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Eur Respir J 2003; 22:227-234
Copyright ©ERS Journals Ltd 2003

Mechanics and crossbridge kinetics of tracheal smooth muscle in two inbred rat strains

F-X. Blanc1,2, C. Coirault1,3, S. Salmeron1,2, D. Chemla1,3 and Y. Lecarpentier1,3

1 INSERM LOA ENSTA, Polytechnic School, Palaiseau, 2 Pneumology Unit, Dept of Internal Medicine, and 3 Dept of Cardiovascular and Respiratory Physiology, Bicêtre Hospital, University of Paris XI, Assistance Publique, Hôpitaux de Paris, Le Kremlin-Bicêtre, France

CORRESPONDENCE: F-X. Blanc, Unité de Pneumologie, Service de Médecine Interne, Centre Hospitalier Universitaire de Bicêtre, 78, rue du Général Leclerc, 94275 Le Kremlin-Bicêtre, France. Fax: 33 145212632. E-mail: xavier.blanc@bct.ap-hop-paris.fr

Keywords: airways, crossbridge, hyperresponsiveness, myosin, smooth muscle

Received: July 17, 2002
Accepted March 25, 2003

S. Salmeron was the recipient of a grant from CNRS and AP-HP (Praticien de Recherche Associé).


   Abstract
TOP
 Abstract
Materials and methods
 Results
 Discussion
 References
 
The aim of the study was to determine whether the nonspecific in vivo airway hyperresponsiveness of the inbred Fisher F-344 rat strain was associated with differences in the intrinsic contractile properties of tracheal smooth muscle (TSM) when compared with Lewis rats.

Isotonic and isometric contractile properties of isolated TSM from Fisher and Lewis rats (each n=10) were investigated, and myosin crossbridge (CB) number, force and kinetics in both strains were calculated using Huxley's equations adapted to nonsarcomeric muscles. Maximum unloaded shortening velocity and maximum extent of muscle shortening were higher in Fisher than in Lewis rats (~46% and ~42%, respectively), whereas peak isometric tension was similar.

The curvature of the hyperbolic force/velocity relationship did not differ between strains. Myosin CB number and unitary force were similar in both strains. The duration of CB detachment and attachment was shorter in Fisher than in Lewis rats (~–46% and –34%, respectively).

In Fisher rats, these results show that inherited, genetically determined factors of airway hyperresponsiveness are associated with changes in crossbridge kinetics, contributing to an increased tracheal smooth muscle shortening capacity and velocity.

Asthma is characterised by chronic inflammatory disorders of the airways associated with increased airway responsiveness to various stimuli 1. The mechanisms underlying nonspecific airway hyperresponsiveness (AHR) remain poorly understood. Although it is clear that airway smooth muscle (ASM) is the key effector of excessive airway narrowing in asthma, it remains unclear whether or not AHR is attributable to ASM abnormality 2, 3. The amount of ASM has been reported to be increased in human asthmatics 4–6 and in animal models of AHR 7. However, recent stereological studies using axial airway sections at high resolution have provided no evidence of increased ASM in large airways of human asthmatics 8. Marked differences in isotonic contractile properties of isolated ASM have been reported in human asthmatic 9, and in human- 10 and animal-sensitised 11–14 tissues. In these studies, isolated ASM was found to shorten more and faster in asthmatic and sensitised tissues than in controls without any differences in the ability to generate force. Taken together, these findings suggest functional changes resulting in increased ASM shortening capacity and velocity in hyperresponsive airways, without increases in the amount of ASM 8 or in ASM force production capacity. Potential mechanisms involved in such an increased ASM shortening capacity and velocity include reduced bronchial stiffness 9, decreased resistance to internal shortening 15, and increased myosin crossbridge (CB) cycling rate possibly related to an increased content of smooth-muscle myosin light-chain kinase (smMLCK) 16.

To study the direct contribution of ASM in the phenomenon of AHR, two complementary approaches are generally used in animals. First, various models dealing with passive or active sensitisation are used to investigate the acquired mechanisms involved in AHR. Second, models of spontaneous, nonspecific AHR can be used to study the inherited, genetically determined factors associated with AHR. To the best of the authors' knowledge, ASM isotonic mechanical properties have never been studied in animal models of spontaneous genetically determined AHR. Two rat strains have been studied extensively because of marked differences in their spontaneous airway responsiveness, the Fisher F-344 strain and the Lewis strain. When compared with the Lewis strain, the Fisher strain exhibited greater airway responsiveness to various inhaled contractile agonists, such as methacholine 17 and 5-hydroxytryptamine 18, 19. This relative AHR was manifested by a shift in the dose/response curve to the aerosolised agonist in both spontaneously breathing 17–19 and mechanically ventilated animals 20.

