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J Appl Physiol 101: 1189-1198, 2006. First published June 15, 2006; doi:10.1152/japplphysiol.00396.2006
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A mechanical model for adjustable passive stiffness in rabbit detrusor

John E. Speich,1 Kevin Quintero,1 Christopher Dosier,2 Lindsey Borgsmiller,3 Harry P. Koo,3 and Paul H. Ratz4

Departments of 1Mechanical Engineering, 2Biomedical Engineering, 3Surgery, and 4Biochemistry and Pediatrics, Virginia Commonwealth University, Richmond, Virginia

Submitted 4 April 2006 ; accepted in final form 7 June 2006


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Strips of rabbit detrusor smooth muscle (DSM) exhibit adjustable passive stiffness characterized by strain softening: a loss of stiffness on stretch to a new length distinct from viscoelastic behavior. At the molecular level, strain softening appears to be caused by cross-link breakage and is essentially irreversible when DSM is maintained under passive conditions (i.e., when cross bridges are not cycling to produce active force). However, on DSM activation, strain softening is reversible and likely due to cross-link reformation. Thus DSM displays adjustable passive stiffness that is dependent on the history of both muscle strain and activation. The present study provides empirical data showing that, in DSM, 1) passive isometric force relaxation includes a very slow component requiring hours to approach steady state, 2) the level of passive force maintained at steady state is less if the tissue has previously been strain softened, and 3) tissues subjected to a quick-release protocol exhibit a biphasic response consisting of passive force redevelopment followed by force relaxation. To explain these and previously identified characteristics, a mechanical model for adjustable passive stiffness is proposed based on the addition of a novel cross-linking element to a hybrid Kelvin/Voigt viscoelastic model.

smooth muscle mechanics; preconditioning; strain softening; passive force; muscle model

IN SMOOTH MUSCLE, FORCE HAS both active and passive components. The passive component is the portion of the total force remaining when cross bridges directly responsible for force development are not cycling to produce active force. In smooth and cardiac muscles, passive force contributes significantly to the total force produced at the optimum length for muscle contraction (16, 43). The degree of passive force also participates in regulating smooth muscle hypertrophy and remodeling by providing a stimulus that activates cell signaling systems (19, 22, 45), which can lead ultimately to pathological conditions, such as hypertension (36) and urinary incontinence (26).

In our previous work, passive mechanical properties of strips of rabbit detrusor smooth muscle (DSM) were examined and found by cyclic loading in a calcium-free solution to display both viscoelastic softening and strain softening (41). Strain softening is irreversible in DSM, even after the return of spontaneous rhythmic tone during 120 min of incubation in a calcium-containing solution. However, 3 min of KCl or carbachol-induced contraction at short muscle lengths permit rapid regeneration of the passive stiffness lost to strain softening, and inhibition of rhoA kinase (ROK) by 3 µM Y-27632 prevents this regeneration. Thus rabbit DSM displays adjustable passive stiffness (APS) that is both strain history and muscle activation history dependent. This component of passive stiffness is adjustable because it can be activated by stimuli such as KCl or carbachol. Whether this stiffness is produced by cross bridges (7), smitin (18), cross-linking proteins such as caldesmon (44), or other structures remains to be determined.

Two simple mechanical models of smooth muscle are the Kelvin (standard linear) model and the Voigt model (Fig. 1) (11, 28). Each of these viscoelastic models contains three elements: a variable damper (D) that generates active force when the actomyosin cross bridges are attached and cycling (termed a contractile component) and acts as a dashpot when actomyosin cross bridges do not actively cycle through a power stroke to cause muscle contraction (i.e., when DSM is maintained under passive conditions); and two elastic components, one in series (SEC) and one in parallel (PEC) with the contractile component. In the Kelvin configuration, the PEC is in parallel with both the SEC and the D (Maxwell configuration), whereas, in the Voigt configuration (28), the PEC is in parallel with the D alone. Although the Kelvin configuration has been used to model the behavior of rubber using hyperelastic stiffness elements and a variable PEC (VPEC) to account for strain softening (5, 21), no studies to date have adapted the Kelvin or Voigt models to account for APS in smooth muscle.


