Prog. Biophys. Molec. Bid. 1975, Vol.

29, No. 2, pp. 105459. Pergamon Press. printed in Gnat Britain.

THEORETICAL FORMALISM FOR THE SLIDING FILAMENT MODEL OF CONTRACTION OF STRIATED MUSCLE PART II TERRELL L. HILL Laboratory of Molecular Biology, National Institute of Arthritis, Metabolism and Digestive Disorders, National Institute of Health, Bethesda, Maryland 20014, U.S.A.

CONTENTS IV.

M ULTIPLE ACTIN SITES A. Introduction

B. Single Group of Actin Sites 1. The formalism 2. Free energy functions in slipping transitions 3. Relative time s&es

4. Numerical example: steady isotonic contraction C. Indefinite Array of Equioalent Actin Sites 1. The formalism 2. A special case (steady isotonic contraction) D. Indejinile Array of Equivalent Groups of Sites 1. The formalism 2. A special case

128 128 129

V. OTHER COMPLICATIONS A. Double-headed

107 107 109 109 111 113 114 118 118 122 126 126 127

cross-bridges

1. Single actin site 2. Groups and arrays of actin sites

131 132 135 138

B. Interfilament Distance as a Parameter C. Passive Elastic Element in Series D. Interactions Between Cross-bridges

140 140 142 146 146 147

FURTHER SKCI-LUTIC CQNSIDERATTONS A. General Diagram

VI. SOME

B. Two-state Cycle C. Related Topics 1. Nerve axon channels 2. Fluctuations and noise

A PPENDIX

Comments on the Point of View of McClare II: Reduction of a Diagram APPENDIX III: Thermodynamics Related to Efiiency

149 152 157

REFERENCES

i59

A PPENDIX I:

105

THEORETICAL FORMALISM FOR THE SLIDING FILAMENT MODEL OF CONTRACTION OF STRIATED MUSCLE PART II TERRELLL. HILL Laboratory of Molecular Biology, National Institute of Arthritis, Metabolism, and Di#estioe Diseases, National Institutes of Health, Bethesda, Maryland 20014, USA

IV. MULTIPLE ACTIN SITES This paper continues Part I (Hill, 1974). To avoid confusion, equation and figure numbers here maintain the sequences begun in Part I, but references will be repeated as needed. The notation is the same as in Part I. Hill et al. (1975) is also a "continuation" of Part I. Part II deals with refinements and complications in the theme of Part I. Since it is too early to tell which, if any, of these somewhat speculative topics are of importance in the actual (unknown) mechanism of muscle contraction, the treatment will be deliberately sketchy rather than exhaustive. Almost all of the material in Part II is previously unpublished. ComraCdoo

-3 -2 -1 I

II 1 - 1

+1 +2 +3

-3

X

-2 -1 11,,0

n-O

+1 +2 +3

-3 -2 -1 n-+l

FIG. 34. Semischematicviewofactin filamentas seen by a cross-bridgeswinging down from above. Groups ofactin sites (monomers)are labelled by n, siteswithin a group by m. Arrow at top indicates direction of motion (relative to myosin filament) in contraction. A. Introduction

I n this section (IV) we remove the "single-site assumption" of Section I l i a 1, but retain all other assumptions. Thus, more than one actin site may be available to a given crossbridge but we still assume here (however, see Section VA) that only one of the two S1 units ("heads") of a cross-bridge m a y be attached to an (any) actin site at any one time. In effect, then, a cross-bridge m a y be treated (as in Section III) as if it has only one head. Figure 34 shows a n actin filament, somewhat schematically. Consider a cross-bridge that m a y swing clown from directly above the aetin filament for possible attachment to 107

