Department of Biomedical Engineering, The Johns Hopkins
University School of Medicine, Baltimore, Maryland 21205
Length-dependent steady-state and dynamic
responses of five models of isometric force generation in cardiac
myofilaments were compared with similar experimental data from the
literature. The models were constructed by assuming different subsets
of three putative cooperative mechanisms. Cooperative
mechanism 1 holds that cross-bridge binding increases
the affinity of troponin for Ca2+.
In the models, cooperative mechanism 1 can produce steep force-Ca2+
(F-Ca) relations, but apparent cooperativity is highest at midlevel Ca2+ concentrations. During
twitches, cooperative mechanism 1 has the effect of increasing latency to peak as the magnitude of force increases, an effect not seen experimentally.
Cooperative mechanism 2 holds that the
binding of a cross bridge increases the rate of formation of
neighboring cross bridges and that multiple cross bridges can maintain
activation of the thin filament in the absence of
Ca2+. Only
cooperative mechanism 2 can produce
sarcomere length (SL)-dependent prolongation of twitches, but this
mechanism has little effect on steady-state F-Ca relations.
Cooperativity mechanism 3 is designed to simulate end-to-end interactions between adjacent troponin and
tropomyosin. This mechanism can produce steep F-Ca relations with
appropriate SL-dependent changes in
Ca2+ sensitivity. With the
assumption that tropomyosin shifting is faster than cross-bridge
cycling, cooperative mechanism 3 produces twitches where latency to peak is independent of the magnitude of force, as seen experimentally.
myofilaments; modeling; twitches; force-calcium relations; isometric relaxation
 |
INTRODUCTION |
IN CARDIAC MUSCLE, steady-state
force-Ca2+ (F-Ca) relations
exhibit apparent cooperativity much higher than that predicted by the
single regulatory Ca2+ binding
site on troponin (reviewed in Ref. 2). There is considerable debate as
to the exact mechanisms of this high cooperativity. One hypothesis
assumes that attached cross bridges increase the affinity with which
troponin binds Ca2+
(hypothesis 1). Experimental
evidence for this type of cooperativity derives mainly from studies
showing that the affinity of troponin for
Ca2+ is decreased by blocking
cross-bridge cycling with the phosphate analog vanadate (27). Also,
manipulations that alter cross-bridge attachment states can produce
changes in the cytosolic Ca2+
concentration ([Ca2+])
transients, presumably by modifying
Ca2+ binding to troponin (1, 2,
30).
A second hypothesis is that attachment of one cross bridge increases
the rate of formation of neighboring cross bridges
(hypothesis 2). Such an interaction
may occur if the formation of the first cross bridge holds tropomyosin
in a permissive state, facilitating the formation of nearby cross
bridges. The experimental evidence for this type of cooperativity
derives from studies in which the number of available cross bridges is
modulated. For example, in vitro studies of the binding of isolated
myosin heads to thin filament show steep (highly cooperative) binding
functions (see e.g., Ref. 44). A second line of evidence derives from
studies showing that the addition of subfragment 1 (S1) rigor-bound
myosin heads can produce activation in the absence of activator
Ca2+ (7, 50, 53).
A third hypothesis is that end-to-end interactions between adjacent
troponin and tropomyosin molecules increase apparent cooperativity (hypothesis 3). This hypothesis
holds that once a tropomyosin becomes permissive to cross-bridge
cycling, it can facilitate the transition of its nearest neighbors into
permissive states. In such a scheme, activation may spread end to end
along the thin filament, thus increasing cooperativity. The evidence
for end-to-end interactions comes from a number of experimental
approaches. Partial extraction of troponin complexes in muscle
preparations produces decreased apparent cooperativity, as shown by
less steep F-Ca relations (43). Likewise, apparent cooperativity is
reduced by end-to-end interactions that are disrupted by truncating
tropomyosin with proteolytic enzymes (46, 52).
Researchers have proposed a number of models of thin-filament
activation and force generation that are based on the cooperativity hypotheses described above. A model proposed by Landesberg and Sideman
(39), in which cross-bridge binding increases the affinity of troponin
for Ca2+, is similar to
hypothesis 1. A Monte Carlo model
proposed by Zou and Phillips (61) incorporates
hypothesis 2 in that cross-bridge formation increases the rate of formation of nearby cross bridges. This
model also incorporates representations of the other two cooperative
hypotheses. Models focusing on hypothesis
3 have been proposed by Hill and co-workers (25, 26)
and by Dobrunz et al. (14). In these models, activated
troponin/tropomyosin units help to activate their nearest neighbors.
Most of these models exhibit highly cooperative behavior as indicated
by steady-state F-Ca relations, indicating that simple F-Ca relations
are inadequate to distinguish between different models of
cooperativity. This study attempts to further resolve the potential
contribution of alternative cooperative mechanisms by testing model
responses against more demanding experimental responses. First, F-Ca
relationships are strongly modulated by sarcomere length (SL), showing
increases in both Ca2+ sensitivity
and maximum force (for review, see Ref. 2). These changes cannot be
explained by changes in thick and thin filament overlap alone,
suggesting that the underlying cooperative mechanisms are enhanced as
SL increases. In this modeling study, steady-state F-Ca relations are
simulated for a range of SLs for comparison with experimentally
determined changes in apparent cooperativity.
A second set of tests involves simulation of twitches. These
dynamically changing force transients represent a more natural activation pattern than steady-state F-Ca relations. Twitches also
provide a more demanding test for models because the dynamic behavior
of muscle during a twitch cannot be predicted simply from the F-Ca
relation but appears to have additional contributions from the dynamics
of thin-filament activation and/or cross-bridge cycling (5). Changes in
SL also modulate dynamic behavior. For example, longer SLs both
increase peak force and decrease relaxation rate, thus prolonging
twitch force (31, 51). This relatively complex dynamic behavior
provides more critical tests for distinguishing between alternative models.
With the use of these tests, this paper explores the behaviors of five
models of force generation in cardiac muscle. The first two models in
this study are derived from existing published models and provide a
baseline of performance for comparison. The next three models
incorporate novel approaches to modeling cooperative activation in
cardiac muscle. These models are developed to progressively incorporate
more cooperative mechanisms that include end-to-end troponin-tropomyosin interactions, neighboring cross-bridge
interactions, and feedback on troponin affinity for
Ca2+. It is hypothesized that
multiple mechanisms of cooperativity may coexist and contribute to the
responses of cardiac muscle.
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MODEL CONSTRUCTION |
The responses of five models are explored in this paper. All models are
similar in that they are structured around a functional unit of
troponin, tropomyosin, and actin. Tropomyosin is assumed to exist in
either permissive or nonpermissive states. Permissive states refer to
tropomyosin for which the accompanying actin binding sites have been
made available for cross bridges to bind and generate force. Depending
on the model version, one or more cross bridges are also assumed to
exist in the functional unit. These are assumed to be either weakly
bound (non-force generating) or strongly bound (force generating). In
the first three models, four states
(N0, N1,
P0, and
P1) are needed to describe the
functional units, as shown in Table 1. The
nonpermissive and permissive tropomyosin states are represented by
Nx and
Px, respectively, where
x denotes the number of strongly bound
cross bridges (either 0 or 1 in models 1-3).