Figure 1
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  		Fig. 1. Kelvin (A) and Voigt (B) viscoelastic models. PEC, parallel elastic component; SEC, series elastic component; CC, contractile component; D, damper.

 
The aim of the present study was to construct a model that reproduces the characteristics of adjustable stiffness exhibited by passive DSM. Development of such a model will be valuable in the identification, characterization, and discussion of the biochemical structures contributing to, and physiological mechanisms regulating, passive stiffness and force in smooth muscle. Also, to facilitate the determination of the appropriate configuration of modeling components, two new experiments were performed to identify tissue characteristics not addressed in the previous study (41). Specifically, experiments were performed to test the hypothesis that calcium-independent cross-bridge cycling is not the primary source of force redevelopment following the quick-release of a DSM strip and to test the hypothesis that strain softening reduces the steady-state force at a given muscle length.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Tissue preparation.   All experiments involving animals were conducted within the appropriate animal welfare regulations and guidelines and were approved by the Virginia Commonwealth University Institutional Animal Care and Use Committee. Tissues were prepared as described previously (33, 38). Whole bladders from adult female New Zealand White rabbits (2–4 kg) were removed immediately after death with an overdose of pentobarbital. Bladders were washed, cleaned of adhering tissues, including fat and serosa, and stored in cold (0–4°C) physiological salt solution (PSS), composed of 140 mM NaCl; 4.7 mM KCl; 1.2 mM MgSO4; 1.6 mM CaCl2; 1.2 mM Na2HPO4; 2.0 mM morpholinopropanesulfonic acid (adjusted to pH 7.4 at either 0 or 37°C, as appropriate); 0.02 mM Na2 EDTA; and 5.6 mM dextrose. For clarity, PSS will be referred to as a "Ca2+-containing solution," while PSS with no CaCl2 and the addition of 1 mM EGTA to chelate Ca2+ will be referred to as a "Ca2+-free solution." Thin strips (~0.2 mm thick) of longitudinal DSM free of underlying urothelium and overlying serosa were cut from the bladder wall above the trigone and close to the dome [upper detrusor (35)] by following the natural bundling clearly demarcated when bladders were in ice-cold buffer, as described previously (35). Each tissue was secured by small clips to a micrometer for manual length adjustments and a computer-controlled electronic lever (model 300H with DMC software, Aurora Scientific) to record force and to induce time-controlled muscle length changes. Each tissue was secured such that its initial (cold) zero preload length was ~3 mm and equilibrated in aerated PSS at 37°C in a water-jacketed tissue bath for 1 h to permit development of spontaneous rhythmic contraction. Tissues were then incubated in a Ca2+-free solution to eliminate spontaneous contractile activity (37), stretched in 0.5-mm step increments, and allowed to stress-relax with each step increase until a stable preload of 0.49 mN (the minimum measurable positive force) was established. This was considered slack length (Ls) at 37°C. In strain-softened tissues, we have found that the optimal length for muscle contraction is ~200% Ls, and the experiments in the present study involve length manipulations along the ascending limb of the active DSM length-tension curve.

Following the determination of Ls, each tissue was incubated in a Ca2+-containing solution for 30 min to obtain spontaneous rhythmic tone, placed in a Ca2+-free solution for 5 min, stretched manually to the maximum strain in the protocol (e.g., 170% Ls) using a micrometer, and allowed to stress-relax at this length for 5 min. Tissues were then returned to Ls, placed in a Ca2+-containing solution for 1 min, and contracted twice with 110 mM KCl (substituted isosmotically for NaCl) for 3 min to determine the maximum KCl-induced force. Isometric contraction at Ls, which permits the restoration of APS lost to strain softening during the stretch (e.g., to 170% Ls) (41), was measured as described previously (34, 38). Force and length signals were digitized (PCI-6024E, National Instruments) and stored electronically for analyses.