The aim of the present study was to determine whether the nonspecific in vivo AHR of the Fisher F-344 rat strain was associated with differences in the intrinsic contractile properties of tracheal smooth muscle (TSM) when compared with Lewis rats. Therefore, the mechanical properties of isolated TSM of inbred Fisher and Lewis rats were studied, focusing on isotonic parameters (extent and velocity of shortening), and the hypothesis of whether potential, inherited, genetically determined mechanical differences could be attributed to differences in myosin CB interactions was tested.


   Materials and methods
TOP
 Abstract
 Materials and methods
Results
 Discussion
 References
 
Tracheal smooth muscle preparation
Experiments were conducted with highly inbred male rats obtained from Charles River (Sulzfeld, Germany). Care of animals conformed to the recommendations of the Helsinki convention and the study was approved by the authors' institution (INSERM, France). Ten adult Fisher F-344 rats (means±sem weight 208±12 g) and 10 adult Lewis rats (weight 215±14 g; p=0.71) were studied after being housed at a conventional animal facility where all animals had free access to food and water. At the time of the experiment, the investigator was not aware of which strain was Fisher or Lewis. For each animal, experiments were performed on TSM obtained from a tracheal ring consisting of four segments. The ring was opened with a dorsal midline section through the cartilage in order to obtain a strip of the posterior membranous portion of the trachea. The epithelium was not removed. The strip was then vertically suspended in a 100 mL organ bath containing a Krebs-Henseleit solution, bubbled with 95% oxygen/5% carbon dioxide and maintained at 37°C, pH 7.4. The lower end of the strip was held by a stationary clip at the bottom of the bath, while the upper extremity was held in a spring clip linked to an electromagnetic lever system 21, 22. Supramaximal electrical field stimulation (EFS; 30 V·cm–1, 50 Hz AC, 3 ms pulse duration, 12 s train duration) was applied every 4 min by means of two platinum electrodes arranged in parallel on either side of the muscle preparation. Mechanical experiments were conducted after a 1-h equilibration period.

Mechanical parameters, force/velocity relationship
Mechanical parameters were recorded at optimal initial muscle length (L0) over the load continuum 21, 22. According to the conventional afterloaded technique, 8–10 EFS-elicited contractions were performed against increasing loads, from zero-load up to the fully isometric contraction. Maximum unloaded shortening velocity (Vmax; L0·s–1), using the zero-load clamp technique 23, maximum shortening velocity at the preload required to obtain L0 (Vc,max; L0·s–1) and maximum extent of shortening at preload ({Delta}L; %L0) were measured. Peak isometric force (F0; mN) was also measured and normalised per cross-sectional area to obtain peak isometric tension (P0; mN·mm–2). For each contraction, peak shortening velocity (V) was plotted against total isotonic load normalised per cross-sectional area (P). Data from the force/velocity curve were fitted according to Hill's 24 hyperbolic equation, -a and -b being the asymptotes of the hyperbola, and G being the curvature of the hyperbola. At the end of the study, cross-sectional area (mm2) was calculated from the ratio of muscle weight (mg) to muscle length at L0 (mm), assuming a muscle density of 1 25. Mean cross-sectional area did not differ between groups (1.17±0.12 mm2 in Fisher versus 0.94±0.08 mm2 in Lewis, p=0.12).

Calculation of myosin crossbridge number, force and kinetics
Huxley's 26 original equations constitute the most widely accepted mathematical model of muscle contraction. This model provides an informative system for estimating the number, unitary force and kinetics of myosin CB in living striated muscles. An adaptation of these equations to nonsarcomeric muscles, such as smooth muscles, was recently developed 27. This approach allows calculation of the unitary force per CB ({Pi}0; pN), the number of CB per mm2 at peak active tension ({Psi}x109), mean CB velocity during power stroke (v0; µm·s–1), total duration of the CB cycle (tc; ms), the rate of adenosine triphosphatase (ATPase) activity (kcat, with kcat=1/tc) and duration of CB attachment (1/f1; ms) and detachment (1/g2; ms), where f1 represents peak value of attachment rate constant and g2 represents peak value of detachment rate constant (another constant of detachment being termed g1; fig. 1Go).



