUB><IT>=</IT>S<SUB>456</SUB><IT>·K</IT><SUB>d</SUB>[(R<SUB>m</SUB><IT>−</IT>R<SUB>min</SUB>)<IT>/</IT>(R<SUB>max</SUB><IT>−</IT>R<SUB>m</SUB>)]](/content/vol284/issue4/fulltext/H1217/img009.gif)
|
(A4)
|
where Rm was derived from the ratio of the
myoplasmic fluorescence, F385m and F456m,
respectively, calculated at each time point by effectively subtracting
mitochondrial compartment Ca2+
([Ca2+]mito) from
[Ca2+]tot and multiplying the remainder by
total end-diastolic fluorescence (as in Eq. A3) so that
![R<SUB>m</SUB><IT>=</IT>[F<SUB>385tot</SUB><IT>−</IT>(f<SUB>385</SUB>)(end-diastolic F<SUB>385tot</SUB>)]<IT>/</IT>](/content/vol284/issue4/fulltext/H1217/img010.gif)
|
(A5)
|
Nonstimulated endothelium does not contribute significantly to
[Ca2+]tot (9, 47).
Governing differential equations of kinetic model.
where d refers to differential. Assuming initial
conditions (4, 38)
| |
ACKNOWLEDGEMENTS |
The authors thank Jim Heisner for technical assistance and Drs.
Srinivasan Varadarajan, Ming Tao Jiang, Enis Novalija, Doug Hettrick,
Dean Jeutter, and Jianzhong An for valuable contributions to this
study. The authors also appreciate the contributions of Lisa Waples,
Becky Bartley, Collin Goggins, Steve Contney, and Anita Tredeau.
| |
FOOTNOTES |
This research was supported in part by National Institutes of Health
Grants HL-58691 and GM-8204-06, American Heart Association Grant
0020503Z, by the Anthony J. and Rose E. Bagozzi Medical Research
Fellowship, and by the Veterans Affairs Administration.
Portions of this study have appeared in abstract form in the
Proceedings of the 2nd Joint EMBS/BMES conference, 2002, p.
252-253.
Address for reprint requests and other correspondence:
D. F. Stowe, Medical College of Wisconsin, 8701 Watertown Plank Rd., M4020, Milwaukee, WI 53226 (E-mail:
dfstowe{at}mcw.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
First published 12 December 2002;10.1152/ajpheart.00816.2002
Received 30 September 2002; accepted in final form 9 December 2002.
| |
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TOP
ABSTRACT
INTRODUCTION
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