Viscoelastic and plastic properties of visceral smooth muscles in vertebrates

VISCOELASTIC AND PLASTIC PROPERTIES OF VISCERAL SMOOTH MUSCLES IN VERTEBRATES* KURTGREVEX Klinikum der J. W. Goethe-Universittit.

Zcntrum der Physiologie. Frankfurt. W. Germany

Abstract-Viscoelastic and plastic properties of the smooth muscles of the viscera in vertebrates which manifest themselves as creep after constant loading or stress relaxation after constant stretch, are biologically highly important. They allow the walls of hollow organs. e.g. the stomach. the urinary bladder, etc. to adapt to increasing volume without increasing internal pressure. The phenomena can be quantitatively treated by Voigt-,Maxwell models composed of elastic springs and plungers which move in dashpots filled with viscous fluids. provided that the resistance of friction in the dashpots is taken to be a function of the degree of creep or relaxation. Under these formal conditions. the mechanics of the above mentioned eRects can be interpreted on the basis of functional changes in well known biological structures of the smooth muscle cells, i.e. in the contractile elements,

IXTRODUCTION

versible changes in shape instead. According to our experiences, no “yield value” in force is necessary to

After-effects of mechanical forces usually manifest themselves in materials which can be deformed by stretch in a twofold manner: (1) creep after constant loading, or (2) stress relaxation as release of tension after being stretched to a constant length. These phenomena, which often are of interest for industrial products, have been thoroughly investigated with

produce plastic phenomena. In living tissues they even occur under a minimal load or by minimal stretch. MODELS

regard to high polymers (Turner and Gurnee, 1957; Oberst. 1964). The deformations observed in non-living materials have mostly been regarded as inconvenient secondary effects. However, in living animals they are of great importance for the mechanics of the walls of hollow organs, i.e. the blood vessels, the intestinal tract, the urinary bladder and others (Alexander, 1957, 1973; Apter, 1964, 1968; Remington, 1957). They allow these organs to adapt to a certain degree to increasing volume without an increase in internal tension. Thus, the compulsion to empty is delayed by slowing the rise in tension of the wall in the rectum or in the urinary bladder. In other words, the organ is better able to store its contents. Hence, it is of greatest interest to physiologists to find out which histological structures in those walls are responsible for these phenomena and which physical laws govern them. However, before entering into physiolo$cal problems. some points regarding terminology should be mentioned. Creep and stress relaxation are mostly preceded by pure elastic phenomena occurring instantly under the influence of external forces and which are instantly and completely reversible as soon as these forces are removed. The subsequent real mechanical after-effects can be divided into firstly “viscoelastic”, and secondly “plastic”. The former occur relatively slowly but are completely reversible.

The latter occur also relatively slowly but cause irre* Rrceirrd

3

March

1911. 39

Models consisting of springs and dashpots are usually taken for a conceptual treatment of our phenomena. The springs symbolize elastic forces; frictional forces can be symbolized by plungers moving in dashpots filled with a viscous fluid. If the dashpots are connected in parallel to springs, the models show a viscoelastic behaviour. Dashpots without parallel springs can simulate plastic properties at least during the elongation by stress in so far as the deformations are not reversed by simple unloading. The so called Voigt-Maxwell elements, combinations of these uhits in higher order sets (Fig. la and b), are to a certain degree able to imitate creep as well as stress relaxation of biological materials. In biological specimens, viscoelastic and plastic properties are often interconnected. Therefore, the so called Winton model (Fig. lc) has a definite value (Winton, 1930). It may be considered as a combination of a Maxwell element without a parallel spring and a Voigt element without a spring in series. With regard to the histology of the walls of the hollow organs, which are divided into very small histological units, a combination of numerous models must be envisaged In models like that of Winton. reversibility which is a necessary quality of biological objects is absent. Real plastic elements, i.e. those without parallel springs, do not show this quality by definition. Lf biological objects are deformed in a plastic manner in oivo, the deformations must afterwards be corrected by contractile activity. To symbolize such a capability, motors or pumps in series have been added to the plastic elements (Riickemann and MiillerOntjes, 1975).