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  		Fig. 1.— Schematic representation of the crossbridge cycle. The power stroke (i.e. the displacement of the myosin head and the force generation) is triggered by the release of the inorganic phosphate (Pi; step 4) and occurs after adenosine triphosphate (ATP) hydrolysis (step 2) and attachment of the myosin head to the actin filament (step 3). Myosin rapidly dissociates from actin (step 1) after adenosine diphosphate (ADP) release (step 5). g2: peak value of detachment rate constant; f1: peak value of attachment rate constant.

 
In fact, if h is termed the molecular step size (h=11 nm in ASM) 28, 29, it can be seen from Huxley's 26 manuscript, that {phi}=(f1+g1)h/2 and that it is assumed that {phi}=b and G=g2/(f1+g1). If w is termed the mechanical work of a single CB (w=3.8x10–20 J) 26, 30, e is termed the free energy required to split one adenosine triphosphate (ATP) molecule (e=5.1x10–20 J) 26, 30, and l is termed the length between two actin binding sites (l=36 nm) 31, then the following is obtained:


(001)


(002)


(003)


(004)


(005)


(006)


(007)
For each TSM, the number, force and kinetics of CB were calculated by substituting the measured values of the force/velocity relationship (namely Vmax, a, b, G) in the seven above-mentioned equations (for details see 26, 27, 32).

Statistical analysis
All data are expressed as mean±sem. Statistical comparisons between groups were carried out by using an unpaired t-test. For the force/velocity curves, a correlation coefficient (r) between measured values of force and velocity and the model estimate was used to test the accuracy of the hyperbolic fit. The correlation between two variables was calculated by linear regression using the least-squares method. A p<0.05 was considered statistically significant.


   Results
TOP
 Abstract
 Materials and methods
 Results
Discussion
 References
 
Classic mechanical parameters in tracheal smooth muscle of rats
During the EFS elicited after loaded contractions, peak isometric force was higher in Fisher than in Lewis rats (fig. 2Go). However, when normalised by cross-sectional area, peak isometric tension (P0) did not differ between Fisher and Lewis rats: P0 was 16.16±2.01 mN·mm–2 in Fisher and 15.10±1.92 mN·mm–2 in Lewis rats (p=0.71). Compared with Lewis rats, Fisher rats exhibited faster Vmax (~46%, p<0.001; fig. 2Go), faster Vc,max (~63%, p<0.05; fig. 3Go) and a greater {Delta}L (~42%, p<0.05; fig. 3Go).



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  		Fig. 2.— Mean±sem a) peak isometric force (F0) and b) maximum unloaded shortening velocity (Vmax) measured in Fisher (n=10) and Lewis (n=10) rats. **: p<0.01; ***: p<0.001.

 


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  		Fig. 3.— Mean±sem isotonic mechanical parameters measured at preload in Fisher (n=10) and Lewis (n=10) rats. a) Maximum shortening velocity of tracheal smooth muscle (Vc,max) and b) maximum extent of shortening ({Delta}L). L0: optimal initial muscle length. *: p<0.05.

 
Force/velocity relationship of tracheal smooth muscle in rats
In both strains, the force/velocity relationship was accurately fitted by a hyperbola: the r value between experimental data and Hill's model was 0.993±0.001 and 0.997±0.001 in Fisher and Lewis rats, respectively. As shown in table 1Go, parameter b (which has units of velocity) was higher in Fisher than in Lewis rats. Conversely, parameter a (which has units of tension) was similar in both strains, as was the G curvature of the hyperbolic force/velocity relationship (table 1Go). Figure 4Go shows representative force per mm2/velocity curves in Fisher and Lewis TSM.


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  		Table 1 Constants derived from the Hill equation in force/velocity curves elicited with afterloaded contractions in Fisher and Lewis rats

 


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  		Fig. 4.— Representative force per mm2 (P)/velocity (V) curves for Fisher (- - -) and Lewis (–––) rat tracheal smooth muscle. a) Absolute P and V values were used to determine the P/V relationships. b) Normalised P and V values were used to determine the G curvature of thehyperbola. L0: optimal initial muscle length; cVmax: calculated maximum shortening velocity; cP0: calculated maximum isometric tension.

 
Myosin crossbridge number, force and kinetics
{Pi}0 and the number of CB at peak active tension ({Psi}x109) are depicted in figure 5Go. No significant differences wereobserved between Fisher and Lewis rats. There wereno strain differences in mean CB velocity during power stroke (v0=0.249±0.040 µm·s–1 in Fisher and v0=0.246±0.046 µm·s–1 in Lewis rats, p=0.95). There was no correlation between the Vmax and v0 (r2=0.056, p=0.32). There was a weak correlation between P0 of the muscle strip and {Pi}0 (r2=0.223, p<0.05; fig. 6aGo). Figure 6bGo shows that there was a strong linear relationship between P0 of the muscle strip and {Psi}.



