Representative families of force responses, F
), to four amplitudes of stretch and four amplitudes of release from rat cardiac muscle fiber bundles reconstituted with troponin containing either WT-cTnT or cTnTS199E/T204E
are shown in . The force response patterns were qualitatively similar when comparing F
) from both groups, which made observable discrimination between the F
) from each group difficult. However, careful examination of the profiles of F
) from each group reveals subtle differences between the contractile behavior of rat cardiac muscle bundles containing WT-cTnT and those containing mutant cTnTS199E/T204E
. Before discussing these differences, we highlight some important features of F
) that have been traditionally identified by previous investigators (Ford et al., 1977
) and which we observed as well in each F
) recorded from both groups.
Figure 1. Representative families of force responses to various amplitudes of quick stretch and quick release. Families of force responses, F(t), to four amplitudes of quick stretch and four amplitudes of quick release in two maximally Ca2+-activated cardiac muscle (more ...)
These features can be illustrated in the force response to large-amplitude quick stretch (). This response exhibited characteristic phases and points that have been previously identified by other investigators (Huxley and Simmons, 1973
; Abbott and Steiger, 1977
; Ford et al., 1977
; Steiger, 1977
), with each event interpreted as follows. During steady-state Ca2+
activation, muscle bundles reach a steady-state force (F0
) as a result of increased acto-myosin interactions. Ca2+
releases the inhibition of the troponin–tropomyosin complex on acto-myosin interactions, and through subsequent interactions between myofilament proteins, it promotes the formation of several force-bearing XBs. Bound XBs contain elastic regions; power stroke–induced distortion of these elastic regions is the fundamental source of muscle force generation (Huxley, 1957
; Huxley and Simmons, 1971
; Ford et al., 1977
). Rapid stretch of the constantly activated muscle fiber bundle resulted in an immediate increase in force (phase 1), to a value F1
, due to rapid distortion of the elastic regions of bound XBs. Force then rapidly decays as distorted XBs rapidly detach to reequilibrate into the nondistorted state (phase 2). A slow, but eventual increase in force production (phase 3) then ensues as the increase in muscle length results in a length-induced recruitment of additional XBs into the force-bearing state. Muscle force slightly overshoots (phase 4) and then approaches a steady-state value, FSS
, as the additional length-recruited XBs equilibrate into the force-bearing state. Some of the characteristics observed in the force response to large-amplitude stretch were also observed in the force response to large-amplitude quick release. Conversely, some of these characteristics of the large-amplitude response were not present or occurred at a different time during the response to release, indicating a departure from linear dependence of the shape of the force response to the magnitude of length change. Thus, there was both linear and nonlinear behavior in the family of force responses of both WT-cTnT– and cTnTS199E/T204E
-containing muscle fiber bundles to various amplitude changes in length. We identified and quantified these behaviors as described below.
Figure 2. Traditionally identified phases and nontraditionally identified nonlinearities in the force-response patterns to varied amplitude step-like changes in length. (A) Force responses to largest amplitude (±2% ML) length change. (B) Force responses (more ...)
Linear response features are identified as those whose amplitudes scale linearly with the amplitude of an input. As a consequence, in linear systems, the response to a positive input of given amplitude will be a mirror image of the response to a negative input of the same amplitude. Examples of linear behavior were that of F1 and FSS responses, whose values scaled linearly with the magnitude of the length change (ΔL) (). The slope of the relationship between F1 and ΔL is an index of the stiffness (ED) of the population of bound XBs at the instance of the length change, and it is proportional to the total number of strongly bound XBs at the time of ΔL. The slope of the relationship between FSS and ΔL (ER) is an index of the sensitivity of length-mediated recruitment of additional XBs into the force-bearing state, and it is proportional to the incremental change in the number of force generators in response to ΔL.