Quick-release protocol.   Tissues were subjected to a quick-release protocol to study the characteristics of isometric force redevelopment and relaxation. Strips of DSM incubated in a Ca2+-free solution for 10 min to eliminate rhythmic tone were stretched manually using the micrometer to 140% Ls. Tissues were then stretched quickly (100 mm/s) using the electromechanical lever to 170% Ls, allowed to isometrically stress-relax for 30 s, quickly (100 mm/s) released to 165% Ls, and held isometrically for 5 min (Fig. 2A), while force redeveloped and then relaxed (Fig. 3A). Tissues were returned to Ls and contracted twice with 110 mM KCl for 3 min to permit the restoration of APS lost to strain softening during the stretch to 170% Ls (41). After washout of KCl to permit relaxation, some tissues were incubated in a Ca2+-free solution containing 10 µM staurosporine for 10 min, stretched using the micrometer to 140% Ls, and subjected a second time to the stretch and quick-release protocol in Fig. 2A while in the Ca2+-free solution containing 10 µM staurosporine. This drug treatment was implemented to seek to determine whether calcium-independent cross-bridge cycling might be responsible for force redevelopment following the quick-release (see DISCUSSION). Control experiments without the staurosporine treatment were performed for comparison.


Figure 2
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  		Fig. 2. Experimental protocols. A: quick-release protocol in which tissues were stretched to 170% slack length (Ls) at a rate of 100 mm/s, allowed to isometrically relax for 30 s, quickly released to 165% Ls at a rate of 100 mm/s, and held isometrically for 5 min while force redeveloped and then relaxed. B: steady-state force protocol in which one-half of the tissues were manually stretched to 200% Ls (dashed line), allowed to isometrically relax for 5 min, and then returned to Ls, while the other half were not stretched, but remained at Ls. After 5 min at Ls, all tissues were manually returned to 130% Ls, and isometric force was measured for at least 1 h.

 

Figure 3
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  		Fig. 3. A: typical isometric force (F) measurement following a quick-release (Fig. 2A), exhibiting both a force redevelopment phase and a force relaxation phase (inset). Fpeak, peak force. B: the normalized peak redeveloped force (0.049 ± 0.009) was significantly higher (*P < 0.05, n = 6) than the normalized 5-min redeveloped force (0.033 ± 0.005). Values are from the first release to 165% Ls and are normalized to the peak force at 170% Ls. C: peak redeveloped forces from a second quick-release in calcium-free solution with and without 10 µM staurosporine (Stauro = 0.89 ± 0.05, Control = 0.92 ± 0.15) were not significantly different (P > 0.05, n = 3). Values are presented as the second release as a fraction of the first release. D: 5-min redeveloped forces were also not significantly different with and without 10 µM staurosporine (Stauro = 0.93 ± 0.03, Control = 0.89 ± 0.16, P > 0.05, n = 3).

 
Steady-state force protocol.   Experiments were performed to examine the relationship between strain softening and the passive force maintained at steady state. Following the determination of Ls, each tissue strip was incubated in a Ca2+-containing solution for 1 min and contracted with 110 mM KCl for 3 min to determine the maximum force and establish APS. After washout of KCl to permit relaxation, tissues were incubated in a Ca2+-free solution for 5 min to eliminate rhythmic tone. Some tissues were manually stretched at ~0.05 mm/s to 200% Ls to cause strain softening, allowed to isometrically stress-relax for 5 min, and then returned to Ls (Fig. 2B). All tissues were then manually stretched to 130% Ls at ~0.05 mm/s, and isometric force was recorded for 1–4 h.

Statistics.   Analysis of variance and the Student-Newman-Keuls test, or the t-test, were used where appropriate to determine significance, and the null hypothesis was rejected at P < 0.05. The population sample size (n value) refers to the number of animals, not the number of tissues.