KCRT GREVEN

T El

E2

CL

K

+

b

a

Fig. 1. Models with viscoelastic and plastic behaviour. a. Voigt model with an additional spring in series (E,). b. Maxwell model with an additional spring in parallel (E2). c. Winton model. K = force, E = stiffness. p = frictional resistance.

HISTOLOGICALPROPERTIES In the first instance one may be probably inclined to explain creep and stress relaxation by a sliding of cells among each other, especially by a sliding of the smooth muscle cells, which may be considered as the mechanically essential elements of the walls. This however was already rejected by Griitzner (1904) for histological reasons. Alternatively, intracellular elements may be made responsible for the phenomena under discussion. Among these intracellular elements the plasmatic ones, i.e. those without structure, may be neglected. This can be easily proven by investigations under various osmotic pressures (Greven et al., 1976a). where smooth muscle cells behave like reliable osmometers over a wide range of swelling and shrinking (Brading and Setekleiv, 1968). Creep and stress relaxation. however. do not change significantly after changes in the osmotic pressure. Thus, the mechanical after-effects must take place in the solid structures of the cells. In the first instance the contractile proteins of the smooth muscle cells may be considered (Bozler, 1952, 1964). Since the sliding mechanism for contractile activity (Huxley and Niedergerke, 1954; Hanson and Huxley, 195.5; Huxley, 1969). i.e. the sliding movements of actin filaments with respect to myosin filaments. can also be postulated for smooth muscles (Somlyo, 1972; Somlyo et al., 1973). the hypothesis that a similar sliding takes place in these structures during creep and stress relaxation has gained a solid base. To support this hypothesis it is irrelevant whether relaxation after contraction and stress relasation have the same time course (Bozler, 1930, 1931. 1976; Lowy and Millman, 1963) or not (Greven, 1951: Schatzmann, 1964). As already mentioned before though. it is important to know that plastic deformations can only be reversed by contractile activity, and that, according to recent observations (Siegman er al.. 1976) cross link attachment between actin and myosin as well as resistance to stretch and viscoelasticity are dependent on the calcium concentration of the medium. Some years ago. a different mechanism could be

conceived where structures deformed by plastic alternations regain their former shape by the contractile activity of elements lying in parallel to the deformed ones (Greven, 1950a.b). The sliding hypothesis of contractile activity however, describing events very similar to plastic deformation, has been recognized to a great extent and modern electron microscopy offers the contractile elements themselves as predisposed structures for creep and stress relaxation due to their situation and their shape. Moreover, quantitative considerations which shall be discussed below are quite compatible with this conception.

QC’~iTIT.~TlvE CONSlDERATIOiVS For the quantitative treatment of the mechanical after-effects under discussion, the spring and dashpot models mentioned above will be a good starting point. To begin with, the factors of friction in the dashpots (pi) are assumed to be constant. Furthermore the stiffness factors of the springs in series (E,,) and those in parallel (Eiz) are supposed to obey Hooke’s law and also remain constant. As is usual in biology, the term “stiffness” applies to the differential quotient dF/dl (F = force, 1 = length). A detailed description of the behaviour of VoigtMaxwell models combined in series or in parallel has already been given in a previous paper (Greven et al., 1973). While neglecting the initial, purely elastic phenomena after loading or stretch the subsequent viscoelastic and plastic deformations have been considered. Then for the Voigt-Maxwell models in parallel-but only for creep and not for stress relaxationthe following function is valid:

N = size of creep K = constant force

t = time, where Pi = pi;

qi = Et2

ai

Properties of ~iscrral smooth muscles in vertebrates

for Voigt models. and

for Maxwell models. The indices 1 or 2 apply as usual to elements in series and in parallel respectively. For Voigt-Maxwell models in series-in the case of stress rehxation but not in the case of creep-a similar equation can be applied:

R = degree of stress relaxation (positive) D = initial stretch (constant), where l-i=

E,,

Pi 7; E;,

+

EiI

si=-z-

for Voigt models, and r = “i. ’ Ef,‘

si = -

1

Eil

for Maxwell models. Models with one or several EiZ = 0, e.g. the Winton model. have irreversible plastic qualities. According to these statements. either creep if the models are in parallel, or stress relaxation if in series. must obey an exponential function in time. In other words: if the models apply, either dN/dt must be a linear function of X. or dR/dt must be a linear function of R. Neither. however, have ever been shown to hold. Further proceedings can be carried out in the way Buchthal and Kaiser (1951) undertook for skeletal muscle or as Oberst (1964) showed in general for high polymers. As a common basis, the exponential functions as won by the integration of equations (1) and (2) are used. Then the retardation times ~~ = pi/qi or ri = ri’si characterizing the time course of creep or stress relaxation with single Voigt or Maxwell models are fitted to a distribution function that may possibly be obtained from experimental results. However, having considered distribution functions rendered by the above mentioned authors and having applied these laws to smooth muscles, it was scarcely conceivable, at least with regard to modern conceptions in smooth muscle histology, how the known intracellular structures should change their shape obeying the alleged functions. Efforts to represent the time course of N or R graphically as a combination of two exponential functions also failed. Since the models in Fig. 1 can scarcely be discarded if a quantitative treatment of the question under discussion is desired. one is obliged to take the values of ~(tor of Ei, or Ei2. formerly regarded as constant, now as functions of X or R. Concerning the stiffness factors Eil and Ei, this is known for many biological objects: the stiffness factors increase with increasing length. However, this must not apply to muscles dur-

ing creep or stress relaxation, as can be shown below. Furthermore. the following points may be considered: if the values of EL, and Ei2 are taken as functions of ,V or R in equations (1) and (2) with a constant pi, both equations will not remain identical in form. With the inverse proceedin,,0 however. i.e. if the cost% cient of friction pi is taken as a function of 5’ or R with constant Ei, and Eil, both equations will show an identical form (Greven. 1975). These considerations are valid, too, when we abstain from a combination of models in Fig. I (a) and (b). and restrict ourselves to a single model. i.e. when we suppose the various E,, and the various Eil to be identical to one another respectively. The experimental results are in favour of the supposition that the Ei should remain constant. the b(, being a function of :V or R. Then. tn, in equations (I) and (2) can be replaced in a conventional manner by the Taylor series and it can be shown that by interrupting these series after the linear link. a satisfying agreement of the formula and of the experimentai results can already be obtained. The resulting function can be given formally by (Greven et nl.. 1973: Greven and Hohorst. 19753: d(.V or R) PC

dt

a’ - b’(N or R)

a + b(N or R) ’

CI,b. a’, b’ = const.

(3)

The differential quotient on the left side is a hyperbolic function of S or R. Attention must be drawn to the facts that according to our results, equation (3) is valid for creep tr& stress relaxation within the same biological object (Greven and Hohorst, 1975; Hohorst et al., 1976). Thus. if the deduction from the physics of the spring and dashpot models, mentioned above, is correct. the various lci concerned as well as the various Ei, or the various Ei3 must be identical to one another respectively. Our objects reacted like combinations of identicni Voigt or Maxwell models in series or in parallel. and the Winton model of Fig. 1 (c) would not apply. This shall be discussed below. It is obvious that a simple formula like equation (3) cannot represent the complicated changes of internal structures during the mechanical after-effects under discussion. The equation can only give an approximation to the facts. rendering some charactetistic features plausible. However. it may be useful in connection with the modern aspects of smooth muscle physiology. ESPERISIENTAL