Figure 3. Relationship between F1 and FSS and ΔL. The magnitude of F1 and FSS scale linearly with the magnitude of length change, ΔL. The dependence of F1 and FSS on ΔL was steeper in fiber bundles containing WT-cTnT (A) when compared with (more ...)
were significantly greater in muscle fiber bundles containing WT-cTnT than in those containing cTnTS199E/T204E
(). The increased ED
observed in bundles containing WT-cTnT correlates well with an observed increase in F1
because both ED
can be correlated to the number of parallel, force-generating XBs bound at steady-state activation before stretch (Campbell et al., 2004
Stiffness estimates from ΔL-F1 relationship and ΔL-FSS relationship
Nonlinear behavior can be identified as that in which the amplitude or shape of a characteristic does not scale linearly to a given input. Examples of nonlinear contractile behavior were observed in the transient of the force response between F1 and FSS. Such nonlinear behavior is illustrated in the observation that the shape and pattern of the force responses to quick stretch were different than that of the force responses to quick release. For example, the shape of the transient in the force response to large-amplitude quick stretch was different than the shape of the transient in the force response to large-amplitude quick release. The transient to quick stretch possessed a well-defined valley or nadir separating a quick component of the transient from a slow component. In contrast, the transient in a quick-release response did not have a corresponding feature, and the transition from the quick component to a slow component was less well defined (). In comparison, the shape of the transient in the force response to small-amplitude quick stretch roughly mirrored that of the transient in the force response to small-amplitude quick release (). These observations are a result of the more global observation that the pattern of the shape of the transients in force responses to stretches was different than the pattern of the shape of the transients in force response to releases. For example, the transients in the force responses to stretch each dipped toward one another at a nadir (point F23 in ), whereas the shape of the transients in the force responses to releases diverged from another and did not possess a common zenith. Therefore, the shape of the transient between F1 and FSS was considered to have a nonlinear dependence on the magnitude of ΔL because the patterns of these shapes in stretch and release were not mirror images of one another.
Nonlinearities can be further demonstrated by examining how specific periods and phases of F
) scale with the magnitude of ΔL. Such analysis was used in a historical study by Ford et al. (1977)
to identify and account for nonlinear behavior in skeletal muscle, whereby both the magnitude of tension approached at an early recovery phase (T2
) and the rate at which tension approached T2
in the force response to quick release varied with the amplitude of length change. In the present study, we note that the time to reach the dip in responses to stretch, T23
, and the time to reach 90% of FSS
, in responses to release also varied with the magnitude of ΔL (). This indicates nonlinear behavior in F
) because the time course, and thus the shape, of the F1
transient was different at different magnitudes of ΔL. Both the ΔL-T23
and the ΔL-T90
relationships were different in rat cardiac muscle bundles containing WT-cTnT and those containing cTnTS199E/T204E
, indicating that the nonlinear behavior was different in these different muscle preparations. Additionally, the magnitude of F23
scaled curvilinearly with the magnitude of ΔL (). shows additional distinction between bundles containing the WT-cTnT and those containing cTnTS199E/T204E
, further indicating that nonlinear behavior was different in these muscle preparations containing different variants of cTnT.
Figure 4. Time to selected features of response waveforms as they varied with amplitude of the stretch, ΔL. Open symbols are data from muscle fiber bundles containing WT-cTnT; filled symbols are data from muscle fiber bundles containing cTnTS199E/T204E (more ...)
Figure 5. The value of the force at the nadir in the response to quick stretches, F23, as it varied with amplitude of the stretch, ΔL. Open points and dashed line are data from muscle fiber bundles containing WT-cTnT; solid points and line are data from (more ...)
Characterization of nonlinear behavior, as presented above, only allowed subtle discrimination between the two muscle preparations (note that the scale and magnitude of variation between groups in and are <10%). Thus, the aim of this study was to characterize the nonlinearities contained in the entire family of force responses to various amplitude step-like length changes in such a way as to easily discriminate contractile behavior of muscle preparations containing different contractile elements. Our goal was to formulate a mathematical model that is able to account for both the linearities and nonlinearities described above, and that is able to discriminate between nonlinear behaviors of different muscle preparations. Both linear and nonlinear features of the family of F(t) represent important aspects of the contractile behavior of the myofilament. Linear features represent linearly elastic properties related to the number of bound XBs based on ΔL-induced distortion and recruitment of force-bearing XBs, whereas nonlinear features represent other mechanisms of length-mediated contractile activation.