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 APPENDIX
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Isometric force redevelopment and relaxation on quick-release.   Strips of passive DSM were twice subjected to the quick-release protocol in Fig. 2A in calcium-free conditions, as described in Quick-release protocol above. Time-dependent redeveloped force values were normalized to the corresponding peak forces at 170% Ls. Tissues exhibited both a force redevelopment phase and a force relaxation phase when held isometrically following a quick-release (Fig. 3A, inset). For the first release, the time required to reach the peak redeveloped force averaged 15.5 ± 1.6 s (n = 6), and the normalized peak redeveloped force (0.049 ± 0.009) was significantly higher (P < 0.05, n = 6) than the normalized redeveloped force 5 min after the release (0.033 ± 0.005) (Fig. 3B). The results of these experiments indicate that the single dashpot element used in the Kelvin and Voigt models is insufficient to model the passive behavior of DSM. Unless the initial force redevelopment involved active cross-bridge cycling causing force production, two dashpots are required to explain this force response because force both increased and decreased during isometric passive conditions.

To determine whether the force redevelopment following quick-release was due to actively cycling, force-developing cross bridges, some tissues were exposed to staurosporine, an agent known to inhibit integrin-linked kinase, zipper-interacting protein kinase, and ROK, three calcium-independent myosin light chain kinases that could potentially activate myosin light chains in these tissues (6, 10, 31). Tissues not treated with straurosporine during their second cycle through the protocol acted as control tissues. Responses produced on quick-release during the second cycle were normalized to the responses produced on quick-release during the first cycle, and the data are presented as a fraction of the response produced during the first quick-release. The second quick-release in calcium-free conditions performed in the presence of 10 µM staurosporine revealed that the peak redeveloped force was not significantly different than the control response obtained in tissues not treated with staurosporine (Fig. 3C, P > 0.05, n = 3). The redeveloped force remaining 5 min after the release was also not significantly different with and without 10 µM staurosporine (Fig. 3D, P > 0.05, n = 3). These results indicate that calcium-independent cross-bridge cycling was not responsible for the force redevelopment following the quick-release in these experiments (see DISCUSSION). We, therefore, have used these empirical results to construct a model of passive DSM in which the first dashpot reflects a fast force redevelopment phase and the second dashpot reflects a slow force relaxation phase (see An APS model below).

Steady-state force reduced by strain softening.   Force-relaxation curves for strips of DSM stretched from Ls to 130% Ls and held isometrically for 1 h at 130% Ls are shown in Fig. 4A. Tissues that had previously been strain softened by stretching to 200% Ls for 5 min (see METHODS) (Fig. 4A, bottom solid line) displayed a significantly weaker time-dependent force profile than those that had not previously been strain softened (Fig. 4A, top dashed line). In addition, the isometric force-relaxation response recorded for 4 h from tissues that had not been strain softened revealed that force had not reached a true steady-state value by the end of the 4-h period (Fig. 4B). These figures demonstrate that isometric stress-relaxation in passive DSM included a very slow component that required many hours to approach steady state. Force values following 1 h of isometric stress relaxation at 130% Ls were compared for tissues with and without prior strain softening to 200% Ls (Fig. 4A, inset). The results show that strain softening significantly reduced the average normalized passive force from 0.260 ± 0.045 to 0.106 ± 0.018 (P < 0.05, n = 3). Without prior strain softening, the slow component accounted for a significant change in force, characterized by an average decrease of 20% between 5 and 60 min in the stress-relaxation phase (Fig. 4A). With prior strain softening, the slow relaxation component was negligible. Therefore, this experiment shows that passive force lost due to the slow stress-relaxation phase is a significant component of the passive force lost to strain softening and suggests that the slow dashpot should be placed in series with a strain-softenable element and in parallel with an elastic element that can maintain passive force as the tissue approaches steady state.


Figure 4
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  		Fig. 4. A: force relaxation curves from typical (n = 3) 1-h isometric holds at 130% Ls for muscle strips with (bottom solid line) and without (top dashed line) prior strain softening to 200% Ls. Forces are normalized to the peak force of the stretch without prior strain softening. Inset: average force values following 1 h of isometric stress-relaxation at 130% Ls for tissues with and without prior strain softening to 200% Ls. Strain softening significantly reduced the average normalized passive force from 0.260 to 0.106 (P < 0.05, n = 3). B: isometric stress relaxation for a 4-h hold without prior strain softening.