Model 1.
This model is shown schematically in Fig.
1A.
Its activation and cross-bridge dynamics were formulated by Peterson et
al. (48). Whereas the full derivation is given in the
original work, a brief description of the rationale is also provided
here (note that a full set of rate constants is provided in the
APPENDIX). The rest state for this
model is N0 (nonpermissive tropomyosin with no strongly bound cross bridge). Binding of
Ca2+ to troponin is assumed to
produce an immediate shift of tropomyosin to a permissive conformation
(P0) that allows a cross bridge to become strongly bound (P1). There is
also a "residual" cross-bridge state
(N1) in which the cross bridge
remains in a strongly bound state even after
Ca2+ has dissociated from
troponin. The "on" rate constants,
kon and k'on, are assumed to
be second order, depending on
[Ca2+]. In
model 2,
k'on is set 40 times
larger than kon.
This is done to capture hypothesis 1 in which the presence of strongly bound cross bridges increases the
affinity of troponin for Ca2+.
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Fig. 1.
State diagrams for models 1-3.
A: model
1 is the simplest, with fixed constants for all
transition rates. Cooperativity is incorporated in that the on rate of
Ca2+ from troponin
(kon) is
increased when a cross bridge is strongly bound
(k'on = 40 × kon).
N0, rest state of nonpermissive
tropomyosin with no strongly bound cross bridge;
P0, shift of tropomyosin to permissive
state on binding of Ca2+ to
troponin; P1, cross bridge becomes
strongly bound during permissive state;
N1, residual cross-bridge state when
cross bridge remains strongly bound after dissociation of
Ca2+ from troponin;
koff and
k'off, off rates of
Ca2+ from troponin;
f and
g, rate constants for cross-bridge
cycling between weakly and strongly bound states;
g', single off rate with
nonpermissive tropomyosin. B:
model 2 incorporates cooperativity by
having koff be a
decreasing function of force. Force is computed as the fraction of
functional units in P1 and
N1 states.
C: in model
3, there are 2 sets of states.
Top states describe
Ca2+ binding to troponin,
represented by change from T (unbound
Ca2+) to
TCa (bound
Ca2+).
Bottom states describe shifting of
tropomyosin and cross-bridge formation. Presence of units in
TCa state is
assumed to cause tropomyosin to shift in a highly cooperative manner.
This cooperativity is implemented as by making forward rate of
tropomyosin shifting
(k1) a
nonlinear function of
TCa, as
represented by dashed arrows. SL, sarcomere length. See text for
details.
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With permissive tropomyosin, the rate constants for cross-bridge
cycling between weakly (P0) and
strongly bound (P1) states are given
by f and
g. These constants were chosen so that
this model would match myosin ATPase rates. Note that, with
nonpermissive tropomyosin, there is a single "off" rate,
g', with no corresponding on
rate. This off rate is about seven times larger than
g, as determined by matching data on
relaxation rates during perturbed twitches (48). This simple two-state
model of cross-bridge cycling lacks explicit biochemical detail of ATP
hydrolysis and force productions. For this reason, the attachment and
detachment rates should be more properly referred to as apparent rates
(i.e., fapp and
gapp). However,
in this and subsequent models, we will use f and
g with the assumption that they refer
to apparent rates, not strict biochemical reaction rates.
In the present paper, force (F) is reported as a normalized value
between 0 and 1. A value of 1 corresponds to the case in which the
maximum possible number of force generators are strongly bound. For
example, assuming full activation in model
1, all units distribute between
P0 and
P1. The fraction in
P1
(Fmax) is computed as
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(1)
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where
f and
g are as described above.
Although not considered in the original model description, the effect
of sarcomere geometry is added to model
1 in this paper. Because of the physical structure of
thick and thin filaments within a sarcomere, zones can exist with no,
single, or double overlap (39). Of these, only the single-overlap zone
is assumed to contribute to force generation. To describe this effect,
an overlap ratio (
) is defined (39). This ratio gives the fraction of thick filament myosin heads in the single-overlap conformation. For
= 1, all myosin heads are able to interact with actin in the
single-overlap zone, whereas < 1 when some of the myosin heads
are in the double- or no-overlap zones. For model
1, as a function of SL is fit to the classic data
of Gordon et al. (19), as shown in Fig.
2A. An
alternate interpretation of is that it corresponds to the maximal
normalized force that can be generated by assuming full activation of
the muscle. With the contribution of the SL included, the normalized
force is computed as
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(2)
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where
P1 and
N1 are the fractions of functional
units in force-generating states (i.e., with strongly bound cross
bridges).
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Fig. 2.
A: sarcomere overlap ratio represents
fraction of thick filament with myosin heads that exist in
single-overlap conformation. See text for details.
B: for model
3, fraction of units in permissive states
(P0 + P1) is plotted as a function of
TCa. Separate
traces correspond to different SLs in 0.1-µm increments from 1.7 to
2.3 µm. C: fraction of units in
permissive states is plotted as a function of
Ca2+ concentration
([Ca2+]). Dashed line
shows TCa, which
is much less steep, as it results from uncooperative binding. Note that
in model 3 there is no change in
Ca2+ binding with SL or force so
that binding of Ca2+ is the same
for all SLs.
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Model 2.
Model 2 is shown in Fig.
1B. The activation and cross-bridge
dynamics were formulated by Landesberg and Sideman (38, 39). As in the
previous model, hypothesis 1 is
assumed to be the major cooperative mechanism. However, there are
important differences between this model and the previous one as to how
force-generating cross bridges affect the affinity of troponin for
Ca2+. In model
1, there is a change in the on rate
(k'on) when a cross
bridge is strongly bound (force generating). In contrast, in
model 2, the off rate of
Ca2+ from troponin
(koff) is
assumed to be a decreasing function of the fraction of units with cross
bridges in strongly bound states (P1 + N1). The effect is to increase the
apparent Ca2+-binding constant of
troponin with increasing developed force (see
APPENDIX for details). Developed force
affects the off rates for both P0 and
P1 equally (in model
1, only the on rate for
P1 is affected). Although a functional
unit in the P0 state does not have a
strongly bound cross bridge, the off rate for this unit is assumed to
be affected by neighboring functional units in force-generating states
(36, 37). The method of changing the off rate of
Ca2+ from troponin used in
model 2 is called
cooperative mechanism 1 to distinguish
it from the more rudimentary cooperativity of having a "residual"
cross-bridge state (as in model 1).
Another difference between the models is in the sarcomere-overlap
function (). As shown in Fig. 2A,
for model 2 is assumed to be a
monotonically increasing function of SL throughout the range from 1.7 to 2.3 µm.
Model 3.