Model development began with an RD formulation that we used successfully to describe the force response to sinusoidal length change (Campbell et al., 2004
; Chandra et al., 2006
). In these sinusoidal force responses, only linear behaviors of the muscle were observed, and a linear combination of independent effects due to distortion and recruitment was used. However, because obvious nonlinear behaviors were observed in the step response, we no longer treated the effects of the recruitment and distortion as independent. Rather, we formulated muscle force to be equal to the product of a recruitment variable, η(t
), and a distortion variable, x
), as is consistent with their physical meaning; i.e., η(t
) represents the net stiffness of a population of parallel stiffness elements, and x
) represents the average elastic distortion among these elements. Thus,
The RD concept arises directly from considering linearly elastic myosin XBs as the elemental force generators in muscle. We refer to earlier publications for the origins of these concepts and for the intellectual bridge between myofilament kinetics and RD fiber bundle dynamics (Campbell et al., 2004
; Chandra et al., 2007
The product of η(t
) and x
) in Eq. 1
to produce force is a nonlinear combination of variables. In our earlier work (Campbell et al., 2004
; Chandra et al., 2006
), we linearized Eq. 1
to predict linear small-signal behavior. Here, however, we will start with the product combination of η(t
) and x
), as this nonlinear combination may be important to explaining some of the nonlinear behaviors we observed in the recorded step response. We now examine dynamic expressions for the length-induced changes in η(t
) and x
After muscle length change during constant Ca2+
activation, net stiffness, η(t
), changes as more or fewer stiffness elements, XBs, are recruited into or out of the stiffness pool. Because of length-dependent activation kinetics and XB cycling, length-mediated recruitment is a dynamic process requiring time to complete after a change in length. In our earlier work (Campbell et al., 2004
; Chandra et al., 2006
), we successfully approximated these recruitment dynamics with the first-order linear differential equation:
is the recruitment rate constant, β0
is a scaling factor, and ld
is a length at which no recruitment occurs.
Also, muscle length change causes a change in the average elastic distortion, x
), of stiffness elements. This change happens immediately; i.e., XB stiffness elements are immediately stretched at the moment the muscle fiber is stretched. However, XB cycling results in the breaking of stretched XBs and the replacement of these broken XBs with new XBs that did not experience the stretch event. We have successfully approximated the dynamics of this transient distortion change with the linear differential equation:
is the distortion rate constant, x0
is a baseline isometric distortion imposed on the XB by the power stroke, and
is the time rate of length change or the muscle fiber velocity.
For an idealized step in which length change occurs instantaneously, analytical solutions of the recruitment and distortion equations can be derived and are helpful in interpreting the response. For an idealized step of magnitude ΔL, the solutions of Eqs. 2
These solutions allow calculation of an immediate response as:
and a steady-state response after all transients have died out as:
where both F1
are normalized with respect to F0
. From Eq. 6
, we see that the starting model predicts that F1
is linearly dependent on ΔL, even though η(t
) and x
) combine nonlinearly to determine F
) in Eq. 1
. This model-predicted linear relationship between F1
and ΔL, which is consistent with what was observed experimentally, demonstrates that distortion by itself is the determinant of the immediate response, and recruitment does not play a role in this very early part of the step response. Further, the starting model also predicts that FSS
too is linearly dependent on ΔL in spite of the nonlinear character of Eq. 1
, which is also consistent with the experimental results. This result demonstrates that recruitment is the determinant of the steady-state response, and stretch-induced distortion does not play a role in the late parts of the response.
The starting model adequately predicts the immediate and steady-state aspects of the step response and now needs to be examined in terms of its ability to predict the transition between these two extremes. Does the nonlinear combination of the recruitment and distortion variables through the multiplication of Eq. 1
produce any of the nonlinear behaviors that were noted in the experimentally observed responses? We examine two cases: (1) the rate constants of recruitment, b
, and distortion, c
, are equal; and (2) the rate constant of distortion is 10 times greater than that of recruitment.