108

TERRELL L. HILL

a site on an actin monomer (alternatively, the sites might be between monomers--i.e, an attachment might involve two neighboring monomers). Index numbers (denoted by m) may be assigned to prospective sites, as shown in the figure. The repeat distance is about 360 A. Correspondingly, the repeat distance of cross-bridges oriented equivalently is about 430 A. Th6 single site assumption of section III would be valid if, because of some very stringent configurational requirement, a cross-bridge (from above) could attach only at sites with index, say, m = 0 (shaded in the figure). It seems somewhat more plausible, however, that geometrical requirements would not be this restrictive and that a cross-bridge (from above) might attach, with significant probability, to any one of a group of sites, say m = 0, ___1, or m = 0, + 1, +2, etc. (Hill, 1973), at appropriate values of x. In general, one would expect the sites of a group to be nonequivalent, because of differing angular orientations 0 (around the axis of the actin filament). With a severe enough angular orientation (e.g. at m -- + 2, -t- 3, or + 4), attachment would presumably become essentially impossible. Contraction

Z+:

X

Myosin

\

~

. .=tin

FIG. 35, Schematic. Relative distances in myosin-actin interactions. (See Fig. 34.) Cross-bridge M can reach two groups of actin sites.

For simplicity, we assume that the actin helix is an integral helix (e.g. with 13 monomers per helical repeat). Otherwise, successive groups would not be quite identical and an additional angular averaging (over 0) would be needed to supplement averaging over x. This would be required even with the single site assumption (section III). But averaging over 0 seems too refined for the present state of the subject, so we omit it (as was done implicitly in section III). The longitudinal relations between equivalently oriented cross-bridges, on the one hand, and groups of aetin sites, on the other, are shown very schematically but with correct relative longitudinal scales in Fig. 35. If only m = 0 sites can be attached to, it is clearly appropriate to assume (as in section III) that a cross-bridge would never be within reach of either of two neighboring (m = 0) sites at any value of x. If only m = O, +_ 1 sites can be attached to, it is almost equally plausible to assume that a cross-bridge would never (at any x) be able to reach either of two neighboring groups (though it could reach neighboring sites

The sliding filament model of contraction of striated muscle--ll

109

within a group). But with a five-(m = 0, + 1, + 2) or seven-site group, a cross-bridge would presumably (for some values of x) have access to two neighboring groups. See, for example, the cross-bridge labelled M in Fig. 35. In fact, with a seven-site group and a thirteensubunit repeat, successive groups overlap (Fig. 34). Thus there are two generalizations of section III of particular interest: first, the "one site at a time" assumption should be extended to "one group at a time"; and, second, an indefinite array of identical groups should be considered in order to take into account the possible interaction of a cross-bridge with either of two neighboring groups. The first generalization is considered in section IVB and the second in section IVD. Section IVC is, in effect, an introduction to section IVD. Section IVC treats an idealized version of Figs. 34 and 35 in which the groups merge into one another (compare the seven-site group case) and all sites are considered equivalent. In this case a cross-bridge "sees" an indefinite array of equivalent actin sites 55 A apart, as shown schematically in Fig. 36 (with arbitrarily assigned index numbers m). Although this is a simplified model, it may be a fairly good approximation. In any case, it provides a useful preliminary to the more complicated problem of an indefinite array of equivalent groups of actin sites. Contraction

-5 -4 -3 -2

± T

-1 m=O +1 +2 +3 +4

*5 Actin

Fro. 36. Idealized model in which all actin sites (55 A apart) are equivalent. Arrow at top = direction of motion in contraction, m = site index number.

