J. theor. Biol. (1975) 52, 1-16

A Sliding Ji’ilament Model for Skeletal Muscle : Dependence of Isometric Dynamics on Temperature and Sarcomere Length JAY E. MMTENTHAL

Department of Biological Sciences, Purdue University, W. Lafayette, Indiana 47907, U.S.A. and Department of Biophysics, The Johns Hopkins University, Baltimore, Maryland 21218, U.S.A. (Received 3 May 1971, and in revisedform

30 October 1974)

The sliding filament model of A. F. Huxley is used to derive a formula for the rate constant for exponential increase of tension to the plateau value in an isometric tetanus. The formula agrees well with measured values of the rate constant for frog sartorius muscle at several temperatures when most sarcomeres have length in the range at which they develop maximum force. The comparison with data suggeststhat the distance over which an attached cross-bridge moves decreasesmarkedly with increasing temperature, from about 100 ,& at 2°C to less than 5 A at 20°C. The relation of the analysis and its results to Hill’s theory of the isometric myogram is discussed. The Huxley theory is extended to treat the rise of isometric force in a muscle fiber of such length that the number of cross bridges opposite thin filaments is less than maximum and changes with time. Predictions of the extended theory do not agree well with observed dynamics of muscle, probably because the extended theory neglects effects of inequalities in sarcomere length within the fiber.

1. Introdwioll The model developed by A. F. Huxley (1957) to describe the mechanochemical interaction of a myosin cross-bridge with a thin filament has served as precursor to the present variety of models for the dynamical processes at a cross-bridge (reviewed by White & Thorson, 1973). Although descender& of the Huxley model have superseded it as tools for exploring 1 1 T.B.

2

J.

E.

MITTENTHAL

the dynamics of molecular interactions, the model is still a useful phenomenological description for many mechanical and thermal properties of vertebrate striated muscle. We have therefore extended the Huxley model to broaden the range of conditions which it can characterize. In the Huxley model the number of cross-bridges able to interact with thin filaments to develop force remains constant during the period of activity. This model is therefore not suited to describing mechanical processes in which the extent of overlap between thin and thick filaments changes with time, such as shortening of sarcomeres stretched beyond the maximum length in vim. Isometric twitches and tetani in fibers containing such extended sarcomeres are of interest because the rise and fall of tension are markedly slower at these lengths than in the range of shorter lengths in which the fiber normally operates (Ramsey & Street, 1940; Hill, 1953; Jewel1 & Wilkie, 1960; Gordon, Huxley & Julian, 1966). In this paper the Huxley model is extended to predict the time course of tension near the isometric steady state in a single fiber when the number of cross-bridges in the region of overlap between filaments changes with time. The model also applies to a whole muscle in which the mechanical effects of connective tissue between fibers may be neglected, so that all fibers in the muscle are effectively joined only at the tendons. The modified theory is used to interpret observations on the rise of tension to its maximum value during an isometric tetanus in frog’s sartorius muscle at several temperatures (Mittenthal & Carlson, 1971). The analysis gives insight into the failure of Hill’s (1938) theory of the isometric myogram to predict the rise of tension in an isometric tetanus correctly (Jewel1 & Wilkie, 1958). 2. Definition of the Model First Huxley’s kinetics for the force-generating processes will be summarized, with modification for variable extent of overlap between thin and thick Filaments. Next the mechanical constraints that sarcomere structure and series elastic elements impose on these processes will be formalized. (A)

FORCE

GENERATOR

Following Huxley (1957) we assume that a cross-bridge in the overlap region of a half-sarcomere is either in a contractile state, C, in which it exerts force on the thin filament apposed to it, or in a non-contractile state, H, in which it does not exert force. Any bridge in the overlap region cycles between states C and H during prolonged stimulation of the fiber. We assume that in a fused isometric tetanus every cross-bridge in the region of overlap

THEORY

OF ISOMETRIC

TETANUS

3

between thick and thin filaments can interact with the thin lilament adjacent to it to develop force.? The cotiguration of a cross-bridge is represented by a variable x. x has dimensions of length and represents translation of the tip of the bridge, that portion nearest to the adjacent thin filament, parallel to this filament. The theory assumes that the force developed by a bridge in the C state is proportional to x, with stiffness k. Let n&x, t) be the density of bridges in the C state having x in the interval (x, xs dx) at time t in one half-sarcomere of a fiber of unit cross-section area. Then the total force which this half sarcomere exerts at t is P(f) = r (kx)n&x,

t) dx.