The starting model predicts that transition between F1 of the immediate response and FSS of the steady-state response is greatly affected by the relative values of b and c (see ). When the rate constants for distortion and recruitment are equal (c = b), the transition occurs monotonically with no feature in the trajectory to distinguish various phases of the response. However, when the rate constants differ by an order of magnitude (c = 10*b), the transition trajectory shows two distinct phases: a fast phase as x(t) quickly recovers to its isometric value and a slow phase as η(t) rises (or falls depending on whether stretch or release) to its eventual new steady-state value as a result of length-mediated recruitment effects. Separation of the fast and slow phases produces a nadir in the response to stretch and a zenith in the response to release. Thus, some aspects of the experimentally observed trajectory between F1 and FSS can be recreated with the starting model by separating the values of the rate constants of distortion and recruitment.
Figure A1. Starting model predictions of F(t) to step changes in length when the rate constants for distortion, c, and recruitment, b, are equal (left) and when they differ by an order of magnitude (right). Insets show time course of recruitment variable, η( (more ...)
However, despite the appearance of a nadir in the trajectory and a clear demarcation of a point in the trajectory that distinguishes the x(t) recovery from the η(t) approach to a new steady state, many of the notable nonlinear features in the experimentally observed step response are absent: (1) for all intents and purposes, the response to a stretch is a mirror image of the response to a release at all step amplitudes, unlike what was observed experimentally where the transition from F1 to FSS was very different for stretches than for releases; (2) trajectories at the various step amplitudes tend to converge on a F23 transition point in responses to both stretches and releases, whereas transitions in the experimentally observed responses to releases followed widely separated trajectories; (3) although the F23 transition point occurs earlier in the response when c is made progressively larger than b, for a given value of c and b it occurs at the same time in the response, regardless of the step amplitude (not depicted); and (4) the force value at the F23 transition point scales linearly with step amplitude and never dips below the initial force, no matter how large the difference between rate constants.
In summary, although the starting model generates many overt features of the experimentally observed step response, the nonlinear product combination of η(t
) and x
) in Eq. 1
does not produce any of the prominent or subtle nonlinear features in the experimentally observed step response.
Model variations to qualitatively reproduce observed nonlinearity
Because the starting model, with its nonlinear combination of η(t
) and x
) in the force calculation (Eq. 1
), did not produce any of the nonlinear behaviors that were noted in the experimental results, we examined variants of this starting model in terms of their ability to reproduce aspects of these nonlinear effects. Results from the starting model, especially in the decomposition of the response into its recruitment and distortion components (see ), led us to focus on the recruitment equation as the most likely place in the model where variations could bring about improvement. Two general schemes for variation were attempted: (1) variants of Eq. 2
in which velocity had a negative effect on recruitment; and (2) variants in which there was nonlinear interaction between η(t
) and x
In brief, all attempts to introduce velocity effects on recruitment were unsuccessful in improving model performance. This was due, in part, to the fact that these velocity effects preferentially and severely degraded the immediate or F1 response turning the F1 versus ΔL relationship from a linear shape (which was experimentally observed) into a curvilinear shape (which was not experimentally observed). In contrast, a nonlinear term with an interaction between η(t) and x(t) had a powerful effect on the transition between F1 and FSS without affecting the F1 and FSS aspects of the response, which were already satisfactorily reproduced by the starting model. The challenge became one of identifying the correct form of the η(t)–x(t) interaction term. We approached this challenge as follows.
The interaction term was incorporated into the differential equation for η(t
) was one of several competing functions of x
) (see ). Actually, x
) was normalized as
which had the effect of confining distortion effects to situations where x
) deviated from its isometric value, x0
(refer to Appendix
for model in normalized form). Further, by dividing this deviation by x0
, physical units were removed from this term and the interaction coefficient, γ, carried units of s−1
, which then allowed γ to be compared with the b
coefficient. As with the starting model, the equation for distortion in these model variants remained as given by Eq. 3
Summary of nonlinear behavior in variants of the recruitment component RD model
Because of the nonlinear interaction term in Eq. 8
, an analytical solution for η(t
) during the transient between F1
could not readily be obtained. Consequently, we used numerical integration methods to solve for η(t
) during this transient. However, the steady-state solution for η(t
) was not affected by the nonlinear term, and the solution for x
) was as before (i.e., Eq. 5
). Thus, analytical solutions were found for both F1
, and these were identical to the linear solutions found for the starting model (e.g., Eqs. 6
). As has already been pointed out, these F1
solutions were consistent with what was observed experimentally.