Model 3 is constructed with the
premise that hypothesis 3 (end-to-end
interaction of troponin and tropomyosin molecules) is the most
important cooperative mechanism controlling force generation. The major
difference between this model and the previous two is the manner in
which Ca2+ binding to troponin
affects tropomyosin shifting. In the previous models, these events are
directly coupled (i.e., binding of
Ca2+ to troponin produced an
immediate shift in tropomyosin). In model 3, these events are assumed to be coupled indirectly,
as represented by the dashed arrows in Fig.
1C. This construct, called
cooperative mechanism 3, allows the
binding of Ca2+ to troponin to be
uncooperative while producing
Ca2+-dependent shifting of
tropomyosin that shows high apparent cooperativity. Cooperative mechanism 3 is a
phenomenological approach to the representation of
hypothesis 3 (end-to-end
troponin-tropomyosin interactions) that produces a low-order system of
equations. There is no attempt to explicitly model end-to-end
interaction because this would require Monte Carlo approaches (25).
This point is addressed further in the
DISCUSSION.
The uncoupling of Ca2+ binding
from tropomyosin shifting required two sets of states. The first set of
states governs only Ca2+ binding
to troponin. In model 3, the first
state (T) represents troponin with
no Ca2+ bound to the regulatory
(low affinity) site.
TCa represents
troponin with Ca2+ bound to the
regulatory site. All functional units are assumed to be in one of these
two states, such that
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(3)
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where
T and
TCa refer to the
probabilities of being in each state. The binding of
Ca2+ is assumed to be simple and
uncooperative with rate constants kon = 40 µM
1 · s1
and koff = 20 s1 (49). Thus the
[Ca2+] for 50%
binding to troponin
(KCa) is
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(4)
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The second set of states, described in Table 1, involves tropomyosin
shifting and cross-bridge formation. In model
3, the shifting of tropomyosin is assumed to produce
all of the apparent cooperativity observed in steady-state F-Ca
relations in cardiac muscle. To achieve this behavior, troponin
shifting must be a highly cooperative function of the fraction of
troponin with Ca2+ bound to the
regulatory site
(TCa). This
relationship is illustrated in Fig. 2B
in which the abscissa shows the fraction of units in TCa, whereas the
ordinate shows the resulting fraction of units with tropomyosin in
permissive conformation (P0 + P1). The increasing steepness and
leftward shift of these traces as a function of SL is assumed to be the
source of the SL-dependent increases in Ca2+ sensitivity and apparent
cooperativity seen in F-Ca relations in cardiac muscle.
The relations in Fig. 2B are Hill
functions with the properties that
1) cooperativity
(N) increases with SL and
2) the value of
TCa producing
half-maximal shifting
(K1/2)
decreases. To produce this behavior, the forward rate of tropomyosin
shifting (k1)
is assumed to be a function of both
TCa and SL, as
shown below
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(5)
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where
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(6)
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![<IT>K</IT><SUB>1/2</SUB> = 1/[1 + <IT>K</IT><SUB>Ca</SUB>/(1.5 &mgr;M − SL<SUB>norm</SUB> × 1.0 &mgr;M)]](/content/vol276/issue5/fulltext/H1734/img007.gif)
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(7)
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where
SLnorm is a dimensionless factor
that ranges from 0 to 1 (see Eqs. 8 and 9). The rate of tropomyosin
shifting from permissive to nonpermissive
(k1) is
estimated from experimental data in reconstituted thin filament (42).
With the removal activator Ca2+,
the thin filament shifts with a rate of ~43
s1, as assessed by a
decrease in fluorescence resonance energy.
With the relations in Eqs. 8 and 9,
the steady-state fraction of units with tropomyosin in permissive
conformation is a Hill function of
TCa.
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(8)
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(9)
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A
more familiar representation is obtained by the substitution of
Eq. 5 for
k1
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(10)
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The plots of "fraction permissive" in Fig.
2B show increasing steepness and a
leftward shift as SL increases. The steepness is a result of increasing
N (in Eq. 8) from 7 at SL = 1.7 µm to 10 at SL = 2.3 µm.
The leftward shift results from the decrease in
K1/2 (in
Eq. 7) at longer SLs. The increasing
steepness and leftward shift are preserved when the fraction of units
in permissive states are plotted versus
[Ca2+] in Fig.
2C. Note that the uncooperative
binding of Ca2+ to troponin (Fig.
2C, dashed line) is much less steep.
In model 3,
Ca2+ binding is simple, with no
dependence on SL or force, so that there is only one binding curve for
all SLs.
As in previous models, the cross-bridge cycling rates are assumed to be
fixed. The cross-bridge attachment rate
f is set to 10 s1, and the detachment rate
g is set to 20 s1. Similar attachment and
detachment rates of 12 and 22 s1, respectively, have been
reported using fluctuation analysis for small numbers of myosin heads
in near-isometric conditions (29).
Model 4.
As in model 3, model
4 is constructed under the premise that
cooperative mechanism 3 is the
important cooperative mechanism controlling force generation. However,
in model 4, up to three cross bridges
are assumed to exist in the vicinity of each functional unit. As shown
in Fig.
3A, up to
six states are associated with tropomyosin and cross bridges in the
functional unit (Table 2). Model 4 also incorporates three new
features: 1) an SL-dependent detachment rate for cross bridges,
2) maintenance of the tropomyosin in
a permissive conformation by strongly bound cross bridges, and
3) cooperative formation of cross
bridges within a functional unit.
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Fig. 3.
State diagrams for models 4 and
5. A:
model 4 is similar to
model 3, except that up to 3 cross
bridges (XBs) are assumed to exist in each functional unit. Two
additional cooperative mechanisms are assumed:
1) multiple attached cross bridges
that maintain tropomyosin in a permissive conformation regardless of
Ca2+ binding (note absence of
N2 and
N3 states); and
2) cooperative cross-bridge
formation so that once a cross bridge becomes strongly bound, the 2nd
(P2 state) and 3rd cross bridges
(P3 state) can progressively bind more
rapidly. B: model
5 is similar to model
4, except that additional cooperativity is incorporated
by allowing k'off to
be a decreasing function of force (i.e., fraction of units in strongly
bound cross-bridge states).
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The SL-dependent detachment rate for cross bridges is suggested by
experiments on skeletal muscle (32, 60). Including this feature in
model 4 allows for more realistic
SL-dependent changes in plateau force at saturating
[Ca2+]. Recall that
the troponin-shifting construct of model
3 is designed to produce steep
Ca2+ sensitivity, with proper SL
dependence. However, this construct alone does not produce SL-dependent
increases in plateau force that are as large as those measured
experimentally (i.e., see examples in Fig.
5A and see data in Ref. 12). In
model 3, the fraction of tropomyosin
in the permissive conformation exceeds 85% for all SLs (see Fig.
2C). Therefore, the effects of
tropomyosin shifting alone can increase maximum force by <15% as SL
increases from 1.7 to 2.3 µm. A second feature that affects plateau
force in model 3 is the
sarcomere-overlap function in Fig. 2A.