B. Single Group of Actin Sites For the most part, we use below a group of three actin sites, with index numbers m = 0, + 1, as a concrete example. But the number of sites in a group, in this "single-group" case, might well be two, four, or possibly five. The formalism does not depend on this number. I. The Formalism The essential point here, which should be emphasized at the outset, is that the formalism required to handle a single group of sites is precisely the same as that needed in the single site case (section III). There is an elaboration of attached states in the diagram, but the fundamental approach is not modified. We again take Fig. 37 (see also Fig. 4) as our basic, illustrative, single-site diagram. The single actin site is located at x, relative to the cross-bridge. We now consider the group of three actin sites nearest a given cross-bridge (see Figs. 34 and 35), with the m = 0 site at x, the m = - 1 site at x - di and the m = + 1 site at x + 6. In general, the ruth site of a group is at x + m6, where 6 = 55 A (longitudinal spacing between actin sites in the same chain). Each site has not only a longitudinal location (x + m6) but also an angular location 0 about the axis of the actin filament. But this angle

110

TERRELL L. HILL 3

2

MD

MT

M

I

AMIxl

AMDlx) 6

....

AMT(x) 5

FIG. 37. Basic single site diagram used for illustrative purposes.

is specified, in effect, by the value of m. Note that a single value of x (arbitrarily assigned to the rn = 0 site) suffices to locate all sites in the group: there is only one variable x. The repeat distance (between groups) is d g 360 A, as in section III. Averages, as for example in calculating the force F, are taken over the interval - d / 2 +2. Here, site m = 0 is too far from the cross-bridge to be accessible but site m = - 1 is one unit closer. In other words, in the pass of a pair of sites by a cross-bridge, site m = - 1 comes into view of the cross-bridge first (X > 3) while site m = 0 recedes from view last (X < - 1.5). Roughly speaking, compared to Fig. 16 (site m = 0 only), the interval of attachment on the X-axis is broadened by one unit (on the right-hand side). This is emphasized in Fig. 45, in which are superimposed the isometric (v' = 0) curves from the various cases. Although F_ t(X) is below Fo(X) (Fig. 44), the right-hand broadening effect on the attachment probability, just mentioned, increases the mean force significantly in Case II over Case I. This can be seen in Table 1. Case III (fast slippin#). This model is the same as in Case II with the additional feature that there is a fast slipping equilibrium indicated by m in Fig. 43b. That is, these transitions are fast compared to the f ' s and g's (which are taken to be the same functions as in Case II). As in section IIIE3, the intrinsic stability of attachment is assumed to be greater on site m = 0 than on m = - 1 (see below). As already mentioned in connection with Case II, this difference is attributed not to the f ' s and g's but to the inverses of the f ' s (detachment).

116

TERRELL L. HILL 1.0

0.8

0.6

p, olr

P-t O,4

0.2

x -- x,~.

FIG. 44. Case II. P r o b a b i l i t y of a t t a c h m e n t o n site m = - 1 (p_ 1) or o n site m = 0 (Po), as a function of X = x/a, for v a r i o u s values of v' (isotonic contraction). The force functions X (m = 0) a n d X - I (m = - 1) are included.

1.0

.

.

.

.

* -- ": " ----g..-7

'"...k--- ,, ÷:,.'

OJ

..L 0.4

-q

= e "l. m /

( F/IOK6 -q=e'~nJ 0.5,

-3

-:t

-I

/ /

/:!

2 x

3

4

§

6

x/G

FIG. 45. I s o m e t r i c (v' = 0) p r o b a b i l i t y functions in v a r i o u s cases. See text for details. T w o force functions for Case I I I (fast slipping) are i n c l u d e d ; see eq. (179). The circles o n the force curves indicate the m i d d l e of the t r a n s i t i o n (P0 = P - ~).

The sliding filament model of contraction of striated muscle--II

117

TABLE 1. VALUESOF PdfkT

0

0.5 1.0 1.5 2.0 2.5

Case I

Case II

2.396 0.981 0.535 0.269 0.085 -0.050

3.274 1.307 0.691 0.336 0.094 -0.085

III,

q = e-



III,

3.380 1.828 1.302 0.966 0.717 0.517

q = em2

4.046 2.351 1.751 1.363 1.072 0.837

In effect, Case III is an explicit example of a steady isotonic contraction using the twostate Huxley-Simmons analysis of section IIIE3. For this purpose the fs and g’s must be specified (as above). The rate constants k, and k- (Fig. 23) do not appear explicitly because these processes are assumed to occur on a faster time scale than the steady isotonic contraction. Only the resulting equilibrium is involved. We denote the (equilibrium) ratio of p. to p- 1 by P:

e-x’l2

4 e-(X-1)2/2

=

4-l

e-x+f

(176)

where 4 has the same significance as in eq. (99) (recall that state 1 -+ - 1, state 2--t 0). We shall take 4 < 1 (attachment more stable on site m = 0). Because of eq. (176), p. and p_ 1 are not independent. We use as independent variable pan = p. + p- 1. In the notation of section IIIE3, P = n&l - n2). The differential equation in pan is

dpon

-v~=(.f-‘+fo)(l-pon)-9-1P-1-SoPo = (f-

1 +

fo)U

-

PO”)-

(177)

Because of the equilibrium pooling of the two attached states, there is only a single differential equation. We were less explicit in Fig. 23: we see from eq. (177) that the effective single f (Fig. 23) is a simple sum and that the effective single g is an equilibrium weighted sum. The mean force is Fd -= kT

s

=

+a, _oo Cp- 1(X - 1) +

~0x1dx

+ou

s -a,

pon(FIK4

(i78f

dx

where F(X)/KG = [(X - 1) -t PX]/(l

+ P).

(179)

The latter function is of the same type as shown schematically in Fig. 8. It is also a disguised form of eq. (98). In the present case, two examples of F(X)/KG are included in Fig. 45. These are for the choices q = e- l and q = e- 2. The circles in Fig. 45 indicate the center of the transition (P, = 1). The slope of F(X)/KS at this point is 3/4 in both cases. This is a consequence of eq. (1 l$and of our particular choice 0 = 6: 1 - (Kd2/4kT) = 1 - (S2/402) = 3/4. The slope would have been zero if we had taken 0 = d/2. The functions p,,,(X) required in eq. (178) for the mean force calculation are obtained by solving eq. (177) numerically. This has been done for q = e- ’ and em2. The q = e- 1 family is shown in Fig. 46. These curves are again (see Fig. 44) a broadened version of Fig. 16 (m = 0 site only). The asymptotic behavior at sufficiently negative X (X < - 1)

118

TERRELL L. HILL

is practically indistinguishable from Fig. 16. The same is true at sufficiently positive X ( X > 3), except for a horizontal shift of one unit. The force calculations (Case III) are included in Table 1. The differences are quantitative rather than qualitative. Figure 45 compares isometric (v' = 0) attachment probabilities in the various cases. The dashed curve in this figure is n(X) from Case I shifted one unit to the right. The dotted curve is Po + P- 1 from Case II. The mean force (Table 1) is larger in the q = e -2 case than in the q = e -1 case primarily because F(X)/K~ and po,(X) are larger in the former example (Fig, 45). That p,,,(X) is larger when q = e- 2 is a consequence of the fact that g_, > go for any X and hence a larger P (q = e- 2 case) reduces the "effective single" g in eq. (177).

0.8

vl:O 0.6

Cmem" q=e -I P~

0.4

02

-3

-2

-1

0

1

2

3

4

S

6

X--x#

FIG. 46. Case IlI. Probability of attachment, p,,,,, as a function of X, in the case q = e - ', for various values of v'.

The effect of velocity on mean force is less in Case III than in Case II because of the instantaneous (equilibrium) distribution between attachments at sites m = 0 and m = - 1. In Case II at high velocities, there is insufficient time for the transitions - 1 --* U ~ 0. Further analysis of these results is probably unjustified because of the arbitrary nature of the various cases.

C. Indefinite Array of Equivalent Actin Sites A possible useful approximate model arises if we idealize Fig. 34 somewhat and assume that a cross-bridge sees an indefinite array of equivalent actin sites, as shown schematically in Fig. 36. The one-site model of section III assumes that the interaction between a crossbridge and an actin site is extremely sensitive to the angular orientation 0 of the sites. Here we assume the opposite: 0 has no effect on this interaction, Also, part of the approximation is to ignore the lack of regular continuity between (with a 13-subunit repeat) site m --- - 3 in one group and site m = + 3 in the adjacent group (Fig. 34).