(1)

The density n&x, t) 2 0 is determined by the equation

an, at + 2,az =f(x)*n,(x, t)-g(x)*n c(x,t). The Huxley theory assumes that the thin filament is rigid and that the tip of a bridge in the C state moves with the same speed u as this filament; hence u = dx/dt. u is positive in a lengthening half-sarcomere. The term u(anc/ax) represents the effect on n&x, t) of translation of the thin filament without transition of bridges between the C and H states. The terms f’nn-gn, determine the rate at which n, changes because of transitions of state. +(x, t) 2 0 is the density of bridges in the non-contractile state. f(x) and g(x) are the probabilities per unit time that a bridge will pass from H to C, or from C to H, respectively. To eliminate nH from the preceding equation it is necessary to specify +(x, t) as a function of n&x, t). The function must satisfy the constraint on conservation of bridges : 7 [n&x,

t>-+n&,

t)] dx = Y[A(t)].

Y[A(t)] is the total number of bridges in the overlap region of a halfsarcomere of length 2, per fiber cross-section area. If 1 is such that the tips of thin filaments are not within the pseudo-I-I zone, Y will vary with time as filaments slide and A changes. t This assumption will be applied to data for the steady state and for the latter part of the transient rise of tension to a maximum in the tetanus. The latter application is not strictly valid, since tension increases somewhat more rapidly to the maximum following a quick release, when the muscle is fully active, than at the onset of the tetanus when activation is incomplete (Jewel1 & Wilkie, 1958).

4

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MITTBNTHAL

The conservation of bridges does not imply a unique dependence of nn on nc; additional hypotheses are required. Huxley (1957) and Julian (1969) assumed that

n,+n, = N, with N independent of X. Clearly, this assumption the conservation of bridges, since

is not consistent with

lim fNdx=a. ‘4-m -A However, these authors also assume that a bridge can only attach to a thin tiament (f # 0) when the bridge is in the “attachment zone” 0 I x I h. Hence their analysis remains valid as long as nC+nH

= N

(in attachment zone)

with N dependent only on the extent of overlap between f?lament arrays. The conservation of bridges then requires -[(n,+nu)dx+Nh+[(n,+nn)dx=

Y,

Clearly Nh must be less than Y, to allow for bridges outside the attachment interval. If changing the extent of overlap alters the scale of n,-(x, t) but not its shape,

Nh=W.

(2)

0 is the fraction of bridges in the overlap region that are in the attachment zone. By our assumptions 0 is independent of the mechanical constraints imposed on the muscle. The equation for n,(x, t) is then

an, x + u2

=fN-(f+g)wc.

Huxley assumed

f($=

fG i0

in the attachment interval outside the attachment interval;

glX

forx>O

Q2

for x c 0.

g(x) =

THEORY

OF

ISOMETRIC

5

TETANUS

In an isometric steady state (~Yn,#t = 0, u = 0), therefore, n&9 0 = n,,(x); in the attachment zone,

(4)

outside the attachment zone;

fi

kh2

P=P,=f+gl’N,.2. (B)

MECHANICAL

(5)

CONkTRAINTS

The structural organization of the fiber and the properties of elastic elements in it determine what speed of shortening u(t) results from the force P(t) developed in each half-sarcomere. We first examine the relation of series elastic elements to half-sarcomere length and fiber length in an isometric tetanus, and then discuss the effects of half-sarcomere length on number of cross-bridges in the overlap region. We assume that all half-sarcomeres in a fiber have the same length, L. Huxley & Peachey (1961) found that the length of sarcomeres was fairly uniform throughout the fiber except at the ends, where sarcomeres were rather shorter than the majority. The significance of variations in sarcomere length for isometric dynamics will be discussed below. To relate v(t) to P(t) note that Iz changes with speed v. The length L, of the series elastic element is a function only of the force P. During the latter part of the increase of P to its asymptotic value, PO, in an isometric tetanus the stiffness of the series elastic element, k,, is constant (Jewel1 6 Wilkie, 1958); hence L,,-L,

