JOURNAL OF MORPHOLOGY 207:211-223 (1991)
Dynamic Shape of Tapered Skeletal Muscle Fibers J.A. TROTTER Department of Anatomy, School of Medicine, Albuquerque, New Mexico 87131
ABSTRACT The muscle fibers of the feline biceps femoris have tapered ends, across which tension is transmitted to the endomysium. The angle of taper of 11ends, measured on scanning electron micrographs, varied between 0.16" and 1.18'. The muscle fibers are highly variable in cross-sectional shape. The shape of the fibers has been quantified as the ratio (form factor [FF]) of the measured perimeter to the calculated circumference of a circle having an area equal to that contained by the fiber perimeter. The FF for 173 terminal portions of fibers varied between 1.06 and 1.85 and was found to have a highly significant negative correlation with sarcomere length. The slope of the regression line suggests that the fibers maintain both volume and surface area as they change length. These studies suggest that isovolumic muscle fibers maintain a constant surface area by changing shape as they change length. Many skeletal muscles, including the feline biceps femoris (Chanaud et al., '901, are composed of short in-series muscle fibers that terminate by tapering within fascicles (Bardeen, '03; Huber, '16; Adrian, '25; Barrett, '62; Richmond et al., '85; Loeb et al., '87; Gans et al., '89; Gaunt and Gans, '90). In such series-fibered muscles, adjacent fibers are not directly coupled by intercellular mechanical junctions, and fiber ends lack the folded morphology of muscletendon junctions (Trotter, '90). Tension, therefore, appears to be transmitted between fiber and endomysium across most or all of the sarcolemmal interface (Trotter, '90). The amount of interfacial area is thus an important mechanical parameter, and any changes in this parameter as fibers change length could have mechanical consequences. A skeletal muscle as a whole (Swammerdam, circa 1660, cited in Needham, '71; Abbott and Baskin, '62), the myofilament lattice (Huxley, ' 5 3 ; Elliott et al., '631, and individual fibers (April et al., '72; Edman and Hwang, '77) are all thought to remain nearly isovolumic a t all sarcomere lengths. An elongated cylinder] such as a skeletal muscle fiber, cannot change length while simultaneously maintaining constant volume, surface area, and shape (see Discussion). For the purposes of this paper, surface area refers to the interface between myofiber and endomysium and does not include caveolae or T tubules. Because the volume of muscle fibers is constant, t? 1991 WILEY-LISS,
INC.
either surface area or shape, or both, must vary with the length of the fiber. If the apparent surface area changes, then mechanisms for removing and adding surface to the muscle fiber must exist. I t has been suggested that transverse folding could serve this function within the physiological range of fiber lengths by storing excess membrane as fibers shorten (Dulhunty and Franzini-Armstrong, '75). That transverse folding actually serves this function has not been demonstrated quantitatively, however. Similarly, it has been argued that surface caveolae function as a reservoir of membrane that can be added to the fiber surface at very long sarcomere lengths (Dulhunty and Franzini-Armstrong, '751, although Poulos et al. ('86) have produced evidence to the contrary. In the studies on the reservoir roles of transverse folding and caveolae, it was assumed that fiber shape did not change as a function of sarcomere length. The latter assumption was apparently validated by the optical sectioning studies of Dulhunty and Franzini-Armstrong ('77), which indicated that the cross-sectional shape of isolated fibers deviated very slightly from a circle and was remarkably constant between fibers and a t all sarcomere lengths. In contrast, Blinks ('65), using similar techniques, observed a large variation in cross-sectional shape both between fibers and as a function of sarcomere length. The differences in the two studies might be due to the use of fibers from different muscles and dif-
212
J.A. TROTTER
ferent species of frogs. No general conclusion concerning whether fiber shape or surface area remain constant or change with sarcomere length could be drawn from these studies. The studies reported here were designed to estimate the amount of fiber surface associated with the tapered ends of cat skeletal muscle fibers in situ a n d to determine whether this parameter changes as the muscle changes length. The data are consistent with a constant surface area and a changing cross-sectional shape as fibers change length. MATERIALS AND METHODS