This feature produces an increase in force with length up to SL = 2.0 µm; however, the effect eventually saturates and decreases for further SL increases. In contrast, the experimental data show larger
and generally monotonic increases.
Consequently, in model 4 the off rate
of cross-bridge binding is assumed to be a decreasing function of SL
![<IT>g</IT>(SL) = <IT>g</IT>* × [2 − (SL<SUB>norm</SUB>)<SUP>1.6</SUP>]](/content/vol276/issue5/fulltext/H1734/img012.gif)
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(11)
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where
g* is the minimal detachment rate and
SLnorm is a dimensionless quantity
between 0 and 1, as described in Eq. 9. Thus the cross-bridge detachment rate increases by a
factor of two as SL decreases from 2.3 to 1.7 µm. Three versions of
model 4, with one, two, or three cross
bridges in the functional unit, have been constructed. The minimal
detachment rate ( g*) changes for each of
the three versions of model 4 (1 cross
bridge: g* = 20 s1; 2 cross bridges:
g* = 27.5 s1; and 3 cross bridges:
g* = 35 s1). The reason for the
higher detachment rates is to promote faster relaxation as the number
of cross bridges increases. This is necessary to counteract the effect
of slowing relaxation by having more cross bridges per functional unit.
The second important feature of model
4 is the cooperative action of multiple cross bridges
to maintain tropomyosin in a permissive conformation. Notice in Fig.
3A that there are no direct
transitions from multiple cross bridge-bound states
(P2,
P3) to nonpermissive tropomyosin
states. The rationale for this feature is that two or more bound cross
bridges within the same functional unit are assumed to hold tropomyosin
in its permissive state. This construction is suggested by experimental
evidence from Ishii and Lehrer (28) showing that one to two bound SL
heads per tropomyosin can trap the thin filament in an activated state
(or "on" state in the authors' terminology). In the same study,
activation of regulated actin (as assessed by fluorescence) decays only
after all cross bridges have dissociated (as assessed by light
scattering). Similar results are obtained with and without
Ca2+, suggesting that the presence
of strongly bound cross bridges can maintain the thin filament in an
activated state without Ca2+ bound
to troponin.
The third important feature incorporated into the multiple cross-bridge
construct is cooperative formation. That is, the rates of cross-bridge
formation are assumed to increase progressively as more cross bridges
form. To understand this construct, consider first the case for three
cross bridges in which formation is not cooperative.
Figure 4A
shows all the permutations of possible states for three cross bridges
(states arranged vertically with 0 = detached and 1 = attached). The
cross bridges are assumed to act independently, and each individual
cross bridge has on rate f and off
rate g. This explicit system can be
represented as a composite system, as shown in Fig.
4B. For instance, the net transition
rate from one to two cross bridges is
2f, which arises from the two separate
paths that form a double-attached composite from each configuration
with one cross bridge attached. Figure
4B shows the complete kinetics diagram
for the composite system in which forward rates are
3f,
2f, and
f. Likewise, the reverse rates are
g,
2g, and
3g. The integral multiples of the basic rate constants are derived from the multiple pathways for association/dissociation of cross bridges, not from any cooperative binding of individual cross bridges.
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Fig. 4.
Construction of cooperative cross-bridge formation.
A: all possible states and transitions
for 3 independent cross bridges.
B: composite model of 3 cross bridges without cooperative formation. Each cross bridge is
assumed to act independently with on rate
f and off rate
g. On rates for cross-bridge formation
in composite system shown are 3f,
2f, and
f. Likewise, off rates are
g,
2g, and
3g.
C: composite model of 3 cross bridges with cooperative formation. Here, formation rates are
3f,
14f, and
10f, making formation rates of 2nd and
3rd cross bridges effectively 7 and 10 times greater than in
uncooperative system. See text for details.
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Next, consider the system shown in Fig.
4C in which cooperative formation is
assumed. These formation rates are 3f,
14f, and 10f, implying that the formation rates
of the second and third cross bridges are effectively 7 and 10 times
greater than in the uncooperative system. Note that, as shown in Fig.
4C, only cross-bridge on rates are
cooperative, and off rates are not cooperative. The net result of this
cooperative formation is to produce a system in which increased levels
of force can be attained if activation is prolonged. This effect works
synergistically with the previous cooperative effect described above
for model 4 (i.e., no direct transition to nonpermissive states when two or more cross bridges are
present) to increase the duration of force generation at high levels of
force. Together, these features, collectively referred to as
cooperative mechanism 2, are designed
to simulate the cooperativity between neighboring cross bridges
(hypothesis 2). Hence,
model 4 contains both
cooperative mechanism 2, described
here, and cooperative mechanism 3,
carried over from the tropomyosin shifting functions of
model 3 (see Eqs.
5-9 and Fig.
2C).
One final issue is the reporting of normalized force. Because of the
multiple-cross-bridge structure, the maximum force
(Fmax) is no longer given by
Eq. 1 but is instead determined by a
more complicated function using the King-Altman rule (35). However, the
normalization factor remains conceptually the same in that Fmax is computed by assuming full
activation (i.e.,
k1 is assumed to
be very large so that all units distribute among
P0 through P3). The steady-state values of
P1,
P2, and
P3 are then multiplied by the weighing
factors 1, 2, and 3, respectively, to account for the number of
force-generating cross bridges represented by each state. Full details
of the calculation of normalized force are provided in the
APPENDIX.
Model 5.
Model 5 is a refinement of
model 4, obtained by adding a feedback
pathway in which attached cross bridges increase the affinity of
troponin for Ca2+. Hence,
model 5 adds
cooperative mechanism 1 to
model 4, which includes both
cooperative mechanisms 2 and
3. Cooperative
mechanism 1 is shown schematically by the dashed arrow
directed from the force-generating cross bridge states
(P1,
N1,
P2,
P3) to the off rate of
Ca2+ from troponin. Similarly to
model 2, this feature is designed to
simulate experimentally measured increases in troponin
Ca2+ affinity in the presence of
cycling cross bridges (hypothesis 1). In model 5,
k'off, the off rate
for troponin, decreases linearly with increasing normalized force
![<IT>k</IT>′<SUB>off</SUB> = <IT>k</IT><SUB>off</SUB>[1 − (1/2 × F)]](/content/vol276/issue5/fulltext/H1734/img013.gif)
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(12)
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where
koff is the same
as in models 3 and
4 and F is normalized by the maximum
value (Fmax).
Fmax is computed by assuming full
activation (i.e.,
k1 is assumed to
be very large so that all units distribute among states
P0 through
P3).
A force-dependent troponin affinity for
Ca2+, as described above,
increases the overall cooperativity in the model. To maintain steady-state F-Ca relations with similar apparent cooperativity, a
compensatory modification is required to make tropomyosin shifting less
sensitive to Ca2+ than in
model 4. This modification is
implemented by decreasing N and
increasing K1/2.