 
An APS model.   Based on the empirical findings presented in Figs. 3 and 4, and our laboratory's previous work (41), we propose a new biomechanical model for passive force in DSM that exhibits the following seven characteristics:
  1. ) Viscoelastic behavior characterized by hysteresis during cyclic loading (Fig. 4 in Ref. 41), isometric stress-relaxation following a stretch (Fig. 4A), and isometric force redevelopment following a quick-release (Fig. 3A).
  2. ) Equilibrium of the length-force relationship approached in a limited number of cyclic stretches (Fig. 2 in Ref. 41).
  3. ) Length-history dependence (i.e., strain softening behavior) (Fig. 4 in Ref. 41).
  4. ) Bimodal isometric force redevelopment/relaxation, distinguished by a fast force redevelopment phase and a slow force relaxation phase when tissues are maintained under isometric conditions following a quick-release (Fig. 3A).
  5. ) Steady-state force maintenance following stress-relaxation, with lower force maintained if the tissue has previously been strain softened (Fig. 4A).
  6. ) Slow isometric stress-relaxation following a quick-stretch, which is significantly reduced by strain softening (Fig. 4A).
  7. ) Activation-history dependence characterized by stiffness reformation following activation, with greater reformation on activation at shorter lengths (Fig. 7 in Ref. 41).


Figure 7
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  		Fig. 7. Simulations of the APS model. A: simulation of a quick-release protocol (stretch to 140%, hold for 30 s, release to 130%, and hold for 300 s) similar to Fig. 2A, which produces results qualitatively comparable to the experimental results in Fig. 3A. B: simulation of a stress-relaxation protocol with (solid line) and without (dashed line) prior strain softening (similar to Fig. 2B) with results qualitatively comparable to the experimental results in Fig. 4A.

 
Our proposed APS model consists of a Kelvin (Maxwell) model in parallel with a VPEC element (Fig. 5A). The variable component of passive stiffness modeled by the VPEC exists at steady state because strain softening reduces steady-state stress-relaxed force following a quick-stretch (Fig. 4A). Thus the VPEC element acts like an elastic element and bears some level of constant force at steady state, like the PEC in the Kelvin model (Fig. 1A). For this reason, the VPEC was placed in parallel with the PEC in the APS model (Fig. 5A).


Figure 5
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  		Fig. 5. Adjustable passive stiffness (APS) model. A: the proposed APS model, consisting of a Kelvin model (PEC, SEC, and D) with an additional variable PEC (VPEC) element. B: the APS model showing the components of the VPEC element (dashed box). C: a cross-linking elastic component (XEC) with the cross-link (XL) attached (closed switch) and detached (open switch). D: an equivalent APS model, with the SECslow and XEC stiffnesses combined into an equivalent stiffness (keq). See text for definitions of SECfast, SECslow, PECslow, Dfast, and Dslow.

 
Figure 5B shows a detailed diagram of the APS model. The D and SEC components from the Kelvin model are relabeled Dfast and SECfast because they model the viscoelastic properties that produce the fast force redevelopment phase following a quick-release (Fig. 3A). To model both strain softening and the slow viscoelastic properties, a novel VPEC element (Fig. 5B, dashed box) was introduced that contains a Voigt configuration (PECslow, SECslow, and Dslow) in series with a serial chain of novel strain-softenable cross-linking elastic components (XEC). This configuration accounts for the elimination by strain softening (Fig. 4A, solid line) of the stiffness responsible for the slow force relaxation phase following a quick-stretch (Fig. 4A, dashed line). The Voigt model configuration was included within the VPEC to model the slow viscoelastic properties, because this configuration contains both a PECslow that can bear steady-state force and a Dslow to produce the slow phases of isometric stress-relaxation (Fig. 4A) and redevelopment (Fig. 3A). What is unique about the XEC element in the VPEC is that each XEC acts as a very stiff, essentially rigid, element when the cross-link (XL) is attached (Fig. 5C) and acts as a SEC when the XL is broken (Fig. 5C) when the force borne by the element exceeds a force threshold. The XL bears essentially all the force in the XEC when it is attached, and the SEC bears all the force in the XEC when the XL is broken. As the number of broken XLs increases, the equivalent stiffness, keq (Fig. 5D), of the SECslow and XEC elements (Fig. 5B) is decreased to effectively model strain softening.