Three female cats weighing between 1.5 and 2.5 kg were anesthetized with ketamine given intramuscularly followed by Nembutal given intravenously. The skin of the right hind leg was removed, and the exposed biceps femoris muscle was kept moist with physiological saline during subsequent procedures. Longitudinal full thickness incisions were made in the anterior and posterior regions of the muscle to produce strips approximately 2 mm wide by 6 cm long. Sutures were tied to each end of a strip, and the limb was positioned to place the muscle in short, long, and intermediate lengths. At each length the sutures of one strip were tied to a wooden stick so that the muscle strip would remain approximately at that length as it was fixed. This procedure produced specimens from each cat in which the muscle fibers were at short, long, and intermediate sarcomere lengths within their normal anatomical range. Each strip was then excised from the muscle and placed in 2.5% glutaraldehyde, 0.2 M sodium cacodylate buffer, pH 7.2, at room temperature. This produced strips of anterior and posterior biceps femoris muscle fixed a t sarcomere lengths representing the full physiological range. After the tissues had been excised, the cats were sacrificed by an overdose of Nembutal. After approximately 30 min the strips were removed from the sticks and sliced with the aid of a dissectingmicroscope into strips about 1mm square by 3 mm long for transmission electron microscopy and about 1mm square by 2 cm long for scanning electron microscopy. Fixation was then continued overnight. Specimens for analysis by transmission electron microscopy were soaked for 1hr in fresh fixative of the same composition containing 0.2% tannic acid. They were then rinsed overnight in several changes of 0.2 M cacodylate buffer. Prior to further processing, the strips were sliced into two pieces.
One piece was later used for cross-sectional analyses; the other was used for longitudinal analyses, including sarcomere length determinations. These pieces were subsequently labeled and processed as matched pairs. The tissues were next postfixed in 1% osmium tetroxide, 0.1 M cacodylate buffer for 1 hr, and rinsed in water for 1 hr. After dehydration in a graded ethanol series the specimens were infiltrated with Spurr’s resin, which was polymerized for 24 h r at 65°C. Transverse and longitudinal ultrathin sections, approximately 100 nm thick, were produced using diamond knives and were imaged a t original magnifications of either ~ 1 , 0 0 0or x 5,000 in a n Hitachi H600 transmission electron microscope after staining with uranyl acetate and lead citrate. To prepare isolated muscle fibers for analysis in the scanning electron microscope, the strips of biceps femoris that had been fixed overnight were rinsed in several changes of cacodylate buffer for several hours and were then placed in ice cold 50% glycerol, 0.02% sodium azide, 0.1M sodium phosphate buffer, pH 7.0 (determined at room temperature). They were gently agitated for several days in the cold room and then stored in this solution at 4°C for up to 6 months. Single fiber segments were prepared from the strips by incubating them for 40-60 min at 60°C in 9.6 M HCl. When individual fibers were seen to have begun to separate from the strips the acid was rapidly diluted with a large volume of water a t room temperature. Fiber segments with tapered ends were selected with the aid of a dissecting microscope and were placed on freshly prepared, gelatin-coated, 12 mm round coverslips. The gelatin was hardened by exposure to 2.5% glutaraldehyde in water, after which the coverslips were passed sequentially through several changes of water and a graded ethanol series. After several changes of absolute ethanol, the fiber segments were dried from CO, by the critical point method. The coverslips were fixed onto aluminum stubs using silver paint, rendered conductive by a thin coating of goldipalladium deposited in a cold cathode sputter coater, and imaged in a n Hitachi S800 field emission electron microscope. The “taper angle” of a fiber end is the angle a t which the fiber surface intersects a line drawn parallel to the longitudinal axis of the fiber. To estimate this angle, scanning electron micrographs were made of the tapered portions of the fibers. Several widths
2 13
TAPERED MUSCLE FIBER SHAPE