Model results for three variants of f
) are summarized in the Appendix
(). Of these, only one variant was successful in generating the prominent aspects of the experimentally observed nonlinear response:
We demonstrate results from this model variant in the following figures.
The time course of the predicted step response of this model-variant is shown in . The outstanding result is that this model-variant reproduced many of the most prominent nonlinear features of the experimentally observed response transient including:
Figure 6. Predicted step response of model variant. The addition of the second-order interaction term produces a family of force responses (A) similar to experimentally observed responses to various amplitude quick stretch and release. (B) Each component of the (more ...)
- The shape of the transient of responses to large-amplitude quick stretches was very different than the shape of the transient of responses to large-amplitude quick releases. However, the response to the smallest-amplitude quick stretch roughly resembled the response to the smallest-amplitude quick release.
- The trajectory pattern among responses to different ΔL amplitudes was very different for quick stretches than for quick releases, with the trajectories of the four transients to quick stretches at different amplitudes tending to all approach one another at a nadir, while the trajectories of the four transients to quick releases at different amplitudes exhibited no nadirs and remained apart and distinct from one another throughout the transient.
- Within the subfamily of the four responses to quick stretches, the value of the force at the time of the nadir in the transient, F23, did tend to converge on a common point and they scaled curvilinearly with the magnitude of length change, and the time to the nadir, T23, varied with length change magnitude.
- Within the responses to quick release, the shape of the response changed from one with an overshoot for the smallest length change to one without an overshoot and a monotonic approach to FSS for the largest length change.
There are only two features of these model predictions that differ from what was seen experimentally: (1) the curvilinearity in the F23 versus ΔL relationship is in a different direction in the experimental data (curvilinear up with increasing ΔL) than what it is in the model predictions (curvilinear down with increasing ΔL); and (2) there is no overshoot in the response to large-amplitude quick stretches preceding the final approach to FSS.
The RD model containing the nonlinear interaction term in the recruitment equation (Eq. 9
) reproduced the qualitative nonlinear features observed experimentally in the step response. Thus, it served as the model for validation using the family of force responses to various amplitude quick stretches and quick releases. To make this validation, we fit the model to each of the records of the F
) family taken from rat cardiac bundles containing either the WT-cTnT or cTnTS199E/T204E
variant. We then compared the fits to the F
) family by the simple RD model with those by the NLRD model predictions. Substantial improvement of fits by the NLRD that could be attributed to reproduction of the observed nonlinear traits was taken as evidence of model validity.
Model validation and application
Model fitting and selection methods.
Model predictions were computed using fourth-order Runge-Kutta numerical integration. Computed model predictions were fitted to data records using a Levenberg-Marquardt regression method to minimize sum of square errors (SSE). Results of fitting were obtained not only for the RD and NLRD models described above, but also for all the competing variants of the model shown in the Appendix
(). Comparisons of models were based on several criteria.
Criterion 1: Reproduction of the shape of F(t).
An important criterion in model validation was that the model, when fit to data, reproduced the essential shape of F
). As shown in and , and , the starting RD and NLRD models were capable of reproducing the general shape of F
). As a qualitative assessment of how well the model was able to reproduce the essential shape, we inspected the likeness of
) and inspected the time course of the residual errors after data fitting.
Criterion 2: Goodness of fit.
Good fit was indicated by low SSE and an R2
value close to 1. SSE and R2
were determined by the following calculations:
is the model predicted force response, F
) is the measured force response, and
is the mean value of F
). Goodness of fit was also estimated by R2
value given by the linear regression of the relationship between F
Criterion 3: Confidence in the model parameter estimates.
Low standard errors relative to the parameter estimates (coefficient of variation <0.05) suggest that the model parameters were independent in their effects on the predicted force response. A fit resulting in a low standard error of the parameter estimate would suggest that the parameters in the recruitment and distortion equations had independent effects on
and were therefore estimated uniquely. Therefore, suitable models were those that, when fit to data, produced low SSE and high R2
values and resulted in low standard errors in the estimation of model parameters.