Model simulations.   Three MATLAB Simulink (MathWorks) simulations were performed to demonstrate the characteristics of the APS model. The equations used in the simulation are provided in the APPENDIX and were solved using the ode45 solver in Simulink 6.0. The parameters listed in Table 1 were selected for the APS model simulations to qualitatively demonstrate its behavior. All of the model elements are linear, except for the PEC, which was assumed to model collagen and elastin and produce the classic nonlinear passive length-tension curve for smooth muscle. For the following simulations, the contribution of the PEC was small (<3.1 mN). We assume that many XEC elements exist in series to explain the actual passive force behavior, but, for convenience in performing the modeling, 10 XEC elements were used, and each XEC element was modeled to have a stiffness of 9.8 mN/mm when its XL was detached and a significantly greater stiffness of 980 mN/mm (essentially rigid) when its XL was attached.


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  		Table 1. Adjustable passive stiffness model parameters

 
The input for the first simulation was a series of six stretch cycles (cycles 1–6) that increased the total length of the APS model, and the output was the total force carried by the model (Fig. 6). The simulation began at Ls (5 mm) with all XLs (XECs) attached. The first three length changes were 1-mm stretches at 0.25 mm/s (Fig. 6B). Each stretch was followed by a 60-s period of isometric stress-relaxation that demonstrated the model's viscoelasticity (characteristic 1), and then the model was quickly returned to Ls. Stretch 1 produced a peak force of ~24 mN, which was modeled to break one XL. Stretch 2 produced less force (~14 mN) and did not break any XLs because the reduced stiffness of XEC1 reduced the overall stiffness of the VPEC. The force produced by stretch 3 was nearly equal to stretch 2 because the number of broken XLs had reached steady state for that particular stretch length; however, there was a very small decrease in force due to the slow viscoelastic properties of Dslow. Thus, stretches 1–3 show that the model can approach an apparent equilibrium in a limited number of cycles (characteristic 2). Stretch 4 was a 2.5-mm stretch at 0.3125 mm/s, which produced a peak force of ~32 mN, and was modeled to break two additional XLs. Stretch 5 was identical to stretch 4, but produced a lower peak force because XEC2 could now lengthen. Stretch 6 was to the same length as stretch 3, but produced a lower peak force because fewer XLs were attached. The length-history dependence of the VPEC stiffness (characteristic 3) can be observed by comparing stretches 1, 2, and 6.


Figure 6
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  		Fig. 6. Simulation of the APS model consisting of six length change cycles (B) and the resulting forces (A) showing strain softening (2nd and 5th curves compared, respectively, to 1st and 4th curves; A), and loss of passive force following stretch to a longer length (6th compared with 2nd and 3rd curves; A). 1–6, Curves 1–6.

 
The second simulation was performed using a protocol similar to the experimental protocol in Fig. 2A. The model was stretched from Ls to 140% Ls, held for 30 s, then released to 130% Ls and held for 300 s. The initial stretch was modeled to break one XL. During the final isometric hold, the tissue model produced both force redevelopment and stress-relaxation (Fig. 7, inset) (characteristic 4), qualitatively similar to the experimental behavior demonstrated in Fig. 3A.

The third simulation (Fig. 7B) was performed using a protocol similar to the experimental protocol in Fig. 2B. The model was stretched from Ls to 160% Ls, both with and without prior strain softening and held isometrically for 60 min. For the case without prior strain softening, the 160% stretch was modeled to break one XL, while for the case with prior strain softening, all 10 XLs were modeled as already broken. The results were qualitatively similar to the experimental results in Fig. 4A and demonstrated that stress-relaxed force was lower if the tissue had previously been strain softened (characteristic 5) and that the slow component of isometric stress-relaxation was significantly reduced by strain softening (characteristic 6).