Specifically, Eqs. 7 and 8 are modified to become
![<IT>K</IT><SUB>1/2</SUB> = 1/[1 + <IT>K</IT><SUB>Ca</SUB>/(1.8 &mgr;M − 1.0 × SL<SUB>norm</SUB>)]](/content/vol276/issue5/fulltext/H1734/img014.gif)
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(13)
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(14)
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where
SLnorm is as defined in
Eq. 9. A second version of
model 5 with stronger feedback on the
off rate for troponin is also developed. In this version, the equation
for k'off, the off
rate for troponin, is given as
![<IT>k</IT>′<SUB>off</SUB> = <IT>k</IT><SUB>off</SUB>[1 − (3/4 × F)]](/content/vol276/issue5/fulltext/H1734/img016.gif)
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(15)
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The
compensatory modifications required in this case are
![<IT>K</IT><SUB>1/2</SUB> = 1/[1 + <IT>K</IT><SUB>Ca</SUB>/(1.8 &mgr;M − 1.0 × SL<SUB>norm</SUB>)]](/content/vol276/issue5/fulltext/H1734/img017.gif)
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(16)
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(17)
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RESULTS |
Steady-state F-Ca relations.
Cooperativity is most directly quantified by the steady-state F-Ca
relation. Effects of SL on experimentally determined F-Ca relations are
shown in Fig.
5A.
Increasing SL changes three key features:
1) plateau force,
2) half-activation point
([Ca]50), and
3) Hill coefficient
(NH). Plateau
force refers to the maximal force at saturating levels of
[Ca2+]. Variation of
SL over the indicated range nearly doubles plateau force.
[Ca]50 refers to the
[Ca2+] producing half
the plateau force value. In Fig. 5A,
[Ca]50 points are
connected by a dashed line. These points show a leftward shift with
increasing SL. Note that the data in Fig.
5A are from a skinned muscle
preparation in which
[Ca]50 ranged from
~3.59 to 13.4 µM. These values are about one order of magnitude
larger than those measured in intact muscle (5, 17, 59). These data do,
however, exhibit the expected SL dependence of
[Ca]50 (i.e.,
[Ca]50 increased by 3.7 times as SL decreased
from 2.15 to 1.65 µm). These experimental data are also fit to Hill
functions. Estimated
NH increases from 3.3 to 5.4 as SL increases from 1.65 to 2.15 µm. Intact preparations are generally thought to show higher
NH values near 6 for medium-range SLs (5, 14, 17). The data shown in Fig.
5A, along with other data sets (13),
show increasing
NH with SL.
However, other experimental results show little change in
NH with
increasing SL (23, 36, 40).
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Fig. 5.
Effects of SL on force-Ca2+ (F-Ca)
relationships where normalized steady-state force is plotted versus
activator [Ca2+].
A: experimental data reprinted from
Ref. 34. Experimental data are normalized by force at an arbitrary
[Ca2+] and SL.
Simulation results are shown for first 3 models. Model
1 (B) lacked
SL-dependent changes in Ca2+
sensitivity with a Hill coefficient
(NH) of 1.1 and
a half-activation
[Ca2+]
([Ca]50) of 0.59 µM for all SLs. Model 2 (C) shows SL-dependent increases in
Ca2+ sensitivity, although
apparent cooperativity is highest during midlevel
[Ca2+] values.
Model 3 (D) produces F-Ca relations with
apparent cooperativity and SL-dependent
Ca2+ sensitivity that are similar
to those for experimental data. However, model
3 fails to show appropriate SL-dependent changes in
plateau force.
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The correspondence between the simulated F-Ca relations and
experimental results are assessed using plateau force,
[Ca]50, and
NH. For simulated
data, NH is
estimated by plotting the F-Ca relations on a logarithmic scale and
then using the relationship
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(18)
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where
S50 is the slope
at [Ca]50 and
Fp is the plateau force.
Equation 18 holds for true Hill
functions for which maximum slope occurs exactly at
[Ca]50. In the
simulated data, the F-Ca relations show minor deviation from true Hill
functions. However, the maximum slope always occurs at or near
[Ca]50, so
NH as computed above
yields a reasonable estimate of the maximum steepness of F-Ca functions.
Simulated F-Ca relations for model 1 are shown in Fig. 5B. These differ
from experimental results in two ways. First, the changes in plateau
force with SL are too small and do not increase monotonically. The
changes in plateau force are a direct reflection of , the
nonmonotonic sarcomere-overlap function (see Fig.
2A and Eq.
2). Second, the simulated relations are less steep
than the experimental results (note that the abscissa in Fig.
5B covers 4 orders of magnitude).
Also, there is no dependence of
[Ca]50 and
NH on SL
(NH = 1.1 and
[Ca]50 = 0.59 µM for
all SLs). There is only rudimentary cooperativity provided by an
increase in the on rate of Ca2+
from troponin when a cross bridge is strongly bound
(k'on > kon in Fig.
1A). Therefore, this feature does
not provide sufficient cooperativity to produce steep F-Ca relations.
The F-Ca relations for model 2 are
shown in Fig. 5C. These are much
closer to the experimental results. Plateau force showed incremental
(termed "graded") changes with SL. As in
model 1, the changes in plateau force
are a direct result of changes in , the sarcomere-overlap function.
Slope and sensitivity of the F-Ca relations for model
2 are also more similar to the experimental data than
for model 1. Note that the F-Ca
relations are steepest for intermediate
[Ca2+] values (0.6 to
1 µM) and less steep at high and low
[Ca2+]. The maximum
slopes, as quantified by
NH, are the
largest for all models, but the changes in both
NH and
[Ca]50 follow the
trends in the experimental data (see Table
3 for summary of
NH and
[Ca]50 values).
The main difference between model 2 (which does produce steep F-Ca relations) and model
1 (which does not) is in a mechanism by which force
affects the Ca2+ affinity of
troponin. In model 2, increasing force
reduces the off rate of Ca2+ from
troponin (koff)
for both the P0 and
P1 states. In contrast, in
model 1, the presence of a strongly
bound cross bridge increases the on rate of
Ca2+
(kon) for the
P1 state only.
The F-Ca relations for model 3 are
shown in Fig. 5D. Plateau forces are
more graded for model 3 than for
model 1, even though the same
sarcomere-overlap function is used in both models. Despite the increase
in gradation, the SL-dependent changes in plateau force
are smaller than those seen in the experimental data. Also, the longest
SL produces a nonmonotonic plateau force, a feature not observed in the
experimental data. However, other experimental data show a decline in
maximum developed force when cardiac muscle is stretched past the
optimal length of 2.2-2.3 µm (15). Although force declines at
these long lengths, Ca2+
sensitivity continues to increase (16). This observation is consistent
with the behavior of model 3 in that
NH increases and [Ca]50 decreases
despite a lower plateau force at the longest SL.