were measured along the tapered regions, and the longitudinal distances between the points of measurement were also determined. The slope of the regression line relating the half-widths to the cumulative length is the tangent of the taper angle. The angle is therefore found by taking the arctangent of the regression slope, as illustrated in Figure 1. The ratio between the measured perimeter of a transversely sectioned fiber and the circumference of a circle having the same area as that of the fiber cross section is a quantitative expression of the shape of the crosssectional profile of the fiber. This measure, designated a form factor (FF), estimates the extent to which the profile deviates from circularity. Because a circle has the minimum perimeter possible for a given area, FF values are always 1 or greater. Because all the fibers analyzed had FF > 1,the use of a single diameter measurement to express fiber size was impossible. The cross-sectional area could have been used as an accurate way to express fiber size, but the values for area are not commonly used for this purpose. Therefare, the length of the diameter of the circle with the same area as the fiber profile was calculated and designated the equivalent diameter (ED). Sarcomere lengths (SL) were determined from longitudinal sections by taking the average distance between at least 10 Z-line pairs in a myofibril. SL were determined from electron micrographs of longitudinal sections; FF and ED were determined from measurements of transverse sections of blocks that were paired with those from which the longi-
1
L4
tudinal sections were taken, as described above. All length and area measurements were made using Sigmascan software and digitizing tablet (Jandel Scientific, Corte Madera, CA) and an IBM PS/2 Model 50 microcomputer. The resolution of the digitizing tablet was adjusted with the software to be either 0.76 or 6.1 mm. Statistical analyses followed standard parametric procedures for linear regression and comparison of slopes by analysis of covariance (Zar, '84). Unless otherwise specified, statements of statistical significance are based o n P I0.05. RESULTS
Taper angles of isolated fibers The tapered ends of isolated anterior and posterior biceps femoris fibers analyzed by SEM showed no prominent surface folds or irregularities (Fig. 2). The tapering was not uniform, and some regions of the tapering portion had no obvious taper. Considering only the tapered portions of the fibers, the range of values for taper angle of the 11 fibers analyzed was 0.16" to 1.18";the mean value was 0.72". The range of taper angle values is shown in Figure 3. Shape of fiber cross sections The cross-sectional shapes of both large and small fiber profiles varied between round and angular, and the outlines varied between smooth and rough. Fibers with all degrees of these characteristics were seen at all sarco-
t
Fig. 1. Method of estimating taper angle from scanning electron micrographs. Arctan I(W,/Z)/LJ = 8, for i = 1 , 2 , 3 , . . . =.
214
J.A. TROTTER
Fig. 2. Scanning electron micrographs of the tapered end of an isolated biceps femoris muscle fiber. Note the absence of surface specializations and the gradual taper
toward the end. Bar = 50 pm in upper panel and 10 p m in lower panel.
TAPERED MUSCLE FIBER SHAPE 1.6
I
1.4
-t
1.2
.)
b 1.0
3
Y
2
3-
' LL: W
0.a 0.6
0.4 0.2
0.0 1
2
3
1
1 ii
4
5
6
7
8
9 1 0 1 1
FIEER NUMBER
Fig. 3. Bargraph of the taper angles of 11 fibers similar to that shown in Figure 1. Error bars show the standard error nf the angle measurements made on each
fiber.
mere lengths. Figures 4 and 5 are tracings of smooth, rough, round, and angular profiles of fibers with short (Fig. 4) and longer (Fig. 5) sarcomere lengths. The profiles from fibers with short sarcomere lengths appeared to be both rougher and more angular than those from longer fibers (compare profiles on the left sides of Figs. 4 and 5). The smaller profiles also appeared to be slightly rougher and more angular than the larger ones (compare the upper left panels of Figs. 4 and 5). The shape of the fibers was quantified using the FF (described in Materials and Methods). The resolution of the digitizing tablet was set at 0.76 mm, and the micrographs analyzed were printed at a final magnification of x 12,700. Hence the measurement resolution was 60 nm. The distribution of FF for the tapered ends (ED between 2.7 and 10.1 +m) of 173 fibers is shown as a histogram in Figure 6a. The distribution is prominently skewed, indicating that a large number of fibers have very large FF. In contrast, a histogram of the form factors of 247 fibers with ED greater than 10 pm shows a distribution that is much less skewed (Fig. 6b). There is thus a tendency for the fibers to assume less circular profiles as they taper toward their ends.