The proposed APS model also exhibited the activation-history dependence characterized by stiffness reformation following muscle activation, with greater reformation upon activation at shorter lengths (characteristic 7). When the model was activated at Ls, each XL was modeled to reattach at its original attachment location (Fig. 8, A and B), which restored the model to its original stiffness. However, if the model was activated at a length greater than Ls, each XL was reattached at a new location (Fig. 8C), because the model XLs have a fixed length. In this case, only a portion of the original stiffness was restored, because only a fraction of the elastic component, SEC1a (Fig. 8C), was bypassed by the XL. The remaining portion of the elastic component, SEC1b, remained free to stretch in series with SECslow, which resulted in a less stiff system compared with the system activated at Ls. In summary, the preceding simulations demonstrate that the APS model exhibits the seven biomechanical characteristics of passive DSM listed at the beginning of this section.


Figure 8
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  		Fig. 8. A VPEC element with a single XL. A: the XL is unattached at the original length. B: the XL is reattached when SEC1 was at the original length. C: the XL is reattached at a length greater than the original length, which divides SEC1 into a cross-linked portion (SEC1a) and a noncross-linked portion (SEC1b), and reduces the equivalent stiffness of the VPEC.

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
APPENDIX
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Passive smooth muscle may be characterized by viscoeleasticity and strain-softening. Early work on taenia coli, vascular and myometrial smooth muscle supports the view that attached cross bridges contribute to the passive viscoelasticity observed when tissues are incubated in an extracellular Ca2+-containing but not Ca2+-free solution (2, 39, 40). Our laboratory showed previously that, in addition to passive viscoelasticity, DSM exhibits strain-softening behavior when tissues are incubated in a Ca2+-free solution (41). Strain softening is a characteristic of natural rubber (1, 27) and living muscle (9, 13) and is proposed to reflect breakage of labile XLs between macromolecules. In DSM, the XLs broken by muscle strain participate in the passive force retained when tissues are in a Ca2+-free solution (41), and the present study represents a first approach to introduce strain-softenable elements into the classical Kelvin muscle model in which the PEC represents intra- and extracellular structures (3, 4). Our goal was to more accurately depict the passive mechanical characteristics of DSM in an analog model by dividing the PEC into two elements, the PEC and the VPEC. The PEC in our model represents extracelluar collagen and elastin, and the new VPEC element represents intracellular structures responsible for the adjustable component of passive stiffness. We identified seven mechanical characteristics of APS in DSM and developed the proposed APS model to qualitatively and semi-quantitatively account for all seven observed characteristics over a broad range of muscle lengths. The APS model consists of specific quantifiable elements, including the novel application of a VPEC element consisting of a series of "switchable" crosslinking structures termed the XEC. The XEC provides APS by softening as XLs are broken and strengthening as XLs are actively reformed. Additional empirical data are necessary to match these elements with specific physiological structures and produce a fully quantitative model for APS in DSM.

We reasoned that the variable stiffness observed in our previous (41) and current data is passive stiffness and not a result of actively cycling, force-generating cross bridges, based on several observations. First, the stretches that produced strain softening were performed in a Ca2+-free solution that eliminated spontaneous rhythm and prevented KCl-induced contraction, which both suggest that the force-generating crossbridges were not cycling (41). Previous studies have shown that incubation for at least 10 min in a Ca2+-free solution is sufficient to reduce cytosolic free Ca2+, abolish spontaneous rhythmic contraction (35, 37), decrease basal myosin light chain phosphorylation (35), and completely prevent KCl from causing contraction (17). Second, stiffness lost to strain softening was not restored, even when tissues regained spontaneous rhythmic contractile tone by incubation in a Ca2+-containing solution for up to 120 min (41). Third, Earley et al. (8) showed force redevelopment following the quick-release of artery strips that is likely due to active cross bridges that are regulated by caldesmon. However, in our experiments, DSM strips produced force redevelopment following a quick-release in a calcium-free solution (Fig. 3A), which should have eliminated any Ca2+-dependent cross-bridge cycling to produce active force. Even so, there still exists the possibility that zipper-interacting protein kinase, integrin-linked kinase, and ROK, three Ca2+-independent activators of myosin light chains that can be inhibited by 10 µM staurosporine (6, 10, 31, 35), could have caused active force redevelopment. Furthermore, Gorenne et al. (12) recently showed that a staurosporine-sensitive kinase that phosphorylates caldesmon may also permit force redevelopment. Therefore, we performed a set of quick-release experiments in Ca2+-free conditions in the presence of 10 µM staurosporine and found that the normalized peak and steady-state redeveloped force values were not significantly different than control values. This provided further evidence that the forces measured in the experiments were due to passive stiffness.