As SL is increased, changes in
[Ca]50 and
NH produced by
model 3 are more similar to the
experimental findings than those for models
1 and 2. The
SL-dependent changes in steepness and Ca2+ sensitivity reflect the way
in which tropomyosin is assumed to shift in response to changes in
[Ca2+] or SL (see
Eqs. 5-9 and Fig.
2C). This construct produces F-Ca relations that more closely resemble Hill relations in that there is
not pronounced steepness for intermediate
[Ca2+] values, as seen
for model 2. In model
2, the increased steepness in the midrange of
[Ca2+] arises from the
force-dependent changes in affinity of troponin (hypothesis 1). Results for
model 3 suggest that another
cooperative mechanism (hypothesis 3)
may play an important role in shaping F-Ca relations.
Figure 6 shows steady-state F-Ca relations
for models 4 and
5. Two versions of
model 4 are shown, a one-cross-bridge
model (Fig. 6A) and a three-cross-
bridge model (Fig. 6B). The data show that Ca2+ sensitivity of both
versions of model 4 is similar to that
of model 3. The similarity results
from the incorporation of cooperative mechanism
3 in each of these models. One difference is that
model 4 produces larger and more
graded changes in plateau force with SL. This effect is a consequence
of SL-dependent detachment rate (see Eq. 11).
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Fig. 6.
Effects of SL on F-Ca relationships for models
4 and 5. Simulation
results are shown for model 4 with 1 (A) and 3 cross bridges
(B).
Ca2+ sensitivity of these models
is similar, which shows that addition of more cross bridges has only
small effect on steady-state responses. See Table 3 for
NH and
[Ca]50 values.
Simulation results are shown for model
5 with weaker (C)
and stronger (D) feedback of force
on Ca2+ binding to troponin. With
increasing feedback of force on
Ca2+ binding, apparent
cooperativity increases during midlevel
[Ca2+] values.
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Figure 6, C and
D, shows the steady-state F-Ca
relations for model 5. The F-Ca
relations resemble a hybrid of the responses produced by
model 2 (Fig.
5C) and by the version of
model 4 with three cross bridges (Fig.
6B). The explanation is that
model 5 contains
cooperative mechanism 3 (tropomyosin-shifting cooperativity) similar to that of model 4 and
cooperative mechanism 1 (feedback on
Ca2+ binding) similar to that of
model 2. When force feedback on
troponin Ca2+ binding is weaker,
F-Ca relations are generally similar to model 4 but exhibit a slight increase for intermediate
[Ca2+] values. As the
degree of feedback increases, the response of model
5 becomes closer to that of model
2 with more marked steepness for intermediate
[Ca2+] values (e.g.,
compare SL = 2.3 µm in traces in Figs.
5C and 6D).
The results thus far have shown that model
1 does not adequately reproduce steady-state F-Ca
relations. The rudimentary cooperativity assumed in this model is
unable to produce sufficiently steep steady-state F-Ca relations, and
there are no SL-dependent changes in apparent cooperativity.
Model 2 produces steeper F-Ca
relations, more similar to the experimental results, as a consequence
of much stronger feedback of force on
Ca2+ binding to troponin
(cooperative mechanism 1). A side
effect, inconsistent with experimental results, is that F-Ca relations are most steep for intermediate
[Ca2+] values.
Model 3 produced steep F-Ca relations
by the method by which Ca2+
binding to troponin is assumed to shift tropomyosin in a highly cooperative manner (cooperative mechanism
3). However, resulting F-Ca relations do not show
graded changes in plateau force as seen in the experimental data.
Model 4 expands on
model 3 with the addition of two
features: 1) variable cross-bridge
detachment rates and 2) multiple
cross bridges (cooperative mechanism
2). Addition of variable cross-bridge detachment rate
produces more graded changes in plateau force in F-Ca relations. The
multiple-cross-bridge formation had little effect on F-Ca relations.
Finally, model 5 expands on
model 4 by adding feedback of force on
Ca2+ binding to troponin
(cooperative mechanism 1).
Model 5 therefore contains
representations of each hypothesized cooperative mechanism. However,
only a modest degree of feedback of force on
Ca2+ binding to troponin may be
added without causing responses that are inconsistent with experimental
data. Specifically, large amounts of feedback cause the F-Ca relations
to become very steep for intermediate
[Ca2+] values, similar
to that seen for model 2.
Dynamic response: twitches.
Figure 7 presents normalized force
transients at different SLs. Experimental data are shown in Fig.
7A; responses of
models 1-3 are shown in Fig. 7,
B-D. The experimental data
exhibit three important changes with SL. First, peak force increases
considerably with SL, more than doubling (0.4 to 1.0 in normalized
units) for the range shown (SL = 1.9-2.2
µm). Second, rate of force development is approximately proportional
to peak force. This observation is more evident after each trace is
individually normalized by its peak value, as shown in Fig.
8A. The
overlap of force traces during the rising phase implies that the rate
of rise is approximately proportional to the peak force. Because of
this proportionality, the time to reach peak force is approximately the
same for each SL (although there is a slight increase at longer SLs).
Finally, as SL increases, the time required for relaxation also
increases. The twitch is fully relaxed by 0.4 s for SL = 1.9 µm,
whereas the twitch is not completely relaxed until 0.6 s at SL = 2.2 µm. This increase in relaxation time is evident by the spread of
traces between the opposing arrows in Fig. 8, indicating the points of 50% relaxation in individually normalized twitch responses.
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Fig. 7.
Effect of SL on force during twitches.
A: experimental data reprinted from
Ref. 31. Different traces correspond to different SLs (in µm) as
indicated in key. Force is normalized so that maximal force corresponds
to a value of 1.0. B:
model 1 shows smaller changes in peak
force than those seen in experimental data. In model
1, only sarcomere-overlap function modulated peak
force. C: model
2 produces larger changes in peak force that are more
consistent with those for experimental data. Model
2 differs from model 1 mainly by feedback of force on
Ca2+ binding to troponin.
D: model
3 also produces large changes in peak force, mainly as
a result of assumed cooperativity in tropomyosin shifting. In all
simulated data, force is plotted for 0.1-µm increments. Activator
[Ca2+] transient is
plotted as a dashed trace and is unchanged for different SLs.
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Fig. 8.
Individually normalized time courses of twitch force development and
relaxation. Data from Fig. 7 were replotted after each trace was
individually normalized so that it had a peak value of 1.0. Opposing
arrows show effects of SL on time to peak force and time to 50%
relaxation. A: experimental data.
B: individually normalized traces for
model 1 overlap for all SLs. Hence,
model 1 produces twitches that show no
SL-dependent changes in kinetics. C:
model 2 produces twitches with
SL-dependent increases in 50% relaxation times, but most changes can
be attributed to change in time to peak force.
D: model
3 produced a relatively constant time to peak but
little prolongation of 50% relaxation times.
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Simulation data for model 1 are shown
in Fig. 7B. The model is driven by the
simulated Ca2+ transient shown by
the dashed trace. The most important difference between simulated and
experimental data is that simulated twitches are simply scaled versions
of each other (see Fig. 8B), with
the scaling provided by the sarcomere-overlap function (Fig.