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  		Fig. 5.— Mean±sem a) calculated total number of myosin crossbridges at peak active tension ({Psi}x109) and b) unitary force per crossbridge at peak active tension ({Pi}0) in Fisher (n=10) and Lewis (n=10) rats. ns: nonsignificant.

 


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  		Fig. 6.— Relationships a) between peak isometric tension of muscle strips (P0) and the unitary force per single crossbridge ({Pi}0), and b) between P0 and the number of crossbridges at peak active tension ({Psi}x109) in Fisher ({circ}) and Lewis ({blacksquare}) rats. {Psi}=0.107 P0+0.247. r2=0.951, p<0.0001.

 
Myosin CB kinetics are depicted in figure 7Go; the duration of both 1/g2 and 1/f1 was significantly shorter in Fisher than in Lewis rats (–46% and –34%, respectively). The tc did not differ between strains: tc was 4.354±0.815 s in Fisher and 4.819±0.590 s in Lewis rats (p=0.65). kcat did not differ between strains: kcat was 0.301±0.046·s–1 in Fisher and 0.242±0.036·s–1 in Lewis (p=0.33).



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  		Fig. 7.— Mean±sem calculated duration of a) crossbridge attachment (1/f1) and b) crossbridge detachment (1/g2) in Fisher (n=10) and Lewis (n=10) rats. *: p<0.05; ***: p<0.001.

 

   Discussion
TOP
 Abstract
 Materials and methods
 Results
 Discussion
References
 
The major findings of the present study are that: 1) the higher genetically determined spontaneous nonspecific AHR of the Fisher rat strain is associated with higher shortening capacity and velocity of isolated TSM when compared with Lewis; 2) myosin CB kinetics differ between strains; 1/g2 and, to a lesser extent, 1/f1 is shorter in Fisher than in Lewis rats, and 3) P0 and the unitary force per CB were similar in both strains, although F0 was higher in Fisher than in Lewis rats.

Validation of the animal models
Both Fisher and Lewis are anatomically well-characterised strains of inbred rats 33. When compared with Lewis, Fisher rats are known to exhibit higher airway responsiveness to various inhaled contractile agonists, such as methacholine 17 or 5-hydroxytryptamine 18, 19. In vivo, this relative AHR is manifested by a shift in the dose/response curve to the aerosolised agonist in both spontaneously breathing 17–19 and mechanically ventilated animals 20. In the present study, in vivo airway responsiveness was not measured in either rat strain. It cannot be excluded therefore that minor differences in airway responsiveness could occur between the rats used in this study and data from the literature. In vitro, the ASM of these two rat strains exhibits marked differences; while isolated TSM develops similar isometric force at baseline in both strains 17, the TSM of Fisher rats has been shown to be more responsive to cumulative concentrations of carbachol 34 or serotonin 35 than the Lewis strain. Lung explants have recently been used to show that this innate relative hyperresponsiveness of the Fisher strain was not restricted to the trachea but extended throughout the airway tree 19, 36. These studies also provided evidence that Fisher rat airways could exhibit a greater capacity for airway narrowing than Lewis. Taken together, these studies support the hypothesis that the relative nonspecific AHR found in the Fisher strain could be related to differences in the intrinsic contractile properties of the ASM 19.

Mechanical contractile properties of tracheal smooth muscle
To the best of the authors' knowledge, no data are available concerning the isotonic contractile properties of isolated TSM of the Fisher and the Lewis strains. The first aim of the present study therefore was to characterise the contractile properties of isolated TSM in both strains, with particular attention paid to isotonic parameters. Vmax was found to be higher in Fisher than in Lewis rats (fig. 2Go), as were {Delta}L and Vc,max (fig. 3Go). These results corroborate previous findings from sensitised dogs 11, 12, guinea-pigs 13, mice 14 and human bronchi 10, which all demonstrated that isolated ASM of sensitised tissue is able to shorten more and faster than ASM of controls. The results could also explain, at least in part, the previous findings reported in lung explants from Fisher and Lewis rats 19, 36.