= $. [P,-P]. c

P(r) is the same in all M half-sarcomeres and in the series elastic element, since all are in series. In the isometric condition the length of the fiber, L = MA+L,,

is constant; hence L CO -L, = Mfb-A,]. Leo and Lo are the lengths of the series elastic element and of a half-sarcomere,

respectively, during the plateau of an isometric tetanus. It follows that

w-no = bM c * L-~o-w)l

(6)

6

J.

or, differentiating

E.

MITTENTHAL

with respect to time, P 2-G;

p

dP =-&

(7)

Equations (6) and (7) relate half-sarcomere length to force. Note that in these relations the stiffness of the series elastic element is assumed to be independent of the change in sarcomere length that occurs during the rise of P to PO. Elasticity associated with attached cross-bridges will vary with length of the overlap region (A. F. Huxley & Simmons, 1973). However, in a whole muscle attached to measuring apparatus through compliant tendons and connections the series elasticity is nearly independent of muscle length, if resting tension (“parallel elasticity”) is absent (Jewel1 & Wilkie, 1958). To formulate an equation for the asymptotic approach of P to P,, from the preceding equations it is necessary to provide an explicit function for the dependence of total number of cross-bridges in the overlap region, Y, on 1. Y changes when bridges enter or leave the overlap region during change of A. If r(n) is the number of bridges per unit length entering or leaving the overlap region when L changes,

y is negative and independent of 1 if the half-sarcomere is sufficiently long that thin filaments have not passed beyond the portion of the thick filaments bearing bridges in the ipsilateral half-sarcomere. When the tips of the thin filaments move in the pseudo-H zone, y = 0. The former condition will be called partial overlap; the latter, full overlap. It is unclear how y varies during further shortening, beyond the length at which thin filaments cross the edge of the pseudo-H zone in the contralateral half-sarcomere (Gordon et al., 1966). If (as we assume) ~(1) is constant during isometric approach of P to PO, r, - Mao]

= I+-& - J(Ol ;

hence, from equation (6), yo - Y[W]

3. Expmedid

= - k+ e

Asymptotk

* [P, -P(t)].

(8)

Approtcn of P to PO

The preceding formalism will now be used to show that the asymptotic approach of P(r) to PO is approximately exponential in a fiber having uniform

THEORY

OF

ISOMETRIC

TETANUS

7

sarcomere length: P = u-p,-P). (9) We observed exponential asymptotic increase of P to PO in frog sartorius muscle (Mittenthal & Carlson, 1971). The present analysis yields the rate constant c1 as a function of mechanical and chemical parameters of the fiber. From equation (l), dx. (10) -CO Use of equation (3) to evaluate &@t, and equation (7) to eliminate I), with integration by parts of the term derived from dn&k, yields i’(r) = i (kx) F

p=

i (kx)(f+g)n, --a0

dx + N i (kx)fdx 01)