Change of fiber shape with sarcornere length The FF data shown in Figure 6 came from three cats. To determine whether there was a
215
correlation between sarcomere length SL and FF, regression analysis was applied to these data from each cat (Table 1). SL values were obtained from electron micrographs of fibers in the same region as those used for crosssectional analysis, but not on the identical fibers, as explained in Materials and Methods. For reasons given in the Discussion, the data were analyzed in logarithmic form. For each cat a significant correlation was seen between FF and SL. Analysis of covariance indicated that the slopes of the three regressions were not statistically different (Table 1).The slope ofthe common data was -0.268. Graphs of the data for the three cats and for the pooled data are shown in Figure 7. There is a large scatter in the data for all sarcomere lengths because of both an inherent variability of profile shape and the tendency of small fiber profiles to have less circular shapes. Nevertheless, there is a highly significant negative correlation between FF and SL (P < 0.005) for both the individual cats and for the pooled data, and the R2 values indicate that between 12% and 24% of the variance in FF can be attributed to the change in SL. The equation of the line calculated from the common data is lnFF = 0.54 - 0.2681nSL. A change in SL from 3 to 2 pm would thus produce a n 11% increase in FF. A similar analysis of the nontapered portions (i.e., profiles with ED > 12 pm) of fibers in the same specimens that were used for the above measurements yielded a common slope of -0.21, with an R2 of 0.25 (N = 143; P < 0.001). Both tapered and nontapered regions thus change shape as a function of SL. Source of shape change with change in SL The higher F F values associated with shorter sarcomere lengths could have been caused by largc-scale or small-scale shape changes. Examples of large-scale changes are increases in the ellipticity or angularity of the profile. A small-scale shape change would be seen as increased roughness in the fiber profile. To discriminate between these possibilities, the FF of the 173 fiber profiles discussed above were determined at a resolution of 480 nm, and the results were compared with the 60 nm determinations described above. An example of the tracings produced by these two settings of the digitizing tablet is shown in Figure 8. Large-scale shape changes are detected at both resolutions, but only the 60 nm tracing detects roughness changes. The relative contribution of shape
216
J.A. TROTTER
FF = 1.6 3D = 5 . 4
?F = 1.1
PD =
4.4
?F = 1.5
Fig. 4. Cross-sectional profiles of muscle fibers fixed at short sarcomere lengths (average SL = 1.9 pm). These profiles were chosen to represent the range of shapes of both small and large profiles. The profiles on the left are
more angular and have rougher surfaces; those on the right are smoother and more nearly round. The form factor (FF) and equivalent diameter (ED) values are indicated in each panel. Bars = 1pm.
217
TAPERED MUSCLE FIBER SHAPE
FF ED
FF = 1.4 ED = 2 . 8
= 1.2 = 3.6
r I
'F = 1.1 1D = 9.9
~
~~~
Fig. 5 . As in Figure 4,except that these fibcrs were fixed at longer sarcomere lengths (average SL = 2.5 km). Bars = 1 km.
218
j!
J.A. TROTTER
,
25
I
1
I
n
a
20 rn w K
15 0
K
w
I
1
I
z 3 1
LL
3z
I
10
20 5 10
0
I]
1 .o
0
1.2
1.4
1.6
1 .o
1.8
FORM FACTOR
FORM FACTOR
Fig. 6. Histograms of the distributions of form factor values for the tapered ends of muscle fibers (a)and the middle regions of fibers (b).Data for a were taken from profiles with equivalent diameters (ED) 5 10.1 IJ-m;fibers used in b had ED 2 10.2 pm.
changes a t these two scales was estimated by comparing the slopes of the regression lines through the lnFF vs. lnSL data for each cat (Table 2). I t can be seen from Table 2 that between 66% (cats 1and 2) and 72% (cat 3) of the change in FF is accounted for by largescale shape changes, and the rest is produced by increases in roughness.