RESULTS

Figures 2 and 3 present the results from smooth muscles with cell strands lying parallel (Greven and Hohorst, 1975; Hohorst et a[., 1977). Figure 2 shows those from the often used taenia coli of the guinea pig, consisting of strands of the longitudinal layer of the animal’s cecum. Eighteen experiments each were

53

Properties of cisceral smooth muscles In bcrtsbrates values of a in this region are not well defined. On the other hand, at the right extremes of the hyperbolas. near the limits of the mechanical after-effects. different structures, e.g. connective tissues, certainly interfere to bring the movements to an end. Hence creep and stress relaxation assume different courses in time here. The zero ordinate cannot be crossed as in our “ideal” hyperbolae. something which would be. of course, physiologically absurd

function of .V or R. an inverse function S orR =

alf(n) - u b’ + bd(V or r)dr = wb’

a’ - ad(N or R)dr

(7)

can be set. Thus a formally similar hyperbolic equation which is nothing but a transcription of equations (1). (2). (3) and (6) d(E or R) (ba’ + ab’) f‘(n)

a’ - h’(4 or R) = -----_.

dr

b’j(n) i- h

’

is)

is obtained. i.e. the factor of friction is a function of the number of transversal bindings. In this way the hyperbolic relations of .V or R to As may be readily seen, the results fit rather well their derivatives in time may perhaps be explained. to equation (3). Since in the case of creep, the parIt may be remarked also that the last term on the ameters Eli of equation (I) are replaced by a linear function of N, the following aspects may perhaps be right side which stands for C(p, or ri) in equations discussed (Greven. 1976b): Because the volume of the (1) and (2) increases with j(n) in a hyperbolic form. beginning with f(n) = 0; C(p, or ri) = 0, and ending preparation remains approximately constant, .V is inversely proportional to the cross section, and it seems with f(n) = z : Z(p, or ri) = (bn’ + ab’)ib’. Thus. our differential equation (3) lends itself quite not unreasonable to think that friction would increase with decreasing cross section, i.e. p = a + b/Q, (where wtell to biological interpretation. The integration of equation (3) gives no further information that would Q = cross section), as the internal - structures approach each other. However. this is not the case clarify biological problems. The question if it is corwith stress relaxation because the cross section is not rect to take the stiffness factors Ei, and Ei2 in equaaltered under these circumstances. To cover both tions (1) and (2) as constants. remains to be settled. cases another approach may be preferred. If Ipi or As already mentioned above, our conception contra5, are written as dicts common experience in so far as the stiffness of biologicai material usually increases with increasing k, stretch. But in the case of the mechanical after-eft’ects rpiorxri= (3 k2 + k,(d,V/dt or dR/dt) ’ considered here, special conditions may prevail: According to Huxley and Simmons (1971) the elastic k,. k2. k3 = const. The former equations (1) and (2) forces during the sliding of actin refative to myosin are also transferred into hyperboIae formally the same are situated in the cross links between both chains as equation (3), i.e. of the molecules concerned. During the movements caused by creep or stress relaxation the bridges are d(N or R) n’kz - b’k?(N or R) (6) stretched, broken off. refastened. and stretched for dt = (k, - a’k,) + b’k,(N or R)’ once again. Hence, many bridges in different states The derivatives dN!dt as well as dR/dt are proporof stretch are connected in series or in parallel. The tional to the sum of the velocities of the plungers resulting elastic forces can remain comtmt with a conin the dashpots of the Voigt and Maxwell models. scant number of bridges, their state of stretch being Such motions may be interpreted in the case of the statistically distributed. However, the forces will inmuscle as movements of the internal structures rela- crease with an increasing number of bridges. Accordtive to one another by means of which transverse ing to our previous comments. the number of bridges bindings may be broken. In this view, which is very increases u-ith .V or R increasing and a consequent decreasing d.V;‘dr or dRdt during the time course of well compatible with the conceptions already mentioned above that mechanical after-effects function by the after-effects under discussion. Hence, the elastic a sliding mechanism of the contractile elements forces defined by the terms NXq, or R,E.si in equation against each other, resistance to friction is nothing (1) and (2) will increase, Zqi or Esi remaining constant. but a force to break bonds which are permanently With this explanation, Hooke’s law seems to have formed during the sliding of internal structures. The lost its validity with respect to the mechanical effects formation of such transverse bonds takes a certain concerned. For. by equation (7) the values of .V and time. and the time of contact T when such bindings R depend on f(n). i.e. the number of cross links can take place will be the longer the slower the relaaccording to a hyperbolic relation. the function f(n) tive velocity of the structures one to the other. Hence, being only characterized by df(n)/dn > 0. But accordif n is the number of bindings that can be formed ing to our results. things can formally be treated as with a certain probability during T, one may define if there were constant stiflhess factors, Hooke’s law a function f(n) = t where dfda > 0. Consequently still being valid. This was already mentioned by the 1 (dS;dr) or l,l(dR/dt) in equation (5) may be replaced comments to equations (1) and (2). byf(n). Moreover. d.V,‘dt or dR:dt being a hyperbolic Besides. there are some recent investigations (Giith DISCUSSlOX