In ASM, two methods are usually used to elicit the force/velocity relationship; conventional afterloaded isotonic contractions or quick releases applied 2 s after the onset of isometric contractions. Since force/velocity characteristics were first described in canine TSM by Stephens et al. 37, the best way of eliciting the force/velocity relationship has been a subject of discussion. In a preliminary study, the Vmax elicited with the afterloaded technique did not differ from the one measured with the quick-release technique 38 in both rat strains. It was also found that experimental data were accurately fitted by a hyperbolic curve, with no obvious deviation of velocities at high load, contrary to results previously reported in dogs 39. The force/velocity relationship has never been studied in the ASM of rats. Interspecies differences may explain, at least in part, the differences observed between the present data and previous reports in canine TSM. As a hyperbola accurately fitted the current experimental data, the modified equation described by Wang et al. 39 was not used. Using the classic Hill 24 equation, parameter b of the equation was higher in Fisher than in Lewis rats, whereas parameter a did not differ between strains (table 1Go). This result was in agreement with the higher shortening capacity and velocity in Fisher compared with Lewis rats. The G curvature was also found to be similar in both strains (table 1Go). Similar results have been reported in sensitised canine TSM when compared with controls 11. In striated muscles, the curvature of the force/velocity relationship has been found to be linked to the myothermal economy of force generation: the more curved Hill's hyperbola (i.e. the higher G), the greater the economy of force generation 30, 40. Assuming that the fundamental properties of myosin CB do not strikingly differ between smooth and striated muscles 26, the present results suggest similar energetic cost of force generation in Fisher and Lewis TSM.

In this study, there were no strain differences in the P0 developed by isolated electrically stimulated TSM, although F0 was higher in Fisher than in Lewis rats (fig. 2Go). In mechanical studies, it is usual to report the results of isometric tension rather than isometric force, as differences in cross-sectional area may interfere with results of force. This was clearly the case in this study. These results are in agreement with those reported previously in the same strains 17 and in animal models of nongenetically determined AHR 10–14.

Myosin crossbridge number, force and kinetics
It has been suggested that the ultimate defect in AHR may be at the muscle cell level itself 14. If so, it should concern structures, mechanisms or pathways involved in or resulting in higher capacity and velocity of muscle shortening. Several approaches have been used to investigate how myosin molecular motors work. Studies using in vitro motility assays 41 and high-resolution single-molecule experiments, such as the optical tweezers technique 42, have provided insights into interactions of elementary contractile proteins. Mathematical models of muscle mechanics provide an informative system for estimating the number, unitary force and kinetics of myosin CB in living muscle 26, 43, 44. Huxley's 26 formalism has been used in many mechanical, as well as biochemical muscle experiments, and can contribute to better understanding of CB interactions 32. The use of Huxley's equations has been applied to nonsarcomeric muscles, such as smooth muscles, allowing a calculation of CB number, unitary force and kinetics from experimental mechanical data 27. This has been validated in ASM from several species, including the rat 27. The reasons for applying Huxley's model in the present study were the following: 1) Huxley's model not only accounts for many macroscopic properties of muscle (force, velocity and heat production), but also relates these properties to its structural and biochemical properties; 2) according to Huxley's model, force and shortening in muscle are generated by cyclic interactions of the myosin heads with specific sites on the thin filaments, with ATP hydrolysis providing the energy; 3) by using mechanical parameters (velocity, length, force) at the level of the entiremuscle and at various load levels, Huxley's theory infers the kinetics and number of CB. Thus, the use of Huxley's formalism seemed to be an attractive way to test thehypothesis that inherited, genetically determined mechanical differences could be attributed to differences in CB interactions.