Contribution of longitudinal ruffling The above measurements consider only the cross-sectional shape of the fibers. To determine whether longitudinal ruffling, i.e., accordion-like folding of the fiber surface, also varies with sarcomere length, 68 fibers were examined in longitudinal sections, representing SLs between 1.75 and 3.58 pm. Two points on the fiber periphery were chosen, and the TABLE 1. Regression Cat
N
length of the sarcolemma between the two points was measured, as was the straight line between the two points. The ratio between these two measurements quantifies the ruffling of the fiber surface and is thus a ruffling ratio (RR). RR as a function of SL is shown graphically in Figure 9. A highly significant correlation exists between these two parameters (P < 0.001). The equation of the regression line is RR = 1.03 - 0.0089SL. From this equation it can be calculated that a change in SL from 3 to 2 pm increases the RR, and consequently the surface area, by 0.89%. In contrast, as noted above, transverse shape changes over this same range of SL increases surface area by 11%over that minimally required to enclose the fiber volume. of
M F F ) on ln(SL)
R
R*
b
SE ofb
P
0.144 0.239 0.119
-0.356 -0.273 -0.229
0.119 0.075 0.073
< 0.005
1
55
0.380
2 3
44 74
0.489 0.345
< 0.001 < 0.005
Analysis of covariance Source Cat 1 Cat 2 Cat 3 Pooled Common
Sum (X') 1.0138 2.5842 2.5603
Sum (XY) -0.362 -0.705 -0.568
6.1584
-1.654
Sum (P) 0.8943 0.8057 1.1238 2.8238
N 55 74
h -0.356 -0.273 -0.229
173
-0.268
44
Res.SS 0.7650 0.6130 0.9893 2.3673 2.3793
Res.DF 53 42
72 167 169
F = 0.4224; P > 0.5. Common slope = -0.268. N, number of fibers analyzed; R, correlation coefiicient;K2,coefficientof determination; b, slope of regression line; SE, standard error of b; P, probability. For an explanation of the terms used in the analysis of covariance, see pp. 300-302 in Zar ('84).
219
TAPERED MUSCLE FIBER SHAPE
2.0 1.8
k
l
1 I
1.6
l i
2
I
f
1. 1.4
1.2
OO 0
@8
-
8
0
0 0
0 0
x%
3
-
0
1.2
O :
-
U 0
0 0
1.0
'
I
I
0
0
I
1
"1 0
-
:
I
SARCOMERE LENGTH bm)
SARCOMERE LENGTH (pm) 1 .a
80
I
3
0 0
0
1.6
0 1
0
e
0
0
008
0 0
K
E E z
1.8
0 0
1.4
1.6 00
0
0
2
0 0
. -
0-0
0
0
0 0
8 8 O
:%O
0
I
5
e
1.4 .
1.2 1.2 0
1 .c
I
I
2.0
2.5
O
'
I
0
0
I 3.0
3.5
SARCOMERE LENGTH (pn)
Fig. 7. Scatter plots of FF values in relationship to sarcomere lengths. 1-3: Data from individual cats; c: combined data from the three cats. The least-squares
linear regression line, together with the 958 confidence interval for the slope of the line, is shown for the combined data.
DISCUSSION
amount of surface area amplification is approximately equal to lisin0, where 0 is the taper angle.' Therefore, the value of this ratio for the 11ends analyzed varies between 49 (for 0 = 1.18") and 358 (for 0 = 0.16"). The tension produced by a muscle fiber is proportional to its cross-sectional area. If the tension is transmitted across the tapered sur-
Functional consequences of smooth tapered ends The absence of any junctional specializations in the cat muscle fibers used in these studies requires that tension be transmitted across the smooth surface membrane of the tapered ends (Trotter, '90). The small taper angle of the ends produces an interface between muscle and connective tissue that is loaded almost completely in shear and also greatly amplifies the surface area for tension transmission relative to the cross-sectional area of the fiber (Trotter, '90). This is because the area of the curved surface of a cone is greater than the area of its base, and the
'Given a right cone with b,aseradius R, and height H, the area of the curved surface is 7iR )/(R'+H2), and the area of the base i s 7iR2.The wrfacebase ratio is then .;iK ~(R2+I12))l.iiR7, which Because d(R2+H') is the hypotenuse of simplifies to b/(R2+HZMR. a right triangle, it can also be expressed as RisinO, where O i s the angle bctween the tapered side and a line parallel to the height. Thus the surfaceibase ratio can be expressed as (RlsinOYR, or l/sinfi.