in,, * c-m To show that the right side of this equation is nearly proportional to PO -P for P near to P,, a solution for nc(x, t) is necessary. Julian (1969) has obtained such a solution by numerical analysis, for the increase of force in an isometric tetanus near rest length and 0°C. The resulting plot of l’ versus P is linear near PO [cf., Julian (1969), Fig. 8; our equation (7) relates P to u, and k, is constant near P]. However, we have sought an analytical approximation which would allow prediction for Q for a range of temperatures and lengths. The classical model of Hill (1938) for an isometric tetanus suggests an approximate form for nc(x, t). Hill proposed that the hyperbolic forcevelocity curve for steady isotonic shortening might also characterize the contractile element during an isometric tetanus. According to this “quasiisotonic” assumption, P(t) changes so slowly during its rise to P, that nc(x, t) can adjust to the steady-state isotonic solution n&x, o(r)] at each instant. Hill’s proposal imphes that v(t) should be chosen from P(t) by use of the hyperbolic force-velocity relation. This stipulation has been shown to be invalid (Jewell & Wilkie, 1958; Julian, 1969). However, ncr[x, v(t)] might still give a good approximation to the isometric &(x, t) if the true v(t) were used. The appropriate u(t) is not known a priori. However, during the period when P(r) approaches PO exponentially, 0 < @)/urn < 0.08 at rest length (Julian, 1969); u, is the maximum speed of isotonic shortening. We therefore evaluated P and P,,-P from the isotonic steady-state solution of the Huxley model, n&x, u), and examined the form and conl+$

8

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E.

stancy of the ratio P/(P,,-P)

MITTENTHAL

in this range of speeds. The evaluation

fi - ___ P=

fi +91

l+LMk

yields

- Nkh-q(u)*o

* - fi

fi+gl

(12)

- Nkh-q(v)

from equation (1 l), and

---

(13)

where, with u = v/v,,

r(v) = (1 -e -1’4”).(l +@13u). Here Huxley’s (1957) finding that and g2 = 3~919~cf,+g,) %l = -2h*Cf,+g,) near 0°C has been used. From equations (2), (8) and (13), 1+YP, Mk, r, N =-. 6 h From the preceding equations P lim “-0 p,-p

= uo9

(14) 4Po * 2h + x e In the following section equation (14) is evaluated using empirical data. The derivation of equation (14) has used the “quasi-isotonic” assumption, that the hyperbolic force-velocity relation Pi(v) characterizes the contractile element, to evaluate P,, -P but not to approximate I? A different route to an approximate formula for CLuses Pi(u) directly in equation (7): uo =-v;

1 df’,(v). ’ = - k,M --&-- ” For small 0 equation (15) may be linearized to yield $= -qv.

(15) (16)

THEORY

OF

ISOMBTRIC

TETANUS

9

ai must approximate a, since u is proportional to 9 [equation (7)] and P decreases exponentially to zero with rate constant a. However, ai should be a less satisfactory approximation to a than ao, since a0 but not ai incorporates the correct equation (according to the Huxley model) for P in an isometric tetanus, equation (11). Equation (13) contains the isotonic force-velocity relation which results from the Huxley model. Use of this Pi(u) in equation (15) leads to l+? -- po Yo Mk . * 4p

ai (Huxley) = -e,

n

(17)

Mk, The equation (P+a)u

= b(P,-P)

with urn=- bpo a

a -=-,

ad

PO

1 4

used by Hill (1938) to represent the force-velocity (y = 0), yields ai (Hill) = -0,

* $-

0

relation for full overlap

(full overlap).

(18)

The difference between ai (Huxley) and ai (Hill) probably reflects the approximate fit of both the Huxley and Hill Pi(u) to empirical data. The significance of the difference between the approximations ai and a0 will be considered in the Discussion. For the present, note that no “microscopic” parameters from the dynamics of a single cross-bridge enter the quasi-isotonic approximation to a, equations (17) and (18); only “macroscopic” parameters of the entire fiber appear. By contrast a0 contains the microscopic parameter h in addition to the macroscopic parameters.

4. Comparison with Data In the preceding section I have suggested that the ratio p/(P,-P), evaluated from equations (12) and (13), is nearly independent of a in the range of speeds occurring during the exponential phase of an isometric tetanus. The limit of this ratio as v + 0 is a0 [equation (14)]. The constancy of the ratio and the validity of a0 as an estimate for the constant may be tested by calculating the deviation of p/[ao*(Po-P)] from unity over the ran@ 0 < u/u, < 0438.

10

J. (A)

E.