1. The Formalism We consider the interaction of a single cross-bridge with the array of actin sites shown in Fig: 36. The longitudinal distance between sites is 6. Some one site is arbitrarily assigned

The sliding filament model of contraction of striated musele--II

119

the index m = 0. The other sites are then labelled accordingly (as in Fig. 36). The variable x locates the m = 0 site relative to the cross-bridge. Usually we would choose x = 0 as the most stable position in some one attached biochemical state (with attachment on the m = 0 site). Site m is at x + m6. Because of the assumed equivalence of sites, the cross-bridge cannot distinguish between, say, x = x' and x = x' + 6. Hence many Properties of this system will be periodic in x, with period 6. For example, the force F(t,x) will be periodic in x so that, in averaging to find the mean force F(t), it will suffice (below) to average over the interval - 6 / 2 ~< x ~< + 6/2. Periodicity is in fact the main new feature that arises in the present subsection. Although the value 6 = 55 A is of primary interest here, the formalism of this section would be necessary and would apply to a model with an array of equivalent single sites separated by, say, 100-200 A but with the possibility that a cross-bridge might be able to reach either of two of these sites for some values ofx. With 6 = 55 A, of course, a crossbridge might well be able to reach out to any one of, say, three or four sites. ~D

2MT

I _o

6AMD(x)

(a)

I

6AMT(x)

2

/ • • • ~







O 6

-2



6

x4 -1

~

§• • •

6

x np.O

w""o... 6

6

x4.~ 1

x+2~ 2

(bl

FIG. 47. (a) Single-site diagram used as illustrative reference case. (b) Expansion of(a) required for an indefinite array of equivalent actin sites. The choice of the m = 0 site is arbitrary.

Concerning biochemical states, for concreteness we shall use as a starting point the single-site diagram shown in Fig. 47a rather than our usual Fig. 37. This simplification is made only in order to keep Fig. 47b relatively uncluttered. In this latter figure, we show the central portion of the corresponding extended diagram that includes the various possible sites of attachment as well as the same biochemical states shown in Fig. 47a. The unattached states 2 and 3 are single states, but each attached biochemical state may involve attachment at any one of the sites m = 0, + 1, + 2...... In principle, for a given x (position of site m = 0), the cross-bridge may be in any one of an infinite number of states: 2,3, im (i = 4,5,6; m = 0, _ 1. . . . ). The corresponding probabilities may be denoted p2(t,x), p3(t,x), pi~(t,x). In practice, though, with the m = 0 site located in the i n t e r v a l - 6 / 2 ~< x ~< + 3/2, the cross-bridge would not be able to reach beyond, say, m = + 2. Since pim(t,x) is the probability that the cross-bridge is in the biochemical state i and attached to site m(at x + m6) when site m = 0 is at x, the p~m(t,x) converge very rapidly to zero with increasing [m[. For example, if all p~(t,x)= 0 for Iml t> 3, the total number of states with non-zero probabilities would be 2 (unattached) + 5 x 3 (attached)= 17.

1'20

TERRELL L. HILL

Because of the equivalence of sites, p2(t,x) and p3(t,x) (unattached states) are necessarily periodic functions of x with period 6. That is, pj(t,x) = pi(t,x + 6). For an attached biochemical state i, the corresponding relation is pim(t,x) = pMt,x + m

Theoretical formalism for the sliding filament model of contraction of striated muscle. Part II.

Prog. Biophys. Molec. Bid. 1975, Vol. 29, No. 2, pp. 105459. Pergamon Press. printed in Gnat Britain. THEORETICAL FORMALISM FOR THE SLIDING FILAMENT...
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