-
220
J.A.TROTTER
Fig. 8. Profile of fiber from Figure 4, traced at ti0 nm (left)and 480 nm (right)resolution. The major shape features are present in the 480 nm tracing, but the roughness of the periphery is smoothed out.
face, the stress in this surface must vary as the inverse of the surface area, since stress = forcelarea. The tapering of the end is therefore an effective mechanism of stress reduction at the interface between muscle fiber and connective tissue. Because of the majpitude of the stress reduction produced by the tapered end, the load applied to the connective tissue is small in comparison with what it would be if it were concentrated in a smaller area. This reduction in load could make it possible for the collagen content of the endomysium to be lower, and for the fibril orientation to be more nearly isotropic, than would otherwise be the case.
TABLE 2. Regression slopes at two resolutions Source
60 nm
480 nm
480160
Cat 1 Cat 2 Cat 3
-0.356
-0.235
-0.273
-0.181 -0.164
0.66 0.66
-0.229
0.72
Slopes of regression line at 60 and 480 nm resolution are shown. Portion of slope at 60 nm present in slope at 480 nm is also gven.
0 O
O
0
1.01
1.00 1.5
1 4.0
2.0 2.5 3.0 SARCOMERE LENGTH (M , )I
3.5
Fig. 9. Scatter plot of the relationship between the longitudinal ruffling ratio and sarcomere length. The least-squares linear regression line is shown, together with the 95% confidence interval for the slope. The R' value for this regression is 0.204, N is 68, and P < 0.001.
221
TAPERED MUSCLE FIBER SHAPE
Shape change and constant surface area The data in this paper suggest that the surface area of a skeletal muscle fiber remains constant a t all SL and that the fiber maintains a constant surface area by varying its shape. The necessity of a shape change to keep both surface area and volume constant in a cylinder that varies in length can be appreciated by considering that, in geometrically similar structures (i.e., structures of the same shape), surface area (S) varies as the second power of the length (L), while volume (V) varies as the third power of I,. The ratio S/V therefore varies as (see SchmidtNielsen, ’84). To maintain both S and V constant while L changes, geometric similarity cannot be maintained: The shape has to change. The amount of shape change expected for a cylinder that remains isovolumic and isoareal as it changes length can be predicted from the following argument. ‘FF, the quantitative measure of shape change in a cylinder that changes length, was defined in Materials and Methods as the ratio of the perimeter of the cross section of a cylinder to the circumference of a circle with the same area. The FF expected for a shortening cylinder that maintains constant volume and surface area is derived as follows. Suppose a muscle fiber a t its maximum length is a uniform cylinder with L, = length (=l), A, = cross-sectional area, Po = perimeter, and LOP, = surface area. The fiber shortens to a new length, X (X < Lo), maintaining constant area and volume. At length X, the cross-sectional area = A,, and the perimeter of the cross section =$ P,. Then, because surface area and volume are constant: lA,
= XA,
LoPo= lP,,
XP,.
L,A,
=
Therefore Ax = A,/X
and Px = P,iX.
FF has been defined previously as P/C, where C is the circumference of the circle with area A. At L = LO= 1,P = Po = ~TR,,,and C = C, = ~ T R , where , R, is the radius, because at ‘This derivation of’FF for a shortening cylinder was provided by Dr. Peter Purslow, AFRC Institute nf Food Research (Bristol Laboratory, Langford, Bristol, U.K.).
L, the cross section is circular. Therefore, FF,=P,/C,= 1 . A t L = X , P = P x = 2 n R , , / X . Cross-sectional area at length X, A,, is equal to the area of a circle, radius R,, such that A, = IT^. The initial area = A, = T R ~ . Since A, = AJX, T~R;= vR;/X
and R,
=
&/Xo’.
Therefore, the circumference, C,, of the circle of area A, is C,
=
‘.
2vRx = XTR,/X~
At L,, then, FF, = P,/C, = (2~rR,/X)/(2aR,/ X”5 ) . Simplification of the last equation yields FF,
=
X-”.