NUMERICAL

MITTBNTHAL VALUJB

FOR

PARAMETERS

Ideally all parameters for the calculation should be evaluated on a single muscle; unfortunately no adequate set of data is available. The ratio was tested on data from two frog sartorius muscles in which most sarcomeres have nearly full overlap. CI and P,, were measured at four temperatures for each muscle; for one muscle (A) the resulting values of CIhave been reported (Mittenthal & Carlson, 1971, experiment of 1 October 1969; the other muscle (B) was that used 25 November 1969 in the same study). For parameters other than P,, needed in the calculations an average value from the literature was used. For the structural parameters Y and y reliable values are available from the morphology of the filament lattice (Hanson & Huxley, 1955; Pepe, 1967): The maximum Y, for full overlap in a half sarcomere, is 5.37. IO” cross-bridges/(cm’ half-sarcomere). Since the length of a thick filament bearing cross-bridges is 0.7 km, and Y changes from zero to its maximum value over this interval, y = -7.67. 1Ol2 cross bridges/(cm2 pm half-sarcomere). From Jewel1 & Wilkie (1958, Fig. 8), k, G (1*33/L,). IO4 dyne/cm’ pm. Lo (cm) is length of the resting muscle in situ. k. was assumed independent of temperature, neglecting a slight dependence which these authors tentatively suggested. Letting s0 = length of central sarcomeres when the muscle length is Lo, M & 2L,/s,,. Hill (1938) reported that the maximum speed of shortening of whole frog sartorius muscle averaged about 1*33.L,/sec, with a 2.05-fold increase per 10°C; these data were used to approximate ZJ,,,at various temperatures. Values for h are calculated from u, = 2h.Y; +gl) (Huxley, 1957). Estimates of fi and g1 are based on data of Siger (1968). The mean lifetimes rn and rc of cross-bridges in the H and C states are related to the average rate R of hydrolysis of ATP during the plateau of an isometric tetanus, R2!i Tc + 2,’ But in an isometric steady state R = -[ d~h&) Equating ncdx),

dx.

these expressions for R, and using equation %+%i =2+2 Bl

(4) to obtain

fl’

Since v = 0 in the plateau, a bridge which enters stage C at x will remain

THEORY

OF

ISOMETRIC

11

TETANUS

at x until it dissociates from the thin filament. Hence the dissociation rate per bridge at + is g(x). The mean breakdown rate for all links is l/rc (Cox & Miller, 1965); hence

E

so that

1 fl . -==El 2 Siger estimated zJO and r JO from the dependence of R on temperature. Comparison of his estimates with other estimates of fi and gi (Huxley, 1957; Julian, 1969) suggest that 6 2 3 to 3. In the following calculations 8 = 3 is assumed. Tables 1 and 2 summarize values used in the calculations. TABLE 1 Estimates of temperature-independent quantities used in calculations Muscle

A

::;5

B

2.3 2.2

3-4.104 209.10’

o-111 0.043

3-4.103 4-1.103

1.53 1.47

TABLE- 2 Estimates of temperature-dependent quantities used in calculations Muscle A

B

2-l 12 18.2

1.82 3.04 3.14 3-12

2.0 7-l 13.0 20.0

2.22 2.67 3.22 3.18

l-8 2.55 3.97 5.7

54 74 112 156

24 142 1400 8800

115 59 13-2 3.2

1: 1280 16600

111 54 13-3 l-9

54 :35 ii8

1:; 176

12

J.

E. (B)

MITTENTHAL FULL

OVBRLAF’

For the data in Tables 1 and 2 the ratio &‘/Lao. (PO-P)] deviated from unity by less than 5.3 % for speeds in the range 0 < U/U,,, < O-08. This small variation is particularly striking because the functions q(v) and r(v) change by 17 and 6 %, respectively, in this range. Thus the ratio p/(P, -P), evaluated from equations (12) and (13), depends negligibly on v near the plateau of the isometric tetanus, and the ratio has the value a, to a good approximation. Although a0 approximates satisfactorily the theoretical function of tr from which it was derived, we must test how well a0 approximates the measured rate constant, a. In Fig. 1 a0 and a are plotted in Arrhenius form against temperature for muscles A and B. The agreement between theoretical and empirical values is good, considering the varied sources of data for some parameters,

20

IO -G 3

s 7 6

a

5 4 3

-- I 3.5

-- I

I 1

34

I

I

3.5

3.6

3.7

T-‘(OK x IO31

FIO. 1. Arrhenius plots of theoretical and measured rate constants, cq, and a, versus temperature for two frog sartorius muscles. Filled symbols and solid lines, measured; dIlled symbols and dashed lines, theoretical.