Others have investigated certain unique characteristics of stiffness at short muscle lengths (14, 15, 25). For example, the mechanical behavior of airway smooth muscle at short muscle lengths is dominated by extracellular compressional forces, which have been modeled by the radial constraint hypothesis (23–25). The strain softenable stiffness examined in the present DSM study reflects intracellular structures (41) and, therefore, may be distinct from that observed at short muscle lengths in airway smooth muscle (23–25).

The proposed APS model supports the hypothesis that XLs contribute to APS in rabbit DSM. Model simulations demonstrate that XL formation, breakage, and reformation can produce the strain-history and muscle activation history-dependent characteristics of APS exhibited by DSM. If XLs are responsible for stiffness formation, they must reside within the cell, because stiffness reformation is activation dependent (41). The model suggests that the cross-linked structures are in parallel with the contractile apparatus and in series with the structures responsible for the slow phase of isometric force relaxation. Further experimental studies are necessary to determine whether very slowly cycling cross bridges [i.e., latch bridges formed by smooth muscle (7) or nonmuscle myosins (20)], or other structures such as smitin (18), caldesmon (44), calponin (42), and other microfilament-binding proteins such as filamin (32), form the XLs that contribute to variable passive stiffness. DSM cell passive force and stiffness would be expected to contribute significantly to overall detrusor function in bladder disorders where DSM hypertrophy is a prominent feature (29, 45); however, our current model is limited to normal tissues and may need to be modified to account for dramatic changes in tissue structure such as that seen in spinal cord injury-induced hypertrophy (29, 30). Thus the clinical relevance of the present study is the potential utility of our modified analog model to formulate experiments that facilitate identification of molecular causes of bladder dysfunction.


   APPENDIX
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
The set of equations corresponding to the APS model (Fig. 5D) is presented below. In these equations, F and L represent force and length, respectively, and subscripts on these variables correspond to the labels on the model elements defined in Fig. 5D. Ftotal is the total force in the model, LPEC is the total length of both the tissue and the PEC element, and {Delta}L is the change in length of the tissue. Parameter values and initial lengths are provided in Table 1.

The total force in the tissue is calculated using the following equation:

Formula 1(1)
Force in the PEC element is calculated using the following pair of equations:

Formula 2(2)

Formula 3(3)
Force in the SECfast element is calculated using the following three equations:

Formula 4(4)

Formula 5(5)

Formula 6(6)
Force in the keq element is calculated using the following seven equations:

Formula 7(7)
where m is the number of attached XEC elements; n is the number of detached XEC elements; u1 is the XEC stiffness with XL attached; u2 is the XEC stiffness with XL detached.

Formula 8(8)

Formula 9(9)

Formula 10(10)

Formula 11(11)

Formula 12(12)

Formula 13(13)


   GRANTS
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
ACKNOWLEDGMENTS
 REFERENCES
 
This study was supported by a National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-59620 (to P. H. Ratz).


   ACKNOWLEDGMENTS
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 ACKNOWLEDGMENTS
REFERENCES
 
We gratefully acknowledge the expert technical assistance of Samer Hijaz and Amy Miner.


   FOOTNOTES
 

Address for reprint requests and other correspondence: J. E. Speich, Virginia Commonwealth Univ., Mechanical Engineering, 601 West Main St., P. O. Box 843015, Richmond, VA 23284–3015 (e-mail: jespeich{at}vcu.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


   REFERENCES
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 

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