KURT GRELEN

54

and Kuhn. 1976; Herzig. 1976; Mulvany and Halpet-n, 1976) confirming the validity of Hooke’s law for myosin bridges. REFEREA-CES

Alexander, R. S. (1957) Elasticity of muscular organs. In: Tissue E[asticit): (Edited by Remington, J. W.). pp. 111-122. Waverly Press, Baltimore, MD. Alexander, R. S. (1973) Viscoplasticity of smooth muscle of urinary bladder. Am. J. Physiol. 224. 618-622. Apter, J. T. (1964) Mathematical development of a physical model of some visco-elastic properties of the aorta. Bull. Math. Biophys.

26. 367-388.

Apter, J. T. and Marquez, E. (1968) A relation between hysteresis and other visco-elastic properties of some biomaterials. Biorheology 5. 255-301. Bozler, E. (1930) Untersuchungen zur Physiologie der Tonusmuskeln. Z. cergl. Physiol. 12. 579-602. Bozler, E. (1931) Die mechanischen Eigenschaften des ruhenden Muskels, ihre experimentelle Beeinflussung und physiologische Bedeutung. Z. rergl. Physiol. 14, 429-449. Bozler, E. (1952) Plasticity of contractile elements as studied in extracted muscle fibres. Am. J. Physiol. 171, 359-364. Bozler, E. (1964) Smooth and cardiac muscle in states of strong internal crosslinking and high permeability. Am. J. PhysioL 207, 701-704. Bozler. E. (1976) iMechanical properties of contractile elements of smooth muscle. In: Physiology of Smooth Muscle (Edited by Biilbring, E. and Shuba, M. F.). pp. 217-222. Raven Press. New York. Brading. A. F. and Setekleiv, J. (1968) The effect of hypoand hypertonic solutions on volume and ion distribution of smooth muscle of guinea pig taenia coli. J. Physiol. (Land.) 195: 107-l IS.

Bronstein, I. N. and Semendjajew, K. A. (1973) ‘Ihschenhrrch der Mathematik, p, 523. Deutsch. Frankfurt. Buchthal. F. and Kaiser, E. (1951) The Rheology of the Cross Striated Muscle Fibre. IMunksgaard, Kopenhagen. Greven, K. (1950a) Die Wirkungen des vegetativen Nervensystems und seiner Mimetica auf den kontraktilen und plastischen Tonus der Magenringmuskulatur. Z. Biol. 103. 301-320. Greven, K. (1950b) Uber die Wiederherstellung des plastischen Tonus der Magenringmuskulatur nach Dehnung. Z. Biol. 103. 321-336. Greven, K. (1951) Sind die Nachwirkungserscheinungen beim glatten Muskel mit den Erschlaffungsvorglngen nach Kontraktion identische mechanische VorgInge? Z. Biol. 104. 373-383. Greven, K., Gotthardt, H. and Hancke, E. (1973) Der zeitlithe Verlauf von Nachdehnungserscheinungen (Nachdehnung und Relaxation) an der Taenia coli des Meerschweinchens unter verschiedenen Bedingungen. PfIiigers Arch. 344. 245-260.