2A). The sarcomere-overlap function
can generate only modest changes in peak force and has no effect on
dynamics in model 1, so there are no
length-dependent changes in the relaxation times, as seen in the
experimental data. The results of model
1 show clearly that the experimentally determined
changes in force transients involve more than simply scaling by means
of a sarcomere-overlap function.
Simulated data for model 2 are shown
in Figs. 7C and
8C. The magnitude of peak force shows
large changes with SL, similar to the experimental data. However,
model 2 does not reproduce the
experimentally observed changes in twitch time course. For example, the
rate of force onset does not increase as fast as the peak force. This
causes a progressive increase in the time to peak force, as shown by
the rightward shift of the rising phases in Fig.
8C. The time to peak increases by
0.064 s.
Results from model 2 and the
experimental data also differ in the relaxation phases.
Model 2 produces an SL-dependent
prolongation of the twitch duration, as seen in the experimental data
(i.e., the traces in Fig. 5C show
incremental rightward shifts much like those in Fig.
5A). However, the rightward shifts
in the simulated data result mainly from differences in the time to
peak force. SL delays the 50% relaxation times by 0.073 s (lower
arrows) and the times to peak by 0.064 s (upper arrows). In contrast,
the experimental results show a relatively constant time to peak but increasing twitch durations. Another difference is that
model 2 produces a slow final phase of
relaxation (as force decays below 25% normalized force) with little
dependence on SL. The experimental data shown in Fig.
5A, and also additional data reported
elsewhere (51), indicated that final relaxation is faster as SL decreases.
Results for model 3 are shown in Figs.
7D and
8D. This model produces twitches that
better match the experimental data in two respects:
1) the peak force shows large
changes with SL, and 2) the rising
phases of force and the time to peak force are relatively independent
of SL. Total twitch duration and the relaxation rate are similar to the
experimental data, at least for shorter SLs in Fig.
7A. However, the responses of
model 3 differ from the experimental
results in that there is little SL-dependent prolongation of twitch
force. Another difference is that model
3 produced a final phase of relaxation with a time
constant that exhibited little dependence on SL.
The deficiencies of relaxation timing in model
3 are substantially alleviated by the new mechanism
introduced in model 4. Figure
9 shows data for the three versions of
model 4 with one, two, or three cross
bridges per functional unit. Assumption of a single cross bridge
produces twitches that are similar to those of model
3 in that there is little SL-dependent prolongation of twitch force. Whereas model 4 also
incorporates SL-dependent cross-bridge detachment rate
(Eq. 11), this feature alone does
not produce dramatic prolongation, as indicated by the 50% relaxation
times. Figure 9, C and
D, shows twitches for the
two-cross-bridge model. The additional cross bridge produces
prolongation of twitch duration with increasing SL (compare Fig. 9,
B and
D). Further prolongation can be
achieved by adding additional cross bridges. As shown in Fig. 9,
E and
F, the three-cross-bridge model
produces larger and more graded prolongation, as shown by the 50%
relaxation times.
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Fig. 9.
Effect on twitch time course of addition of multiple cross bridges with
cooperative formation. Twitches at varying SL are shown for 1- (A), 2- (C), and 3-cross-bridge versions
(E; *, 2.0-µm SL;  , 2.3-µm
SL) of model 4. Same data are
replotted as individually normalized traces in
(B),
(D), and
(F), respectively. As number of
cross bridges increases, twitch force is prolonged, as indicated by
increase in 50% relaxation times (opposing arrows at 0.5 level of
force). In contrast, there are only small changes in times to peak
force (opposing arrows at 1.0 level of force). Also shown are time
constants ( ) of final relaxation phase from 25 to 0.5% of
normalized force. See text for details.
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To quantify the final relaxation rate, the relaxation is fit to an
exponential. The time constants () are shown for traces corresponding to SLs of 1.7 and 2.3 µm. For the one-cross-bridge version of model 4, the time constant
increases only slightly (from 0.042 to 0.054 s) over the SL range. This
increase is mainly the result of the decreasing cross-bridge detachment
rate as SL increases. For the two-cross-bridge version of
model 4, there is almost a doubling of
the time constants (from 0.055 to 0.094 s). In the three-cross-bridge
version, there is little further change in the time constants (compare
Fig. 9, D and
F). Therefore, the third cross
bridge produces a larger and more graded prolongation in twitch force
but little change in time constant of the final relaxation phase.
The mechanism by which multiple cross bridges prolong force is shown in
Fig. 10. The fraction of functional units
in the force-producing states (P1,
N1,
P2, and
P3) are shown for SL = 2.0 µm
(Fig. 10A) and 2.3 µm (Fig.
10B). These data correspond to the
2.0-µm (closed circles) and 2.3-µm (asterisks) traces in Fig.
9E. First, consider the 2.0-µm case
(Fig. 10A). The fraction of units in
the P1,
P2, and
P3 states peaks at progressively later
times. For example, P1 (solid trace)
peaks at 0.9 s, whereas P3 (dot-dash
trace) peaks at 1.2 s. The last to peak is
N1 (long-dash trace) at 1.4 s, because this state becomes most highly populated during relaxation. As [Ca2+] falls and the
population of TCa
decreases, there is a corresponding decrease in
k1. With a small
k1, the
nonpermissive states (Nx) are
favored over permissive states (Px)
(see Fig. 3A). This decreases the
probability of being in P1,
P2, or
P3 states, with a corresponding increase in the probability of being in the
N1 state.
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Fig. 10.
Mechanism by which twitch duration can be progressively prolonged when
multiple cross bridges attach within same functional unit. Fraction of
functional units in force-producing states
(P1,
N1,
P2, and
P3) are shown for 2.0- (A) and 2.3-µm SL
(B). Data correspond to 2.0- and
2.3-µm traces in Fig. 9E. See text
for details.
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The mechanism by which force is prolonged at longer SLs is illustrated
with data for SL = 2.3 µm (Fig.
10B). The fraction of units in
strongly bound states (P1,
N1,
P2, and
P3) is now larger, as would be
expected for the greater total force generation. Except for
P1, the time to the peak of each state
is also later than the corresponding data for 2.0 µm (Fig.
10A). The proportion of units in the
different states also changes. For example, at the shorter SL, the peak
of P1 exceeds the peak of
P3, whereas the opposite occurs at the
longer SL. At both lengths, the N1
state peaks later than the other force-generating states. However, at the longer length, the peak of N1 is
more delayed as the persistence of the other force-generating states is
fed more slowly into the N1 state.
Model 4 demonstrates clearly that
multiple cross bridges prolong the duration of force as SL increases.
Therefore, model 4 can simulate, at
least qualitatively, the experimental data considered so far (both F-Ca
relations and twitches). Cooperative mechanism 1 (attached cross bridges modify the
Ca2+ affinity of troponin) is not
included in model 4. However, there is
considerable experimental evidence that the presence of cycling cross
bridges can increase the affinity of troponin for
Ca2+ (21, 27).