The derivation of FF for a right cone, assuming that it is also circular in cross section at its maximum length, Lo, differs only in the beginning steps. Surface area is equal to aR,,(R~+ L;)”, where R,, is the radius of the base of the cone; for large ratios of L,/R,, such as those that are characteristic of the muscle fibers used in these studies, this expression is approximately equal to TR,L,, or to P,L,/2. Volume is equal to nR;L,,/3, OF A&,,/3. Then, because volume is constant, L,A,/3
=
1&/3
=
XAJ3
and thus L,A, = lA,
= XA,.
Similarly, because surface area is constant LOPOI:!= 1P0/2 = XP,i2
and thus L”P, = lPo = XP,
Therefore, A, =.AJX, and P, = P,/X, and the derivation continues as above for cylinders. F F for a cone is thus identical to that for a cylinder. It can be shown that these relationships hold for the volume and surface area of cones of any cross-sectional shape. For both cylinders and cones, then, the relationship between FF and X is a power function, the slope of which is -0.5. In logarithmic notation: lnFF = 0.51nX. Hence, as the muscle fiber shortens, FF must become greater. For example, if the cylinder were to shorten from 3 units to 2 units, X would be 0.67 and FF would be 1.22. That is, F F would increase by 22%. The lnFF vs. lnSL regression of the common data in the present study yielded a slope of -0.268. Both the sign of this slope and its
222
J.A. TROTTER
approximate magnitude are consistent with a change in shape in order to maintain constant surface area. The discrepancy between the calculated slope and the theoretical one may be accounted for in part by the large scatter in the FF data caused by the wide variation in fiber shape at all sarcomere lengths and by the inclusion in the FF vs. SL graph of all fibers with equivalent diameters I 10.1 p,m, although there was a n increased FF associated with smaller ED. It could also be true that constancy of surface area is only approximately maintained. The contribution of ruffling to a constant surface area is very small. Previous investigators have described "festoons" along the fiber surface of much greater magnitude than those seen in the present study (Street, '83; Shear and Bloch, '85). In both of the cited papers, however, the large festoons were seen in fibers with sarcomere lengths of about 1 km and were not seen in fibers within the physiological range of lengths. In the present study there was a n order of magnitude greater change in transverse fiber shape than in longitudinal surface ruffling as fibers changed length. Approximately two thirds of the shape change appears to be due to large-scale adjustments in shape. How this comes about is unknown. Although it was not quantified, there did not seem to be much myofibril-free sarcoplasm beneath the sarcolemma, and shape changes would therefore seem to involve myofibrils as well as the nonfibrillar sarcoplasm. That is, the sarcolemma appears to be a tight-fitting skin that conforms to the shape of the underlying contractile structures rather than a loose bag that changes shape independently of the underlying structures. Thus there may be a mechanism by which the packing of myofibrils, or even the myofilament lattice, changes as a function of fiber length. It is also possible that the magnitude of the shape changes seen in these studies is partly due to constraints resulting from the packing together of the fibers in a tissue. There are presently no data on these questions.