Greven, K. and Hohorst, B. (1975) Creep after loading in relaxed and contracted (KC1 or KlS04 depolarized) smooth muscle (taenia coli of the guinea pig). P’iigers Arch. 359, 11l-125. Greven, K., Rudolph, K. H. and Hohorst. B. (1976a) Creep after loading in the relaxed and contracted smooth muscle (Taenia coli of the guinea-pig) under various osmotic conditions. Pjiigers Arch. 362. 255-260. Greven, K. (1976b) The time course of creep and stress relaxation in relaxed and contracted smooth muscles (taenia coli of the guinea pig). In: Physiology of Smooth Muscle (Edited by Btilbring, E. and Shuba, M. F.), pp. X3-228. Raven Press, New York. Greven, K. (1976~) Plastic properties of vertebrate smooth muscle (taenia coli of the guinea pig). Pj%gers Arch. 362. 289-290.

Grimmer, P. (1304) Die glatten Muskeln. Erg. Phjsioi.

3.

12-88.

Giith, K. and Kuhn. H. J. (1976) Cross bridge elasticity in the presence and absence of ATP. Pfiiigers Arch. Suppl. 362. 25.

Ha&n. J. and Huxley, H. E. (1955) The structural basis of contractions in striated muscle. Symp. Sot. e,xp. Biol. 9. 228-26-t. Herzig. J. W. (1976) Mechanical evidence for a conformational change of attached cross bridges during muscle contraction. PfIiigers Arch. Suppl. 362. 25. Hohorst. B.. Krohnert, U. and Greven. K. (1976) Passive stress relaxation followed by active contracture in vertebrate smooth muscles (taenia coli oi the guinea pig). Pfliigers Arch. 336. 137-142. Hohorst. B.. Greven, K.. Kriihnert. C. and Schober, 0. (1977) Stress relaxation compared with relaxation after contraction in smooth muscles. Pfiiigrrs Arch. 368. 177-179. Huxley, H. E. (1969) The mechanism oi muscular contraction Science 16-t. 1356-1364. Huxley, A. F. and Niedergerke, R. (1954) Structural changes in muscle during contraction. Interference microscopy of living muscle. :Vrlrure (Land.) 173. 971-973. Huxley, A. F. and Simmons. R. IM. (1971) Proposed mechanism of force generation in striated muscle. Nature 233. 533-538.

Lowy. J. and Millman, B. hl. (1963) The contractile mechanism of the anterior byssus retractor muscle of Mytilus edulis. Phil. Trans. R. Sot. Land. (Biol. Sci.) 246. 105-145.