THEORY

OF

ISOMETRIC

TETANUS

13

The energy of activation for a0 (or ct) and for v, is about 13 kcal/moIe (Hill, 1938; Mitten&al k Carlson, 1971). Equation (14) and Table 2 show that u, does play the dominant role in determining the dependence of a on temperature. h is small and decreases markedly with temperature by comparison with 2P,/Mk,, which increases slightly with temperature. This close alliance of a and u, is not surprising, since both depend on the rate at which the actin-myosin interaction drives thin and thick filaments past each other. Considering the agreement of a and a0 for full overlap, it is desirable to interpret the form of ao: a0 =

fi +81

(full overlap).

(14)

l+&.;

Since a0 is a &St-order sum of two terms:

c rate constant, cc;’ is a time constant. a;’

is the

which depends only on the kinetics for attachment and detachment of crossbridges, and

(fi+sl>-lg

e

;

which depends on cross-bridge kinetics and the characteristic of the series elastic element. Po/Mk, is roughly the distance that each half-sarcomere must shorten to develop force PO in the series elastic element; nonlinearity of the forcelength characteristic of this element makes the estimate inaccurate. h/2 is roughly the average distance that a thin filament will slide relative to a thick one as a result of motion at one cross-bridge. Thus Po/Mk;2/h is roughly the number of sequential contractions by cross-bridges facing each thin filament that are required to translate the thin filament far enough to develop force PO in the series elastic element. From Tables 1 and 2, Po/Mk;2/h varies from about 6 at 2°C to more than 500 ar 20°C. Since Po/A4k;2/h is the ratio of elasticity-dependent to elasticity-independent terms in a; ‘, evidently the latter term, which depends only on attachment-detachment kinetics, plays a minor role in determining the value of ao. The quasi-isotonic approximation used in Hill’s (1938) model assumes this term is completely negligible [cf. ai, equation (17) and (18)]. Evidently this is not the case at 2°C but becomes more nearly so at high temperatures.

14

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MITTBNTHAL

(C)

PARTIAL

OVERLAP

Inclusion of the term for partial overlap in a0 [equation (14)] only increases the maximum deviation of ~/[uo~(Po-P)J from unity slightly, to about 5.4% for 0 < 11/o, < O-08. However, the theoretical rate constant a0 corresponds poorly to the measured time course of increase in isometric tension in partial overlap. The preceding theory predicts that for any degree of partial overlap a0 should assume a value about 8% less than the value for full overlap. However, Mittenthal and Carlson (1971) found that the mean rate constant for increase of force declined from 7.1 set-’ to 2.2 see-’ when the length of central sarcomeres was increased from 2.2 to 2.8 pm at 1°C. Observations of Huxley & Peachey (1961) suggest an interpretation for this discrepancy. During tetanization of an isometric fiber in which sarcomere length is not uniform and the longer central sarcomeres are in partial overlap, the shorter sarcomeres shorten progressively, stretching the longer ones. During this process tension “creeps” slowly upward at a low rate (Ramsey & Street, 1940; Hill, 1953; Huxley & Peachey, 1961; Gordon, Huxley & Julian, 1966; Mittenthall & Carlson, 1971). On this interpretation the apparent decline of a for isometric tetani at muscle lengths greater than Lo (Mittenthal t Carlson, 1971) results from estimating an exponential rate constant during the period of creep. The inadequacy of the preceding model for predicting this creep of tension thus seems to be a consequence of assuming uniformity of sarcomere length throughout the model fiber. 5. Discussion