Distinguishing between competing models.
From the group of competing models ( and ), the best model was determined based on respective Akaike’s information criterion (AIC). AIC of each model prediction was computed using the following equation:
is the number of parameters in the model, and nobs
is the number of observations in the entire F(t)
data record. AIC was calculated for each model fit and was used to rank the models for each data record. The model that consistently gave the lowest AIC value was considered the best model out of the group of competing models.
The models were used to describe the contractile behavior of the experimental preparations. Fitted parameters were used to distinguish contractile behavior between rat cardiac muscle bundles containing either WT-cTnT or cTnTS199E/T204E. This was done by comparing parameter estimates obtained in each group using a two-tailed t test (α = 0.05).
Model validation and application results
Criterion 1: Reproduction of the shape of F(t).
Representative fits of RD and NLRD models to F
) records collected from WT-cTnT– or cTnTS199E/T204E
-containing muscle fiber bundles are shown in . The waveforms of the responses in the NLRD predicted responses were qualitatively closer to those in observed data, reproducing the essential phases and nonlinearities of F
) described previously, than were those of the RD prediction (). This was substantiated by the observation that the time course of residual errors showed lesser variation when the NLRD model was used to fit data than was observed in the time course of residual errors when the RD model was used to fit data (not depicted). This decrease in residual error due to the addition of the nonlinear interaction term into the recruitment component of
suggests that the shape of NLRD predicted
was closer to that of F
) than was the shape of simple RD model–predicted
. Therefore, the NLRD model was more accurate in the reproduction of the essential shape of F
) than was the RD model, suggesting that the NLRD model is more appropriate for describing data based on criterion 1.
Figure 7. Representative model-predicted F(t) when fit to data from muscle fiber bundles reconstituted with troponin containing WT-cTnT or cTnTS199E/T204E. When fit to data, the NLRD model reproduced the essential shape of the waveform of F(t) in both fiber bundles (more ...) Criterion 2: Goodness of fit.
Models were fitted to 11 families of force responses to various amplitude stretches and releases of maximally activated bundles (n
= 6 from WT-cTnT–containing bundles; n
= 5 from cTnTS199E/T204E
-containing bundles; nobs
= 8,800 in each F
)). Both the RD and NLRD models met the first criterion in model selection and validation reasonably well, producing high R2
values and low sum of square residual errors ( and ). R2
ranged from 0.881 to 0.966 in simple RD model fits and ranged from 0.971 to 0.993 in NLRD models fits. The NLRD model resulted in lower overall SSE and higher R2
values than did the simple RD model when fit to data records from rat bundles containing either WT-cTnT or cTnTS199E/T204E
variants. The observed decrease in SSE and increase in R2
from the NLRD model fits compared with fits from the RD model was highly significant (). This suggests that the addition of the nonlinear interaction term into the recruitment component of
resulted in a better descriptor of the data from both experimental groups because doing so improved the goodness of fit.
Figure 8. Relationship between observed F(t) and model-predicted F(t). The relationship between observed F(t) and RD-predicted F(t) values is nearly linear from cardiac muscle fiber bundles containing either WT-cTnT (A) or cTnTS199E/T204E (C). Adding the nonlinear (more ...)
SSE and R2 values from fitting of RD and NLRD models to F(t)
The result of the F(t)
regressions are shown in . In WT-cTnT–containing muscle fiber bundles, regression fits are
) + 0.0778 for the RD model and
) + 0.0326 for the NLRD model. In cTnTS199E/T204E
-containing bundles, regression fits are
) + 0.1291 for the RD model and
) + 0.0493 for the NLRD model. Therefore, the slope of these regressions approached 1 and the intercepts approached 0; slope and intercept were more close to ideal values in the NLRD model than in the RD model (P < 0.0001). Collectively, these results suggest that the addition of the nonlinear interaction term to η(t
) was appropriate for fitting to data because NLRD resulted in better fits of
and, therefore, satisfied criterion 2 better than the RD model did.
Criterion 3: Confidence in the model parameter estimates.