Model 5 adds this feature to the
three-cross-bridge version of model 4.
The off rate of Ca2+ from troponin
is decreased in proportion to the number of strongly bound cross
bridges (see Eq. 12). Figure
11, A
and B, shows the twitch responses for
model 5 with a relatively modest
feedback. The responses of this model are similar to those of
model 4, but the latency to peak force
increases as SL increases. This increasing delay results from the
feedback of force on Ca2+ binding
to troponin. Recall that this behavior is also evident in the responses
of model 2 (see Fig.
8C), which had feedback of force on
Ca2+ binding to troponin as its
major cooperative mechanism.
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Fig. 11.
Force and normalized twitch time courses for model
5. Twitches at varying SL are shown for
model 5 with weaker
(A) and stronger
(C) feedback of force on
Ca2+ binding to troponin. Same
data are replotted as individually normalized traces in
(B) and
(D), respectively. In
(A) and
(C), attached cross bridges can
change troponin off rate
(koff; see
Eq. 12) by a factor of 2. In
(B) and
(D), attached cross bridges can
change koff by a
factor of 4. With increasing feedback of force on
Ca2+ binding, latency to peak
force also increases with SL. Also shown are time constants () of
final relaxation phase (below 25% of normalized force). See text for
details.
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As the degree of feedback increases, the responses of
model 5 (Fig. 11,
C and
D) more closely resemble those of
model 2 (Figs. 7C and
8C). The most prominent feature is
an increasing latency to peak force as SL increases, as was also seen
for model 2 (Fig. 8C). A more subtle effect is a
slowing of the final relaxation phase at short SLs. This effect is
indicated by the time constant of the final relaxation phase for SL = 1.7 µm. The time constant increases from 0.064 to 0.085 s when the
degree of feedback is increased (compare Fig. 11,
B and
D). In general, slowing of the final relaxation phase occurs when the steady-state F-Ca relations have
low apparent cooperativity at short SLs. For example, the trace at SL = 1.7 µm for model 5 (Fig.
6D) indicates that there is small
but nonnegligible force at
[Ca2+] 
0.3 µm,
compared with the same trace for model
4 (Fig. 6B) with
higher apparent cooperativity. The low apparent cooperativity results
in a tendency to maintain force production late in the twitches, when
[Ca2+] is slowly
returning to diastolic levels. Such an effect is more apparent in the
second version of model 5 because it
is less cooperative at short SLs and low
[Ca2+]. The two
effects of feedback of force on
Ca2+ binding to troponin described
here (increased latency to peak force and decreased relaxation rate)
are both counter to the effect observed in experimental data. Hence,
the results with model 5 suggest that
there may be only a modest amount of feedback of force on
Ca2+ binding to troponin.
| |
DISCUSSION |
The responses of the five models described are closely tied to the
cooperative mechanisms assumed for each model. Clearly, all the
cooperative mechanisms are not equal in how they affected the
steady-state and dynamic responses of the models. The following discussion will focus in more detail on these cooperative mechanisms and will also cover cross-bridge cycling and limitations of the models.
Cooperative mechanism 1.
Cooperative mechanism 1 holds that the
presence of strongly bound cross bridges increases the affinity of
troponin for Ca2+. This
cooperativity is incorporated in models
2 and 5, and the effects are similar in both models. This mechanism of cooperativity increases Ca2+ sensitivity by
increasing steepness in F-Ca relations, especially in the midlevel
ranges of force (see Fig. 5C and 6,
C and
D). Note, however, that this
mechanism does not produce length-dependent changes in plateau force in
F-Ca relations. In the simulations, Ca2+ can always be made large
enough so that full activation is achieved with or without feedback.
Therefore, other features are necessary to control plateau force (e.g.,
the sarcomere-overlap function in model
2).
During twitches, feedback on Ca2+
binding can produce the length-dependent increases in peak force (i.e.,
Fig. 7C). However, such feedback
also increases latency to peak force (see Figs. 8C and
11D). The mechanism of the increase
in latency can be explained intuitively. We assume that thin-filament
activation (i.e., making actin sites available for cross-bridge
binding) is fast relative to cross-bridge cycling (this assumption is
considered later). As
[Ca2+] rises during a
transient, the thin filament becomes activated rapidly. Cross bridges
then become strongly bound after a delay as governed by their own
slower kinetics. After cross-bridge attachment, troponin affinity
increases, producing more activation, which in turn allows more cross
bridges to form at their slower attachment rate. As a consequence of
this feedback, the rate of activation is slowed so that it becomes
similar to that of cross-bridge cycling.
Several researchers have suggested that the feedback of cycling cross
bridges on the troponin affinity for
Ca2+ may play a role in
prolongation of force at high force levels (2). The modeling results
presented here agree with this suggestion in part. When feedback on
troponin affinity is the major cooperative mechanism
(model 2), there is no dramatic
prolongation of force duration. In contrast, model
5 produces a prolongation of force. Here, changes in
troponin affinity appear to work synergistically with the other
cooperative mechanisms to produce prolongation of force. Feedback of
force on the affinity of troponin for
Ca2+, in conjunction with other
cooperative mechanisms, may play a role in prolongation of the force
transient, but this mechanism alone does not seem capable of producing
significant prolongation of force.
An important finding of this study is that cooperative
mechanism 1 is not crucial to reproduce any of the
experimental results. In model 5,
cooperative mechanism 1 is added, but
only a small degree of feedback of force on the affinity of
Ca2+ binding to troponin could be
added before simulated results became inconsistent with experimental
findings (i.e., latency to peak force increased too greatly with SL,
see Fig. 11D). These results suggest
that feedback of force on Ca2+
binding may not play a major role in controlling force generation in
cardiac muscle. This conclusion is surprising given that feedback of
force on Ca2+ binding has been
presumed to be an important cooperativity mechanism (2). Contrary to
this finding, we note that not all researchers have reported that
active contraction has a large effect on
Ca2+ binding to troponin (see
e.g., Ref. 47). Also, force generation generally produces small changes
in the free Ca2+ transient (3,
30). Such an observation may suggest little change in
Ca2+ bound to troponin, especially
because changes in bound Ca2+
should be amplified to produce a relatively larger change in free
Ca2+ (20). The fraction of free
intracellular Ca2+ is estimated to
be only a small percentage of the much larger pool of bound
Ca2+ (20, 47).
Cooperative mechanism 2.
TOP
ABSTRACT
INTRODUCTION
![= <IT>k</IT><SUB>1</SUB>/(<IT>k</IT><SUB>1</SUB> + <IT>k</IT><SUB>−1</SUB>) = 1/[1 + (<IT>K</IT><SUB>1/2</SUB>/<IT>T</IT><SUB>Ca</SUB>)<SUP><IT>N</IT></SUP>]](/content/vol276/issue5/fulltext/H1734/img011.gif)