The preceding analysis provides an analytical description of the increase in tension to the maximum value, PO, in isometric tetanus, based on the sliding filament model of Huxley (1957). The formalism predicts values for the exponential rate constant of tension increase, a, which are in reasonable agreement with measured values of a over a range of temperatures if most sarcomeres have full overlap between thin and thick filaments. The dependence of a on temperature arises mainly from the temperature-dependence of u,,,, the maximum speed of shortening. In this test of the formalism values for h, the length of the interval in which a cross-bridge can form an attachment to a thin filament, were calculated for several temperatures. h decreases sharply with increasing temperature, from about 110 A at 2°C to 2 A at 20°C (Table 2). It will be interesting to see whether the most successful theory which emerges from current alternative ideas of cross-bridge dynamics also shows such’ a marked variation in scale of cross-bridge motion with temperature.

THEORY

OF ISOMETRIC

TETANUS

15

The present modification of the Huxley (1957) theory, when applied to a fiber having all sarcomeres of uniform length and in partial overlap, predicts a small decrease in o!which is independent of sarcomere length. In all reported measurements of isometric tetani under conditions where most sarcomeres were in partial overlap, a prolonged slow rise of tension (“creep”) has masked the increase of force for which # should be measured. Creep even occurs in tetani of regions of fibers selected for uniformity of sarcomere length (Gordon et al., 1966). It is not clear whether some normalization can remove the effects of creep to permit estimation of CLfor comparison with the present theory, or whether a revised theory which takes account of creep and nonuniformities of sarcomere length is required to describe tetani in long sarcomeres. Hill (1938) suggested that the force-velocity relation for steady-state isotonic shortening should characterize a fully active muscle under all mechanical constraints. Several lines of evidence now show that this suggestion is incorrect and indicate the need to consider history-dependent aspects of cross-bridge dynamics. A step- change in tension or Iength of a fiber results in transient changes in length or tension (the unregulated variable) for many milliseconds. These transients reflect adjustment of cross-bridges to the altered mechanical constraints; the time course of adjustment has beeh used to suggest and test new models of cross-bridge dynamics. (Mechanical transients and models for them are reviewed by White & Thorson, 1973.) In an isometric tetanus in whole muscle, force and sarcomere length change gradually rather than abruptly as sarcomeres shorten, stretching series elastic elements. Our analysis complements the simulations of Julian (1969) in showing that even in such a gradual alteration of constraints, crossbridges adjust over distances and times long enough to require explicit consideration in modelling the dynamics of the muscle. Near 0°C the rate constant ct is of the same order of magnitude as I; and gl, the maximum rate constants for attaching and detaching bridges. Also, as noted above, near 0°C h is comparable in magnitude to the distance that each half-sarcomere must shorten to develop maximum isometric force in the series elastic element. Hence h, the length scale for operation of a single cross-bridge, enters the formula for a along with parameters for the whole muscle (e.g. P,, ~3. Under some circumstances, however (e.g. near POat high temperature) the time and distance scales for cross-bridge operation are apparently negligible compared to the corresponding scales for a fiber; in such cases Hill’s model should describe the fiber’s mechanical behavior adequately. Dr Francis Carlson aided this work with patient interest and encouragement. I thank Dr A. T. Winfree for helpful criticism of the manuscript.

16

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H. D. (1965).

77mr.~

of Stochastic Processes. New York:

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JBWELL, B. R. & WILKIE, D. R. (1958). J. PhysoL, Land. 143, 515. JEWELL, B. R. & WILKIE, D. R. (1960). L Physiol., Land. 152,30. JULIAN, F. J. (1969). Biophys. J. 9, 547. Mm, J. E. & C-N, F. D. (1971). J. gen. Pbysiol. 58,20. Papa, F. A. (1967). J. nwlec. Biol. 27, 203. RAMSEY, R. W. 8~ STREET, S. F. (1940). J. cell. camp. Physiol. 15, 11. SIGER, A. (1968). Ph.D. the&. WHITE, D. C. S. & tiORSON,

The Johns Hopkins University, J. (1973). Prog. Biophys. molec.

John

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Biol. 27, 175.