The standard errors of the parameter estimates relative to the respective parameters (coefficients of variation [CoV]) were satisfactorily low, as shown in . For each parameter estimate, the relative standard error was always <5%. This was consistent across fits to all data records using both RD and NLRD models. Because CoV were consistently low for each parameter in each model, we were able to assume that each of the model parameters had independent effects on the overall model-predicted
. Therefore, both models provided confidence in the uniqueness of the estimated parameter values.
Between the two models, the NLRD model obtained significantly lower CoV of each parameter estimate when compared with those obtained by the RD model, despite the presence of an additional parameter. Over-parameterized models, when fit to data, often result in non-unique solutions and contain parameters that are estimated with little confidence. In this case, however, the addition of the nonlinear interaction term was not damaging to our confidence in parameter estimation, but instead increased the confidence by which parameters were estimated.
Discriminating between competing models.
Because both models to a large degree satisfied criteria 1–3, we looked at the AIC to provide an objective index to rank the competing models ( and ). In every fit of F(t) from both WT-cTnT– and cTnTS199E/T204E-containing cardiac muscle fiber bundles, the AIC index was less for the NRLD model than it was for the RD model. Based on this index, the NLRD model was always a better descriptor of F(t) than was the RD model.
Fitted model parameter estimates from RD and NLRD are shown in and , respectively. The RD model was unable to distinguish any difference in the rate constant of length-mediated XB recruitment, b, between groups containing either WT-cTnT or cTnTS199E/T204E (). The RD model was, however, capable of distinguishing a faster rate constant of strain-induced XB detachment, c, in cardiac muscle fiber bundles containing the cTnTS199E/T204E when compared with muscle fiber bundles containing WT-cTnT ().
Figure 9. Comparison of fitted RD model parameters between fiber bundles reconstituted with troponin containing WT-cTnT or cTnTS199E/T204E. Error bars are SEM. (A) The rate constant of XB recruitment. (B) The rate constant of XB distortion. The RD model distinguished (more ...)
Figure 10. Comparison of fitted NLRD model parameters between fiber bundles reconstituted with troponin containing WT-cTnT or cTnTS199E/T204E. Error bars are SEM. (A) The rate constant of XB recruitment. (B) The rate constant of XB distortion. (C) The nonlinear (more ...)
Similarly, the NLRD model was not able to distinguish differences in the XB recruitment rate constant between muscle fiber bundles containing either WT-cTnT or cTnTS199E/T204E, but it readily distinguished a faster dynamic rate constant of strain-induced XB detachment in muscle fiber bundles containing cTnTS199E/T204E (). However, the NLRD model provided further distinction between the contractile behavior of the two muscle groups in that the NLRD model predicted that XB recruitment was affected differently in the two groups of muscle fibers by the XB strain-mediated effect on the recruitment of strong XBs (i.e., γ; ). XB strain had a much greater negative impact on XB recruitment in bundles containing the cTnTS199E/T204E than in muscle fiber bundles containing the WT-cTnT, contributing to a more pronounced nonlinearity in F(t) that was observed in bundles containing cTnTS199E/T204E. Although this unique characteristic of the cTnTS199E/T204E-containing group was easily identifiable using the NLRD model, the RD model was unable to distinguish between the nonlinear behaviors of groups containing either WT-cTnT or cTnTS199E/T204E.
Therefore, the NLRD model can be used to provide a more detailed interpretation of the effects that alterations in the structure of contractile proteins have on the contractile function of cardiac muscle fiber bundles. The NLRD model contains the nonlinear interaction term, which is formulated as an effect by which the state of bound XBs influences the recruitment of other XBs. Because XBs do not directly interact with one another, this interaction describes cooperative and/or allosteric mechanisms translated along the thick or thin filaments. Here, our study shows that the thin filament is involved in the transduction of the XB strain-dependent effect on other XBs because different variants of cTnT affect this nonlinear interaction process differently. Modifying cTnT and, in turn, the thin filament structure modified the mechanisms by which XBs influence the recruitment of other XBs, as measured by γ. This suggests that thin filament proteins (e.g., cTnT) play an important allosteric regulatory role in XB-mediated XB recruitment, a process that underlies prominent nonlinear contractile behavior in cardiac muscle.