Mulvany, M. .I. and Halpern, ‘W. (1976) Mechanical properties of vascular smooth muscle cells in situ. IVatrrre 260, 617-619. Oberst, H. (1961) Das elastische Spektrum von Hochpolymeren. Phys. Verb. DPG _(. 13-26. Remington, J. W. (1957) Extensibility behavior and hysteresis phenomena in smooth muscle. In: Tisslre Elasticir) (Edited by Remington. J. W.), pp. 138-153. Waverly Press, Baltimore MD. Rockemann, W. and Miiller-Ontjes, J. (1975) Force-velocity-relation of depolarized smooth muscle described by a model of interaction between viscous flow and contraction. PfIiigers Arch. Suppl. 359. 64. Schatzmann, H. J. (1964) Erregung und Kontraktion glatter Vertebratenmuskeln. Erg. Physiol. 55. 28-130. Siegman, M. J.. Butler, T. M.. Mooers. S. U. and Davies. R. E. (1976) Crossbridge attachment. resistance to stretch, and viscoelasticity in resting mammalian smooth muscle. Science 191. 3%3S5. Somlyo, A. P. (1972) Excitation-contraction coupling in vertebrate smooth muscle: Correlation of ultrastructure with function. The Physiologist IS. 3X-34% Somlyo, A. P., Devine, C. E., Soml;o. A. V. and Rice. R. V. (1973) Filament organization m vertebrate smooth muscle. Phil. Trans. R. Sot. Lond. B. 265. 223-229. Turner, A. Jr. and Gurnee, E. F. (1957) Slolecular structure and mechanical behavior of macromolecules. In: Tissue Elasticity (Edited by Remington, J. W.), pp. 12-32. Waverly Press. Baltimore MD. Winton, F. R. (1930) Tonus in mammalian unstriated muscle-I. J. Physiol. (Lond.) 69. 39:--110. APPESDIX:

PLASTIC

PROPERTIES

There are some difficulties concerning the theoretical treatment of plastic properties. Since equation (3) in our experiments applies to creep as well as to stress relaxation. our preparations behave as if there would be only one single Voigt or Maxwell model with one variable p = a + b(.\or R). However in the Winton model (Fig. lc), characterized by plastic properties with the restrictions mentioned in Section 2, two diflereerentelements in series are combined Consequently, equation (3) can be used for stress relaxation

Properties of visceral smooth muscles in vertebrates

:23

_______-

----

0

-----

:23-

N

!SC -

‘E ”

3c -

i

at,

55

ively. two equations equal in form to equation (3) are obtained. provided that in transforming equation (9) b’ of equation (3) is taken as zero. d.Y dr is the sum of two hyperbolic equations dl, dr + dll Jr. However. if K = E: Al2 during the creep equation (10) vanishes only one hyperbola according to equation (9)

(II) -

with asymptotic approximation to the zero ordinate remains. Creep is practically limited because dN,‘dt approaches zero. After unloading the model shortens only I i I ‘1I I I ,c,j 2 5 10’ 2 ’ IO” 2 5 IO’ by Al, + Ai, (plastic effect). It must be emphasized that b’ of equation 13). repref. s.?c sented here in equation (IO) by El. i.e. an elastic stiffness Fig. 1. (a) time course of creep !\; after loading according in parallel, cannot be set equal to zero for the upper part to integration of equation (3) using the data of depolarized of the curves in the Figs. 3 and 3. because in these cases and contracted taeniae coli of Fig. 2. the experimental results on the one hand and the theoreti(b) theoretically computed curve of shortening in time after cal hyperbolae on the other hand would strongly differ. unloading using identical values of the parameter h of One might try to avoid the difficulties just mentioned equation (3) as in curve a. and still strive to obtain plastic effects by other means (c) same but tenfold value of parameter 6. Shift to the with a single Voigt Maxwell model provided with a right approximately with a power of ten on the time scale. parallel spring. This might perhaps be brought about by For further explanation see text. conferring a greater value on the parameter b in the denominator of equation (3) during the shortening after unloading than during stretch: the resistance of friction but not for creep. With the Winton model. the time course would behave asymmetrically. Figure 4 shows the results of creep is given by the sum of two differential equations of the purely theoretical conception. If the parameter b (Greven. 1976~) during stretch and during the following shortening after unloading are identical, the curves are symmetrical with dl, K (9) regard to time. However. if the parameter h_is raised to dr = z(,’ a power of ten during the shortening, the time until the dll K - E, A/z original length is restituted also increases tenfold (Fig. 4~). (10) The respective time would have to be prolonged infinitely dt= p(1 to realize a true plastic effect. This however. would only If the constant factors ,ni and p2 are replaced by the be obtained by a complete blockade (h = x) during shortening after unloading. linear functions (ai + b, AI,) and (a2 + b2 AI:) respect23 -

._ I