Beside the power stroke, two important steps occur during the CB cycle, i.e. the attachment and detachment of the myosin head from actin (fig. 1Go). Attachment of myosin to actin occurs after ATP hydrolysis 45. Myosin rapidly dissociates from actin after ADP release. Thereafter, one molecule of ATP binds to the nucleotide site of the myosin head (ATP-binding pocket). In this study, the attachment step was found to be shorter in Fisher than in Lewis rats (fig. 7Go), and the detachment step was markedly shorter in Fisher than in Lewis rats (fig. 7Go). A shorter detachment step could account, at least in part, for increased velocity of shortening at the muscle level. In fact, according to Huxley's CB model 26, Vmax is proportional to the maximum value of g2; the higher Vmax, the higher g2, thus the lower 1/g2, i.e. the shorter the detachment step. Therefore, the detachment step is a crucial step in the determination of maximum shortening velocity in ASM. 1/g2, rather than tc, would thus strongly influence the shortening capacity of the whole muscle. Further studies are needed to investigate which structural, biochemical or mechanical factors influence g2 and, to a lesser extent, f1 in the Fisher strain. Potential candidates are as follows: 1) differences in the three-dimensional configuration of the ATP-binding pocket of the myosin head, or the presence of a seven amino acid insert near the ATP-binding pocket 42, 46, or differences in the configuration of the actin-binding site; 2) higher expression of the smooth muscle-B isoform of the myosin heavy chain in Fisher trachea (the presence of this isoform has recently been reported in TSM of adult Fisher rats) 47; 3) differences in the 20 kD regulatory light chains of myosin; 4) differences in the structure, content or activity of myosin light chain kinase, especially the smMLCK isoform 16; 5) structural or functional differences in the calcium/calmodulin complex; and 6) differences in the activity of the Rho-associated kinase that can directly phosphorylate the 20 kD regulatory light chains of myosin independently of calcium 48. However, it should be mentioned that increased intracellular calcium mobilisation has recently been described in Fisher rats when compared with Lewis rats 19, suggesting that calcium-dependent pathways are at least partly involved in the mechanisms underlying the nonspecific AHR of the Fisher strain.

Using Huxley's equations adapted for smooth muscles 27, no interstrain differences were found between the number, unitary force and mean velocity of myosin CB at peak active tension. Moreover, no relationship was found between Vmax and v0. Therefore, greater velocity of muscle shortening observed in Fisher rats could not be attributed to higher velocity of CB during power stroke. Potential differences in tc were also sought but no such differences were found between Fisher and Lewis rats, indicating that higher velocity of shortening is not necessarily associated with a shorter time cycle.

In conclusion, the isotonic and isometric contractile properties of isolated tracheal smooth muscle of Fisher and Lewis inbred rat strains were analysed. The higher genetically determined spontaneous nonspecific airway hyperresponsiveness of the Fisher rat strain was associated with higher shortening capacity and velocity of tracheal smooth muscle when compared with Lewis rats, with no difference in the ability to produce isometric force. These mechanical differences were not attributed to differences in the number, unitary force or mean velocity of myosin crossbridge during power stroke. In contrast, differences in myosin crossbridge kinetics, i.e. shorter duration of crossbridge detachment and attachment in Fisher than in Lewis, may help explain the higher shortening capacity and velocity in Fisherrats. These results highlight the crucial role of the crossbridge itself in the phenomenon of innate greater airway responsiveness of the Fisher rat strain.

Appendix: Calculations

Using Huxley's 26 manuscript, the authors first expressed the rate of total energy release (


(024)
) and the isotonic tension (PHux) as a function of velocity (V). Thus,


(025)
is given as:


(008)
The minimum rate of total energy release (


(026)
) occurs under isometric conditions (V=0 in equation 8Go), is equal to the product of axb 24, 26 and is given by:


(009)
The crossbridge (CB) number per·mm2 at {Psi} is:


(010)
The isotonic tension PHux is given by equation 11Go 26:


(011)
PHux,max is reached for V=0 and is assumed to be equal to total isometric tension (P0), which was experimentally determined. The mean unitary force per CB in isometric conditions is


(012)

Calculation of peak value of detachment rate constant (g2)

As fe=(f1+g1)h/2, fe=b, G=g2/(f1+g1) 26 and Vmax=Gb, it is deduced that


(013)

Calculation of detachment rate constant (g1)

From equations 9Go (with


(027)
) and 11 in isometric conditions (with V=0 and PHux,max=Ga), it was deduced that


(014)
Thus,


(028)

Calculation of peak value of attachment rate constant (f1)



(029)
was linearised as a function of PHux.

Let


(030)

By replacing A and B in equations 8 and 11GoGo, equation 8Go becomes:


(015)

equation 11Go becomes


(016)
where d=1/G.

From equation 13Go, it was deduced that:


(017)

From equation 14Go, it was deduced that


(018)
From equations 13 and 14GoGo, it was deduced that


(019)

From equation 15Go and near isometric conditions, where V can be neglected (V{approx}0),


(031)
can be linearised as a function of PHux as follows:


(020)

The slope of this relationship between


(032)
and PHux has been shown to be equal to b 26, 27, then,


(033)
so that


(021)

From equations 1 and 17 it was deduced that


(022)

As G=g2/(f1+g1) 26 and f1=Gg1, by solving the quadratic equation f12+g1f1–g1g2=0, f1 as a function of g1 and g2 was obtained:


(023)


   References
TOP
 Abstract
 Materials and methods
 Results
 Discussion
 References
 

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