Significance of a constant surface area The tendency to maintain a constant surface area is important because tension is transmitted across the surface of the tapered end to the endomysium (Trotter, '90). The perifiber matrix (endomysium) is a sheath composed of small collagen fibrils, and other connective tissue macromolecules, that sur-
rounds the muscle fibers (see Mayne and Sanderson, '85). If the endomysium is tightly adherent to the surface of the fiber, as appears to be the case (see Fields, '70, and Rapoport, '73, and references therein), it is to be expected that it will conform to the fiber shape. Furthermore, if the endomysial volume remains constant at all SL, as seems likely, then its thickness will also remain nearly constant, because thickness of the endomysial sheath is equal to (constant) volume of endomysium divided by (constant) area of fiber surface. If the endomysial thickness remains constant, then the separation between muscle fibers should also remain constant, a t all SL. Considering series-fibered muscles as fiber-composite materials, with muscle fibers representing the fibrous (or anisotropic) phase and the endomysium representing the amorphous (or isotropic) phase (Piggott, '80), the constancy of the distance between fibers may make important contributions to the mechanical stability of the muscle. ACKNOWLEDGMENTS
The technical aspects of these studies were all expertly performed by Rayanne Ozbaysal. Conversations with Carl Gans were very helpful in clarifying the arguments concerning constant surface area. This paper was greatly improved by the suggestions made by Peter Purslow and by four anonymous reviewers. Statistical aspects of this work were discussed with Dr. Betty Skipper, Professor of Biostatistics (University of New Mexico School of Medicine). This work was supported by NIH grant AR39922. LITERATURE CITED Abbott, H.C., and R.J. Baskin (1962) Volume changes in frog muscle during contraction. J. Physiol. (Land.) 161:379-391. Adrian, E.D. (1925) The spread ofactivity in the tenuissimus musclc of the cat and in other complex muscles. J. Physiol. (Lond.) 60:301-315. ADril. E.W., P.W. Brandt. and G.F. Elliott (1972) The myofilament lattice: Studies on isolated fibers. 11. The effects of osmotic strength, ionic concentration, and pH upon the unit cell volume. J. Cell Biol. 53r5345. Bardeen, C.R. (1903) Variations in the internal architecture of the m. obliquus abdominis externus in certain animals. Anat. A n z . 23t241-249. Barrett, B. (1962) The length and mode of termination of individual muscle fibres in the human sartorius and posterior femoral muscles. Acta Anat. 48,242-257. Blinks, J.R. (1965) Influence of osmotic strength on cross-section and volume of isolated single muscle fibers. J. Physiol. (Lond.) 177r42-57. Chanaud, C.M., C.A. Pratt, and G.E. Loeb (1990) Functionally complex muscles of the cat hindlimb. 11. Me-
TAPERED MUSCLE FIBER SHAPE chanical and architectural heterogeneity in the biceps femoris. Exp. Brain Res. (in press). Dulhunty, A.F., and C. Frdnzini-Armstrong (1975) The relative contributions of the folds and caveolae to the surface membrane of frog skeletal muscle fibres at different sarcomere lengths. J. Physiol. (Lond.) 250: 513-539. Dulhunty, A.F., and C. Franzini-Armstrong (1977) The passive electrical properties of frog skeletal muscle fibres at different sarcomere lengths J. Physiol. 266: 687-711. Edman, K.A.P., and J.C. Hwang (1977) The forcevelocity relationship in vertebrate muscle fibers at varied tonicity of the extracellular medium. J. Physiol. (Lond.) 269;255-272. Elliott, G.F.. J. L o w . and C.R. Worthineton (1963) An X-ray and light-diffraction study of the filament lattice of striated muscle in the living state and in rigor. J. Mol. Biol. 63295-305. Fields, R.W. (1970) Mechanical properties of the frog sarcolemma. Biophys. J. 10:462-480. Gans, C., G.E. Loeb, and F. De Vr6.e (1989) Architecture and consequent physiologicalproperties of the semitendinosus muscle in domestic goats. J. Morphol. 199:287297. Gaunt, A.S., and C. Gans (1990) Architecture of chicken muscles: Short fiber patterns and their ontogeny. Trans. R. SOC. Lond. [Biol.]2401351-362. Huher, G.C. (1916) On the form and arrangement in fasciculi of striated voluntary muscle fibers. Anat. Rec. 11:149-168. Huxley, H.E. (1953) X-ray analysis and the problem of muscle. Proc. R. Soc. Lond. TBiol.1 141159-62. Loeb, G.E., C.A. Pratt, C.M. Chanaud, and F.J.R. Richmond (1987) Distribution and innervation of short,
223
interdigitated muscle fibers in parallel-fibered musclcs ofthe cat hindlimb. J. Morphol. 19111-15. Mayne, R., and R.D. Sanderson (1985) The extracellular matrix of skeletal muscle. Collagen Rel. Res. 51449468. Needham, D.M. (1971) Machina Carnis. Cambridge: Cambridge University Press. Piggott, M.R. (1980) Load Bearing Fiber Composites. Oxford: Pergamon Press. Poulos, A.C., J.E. Rash, and J.K. Elmund (1986) Ultrarapid freezing reveals that skeletal muscle caveolae are semipermanent structures. J. Ultrastruct. Mol. Struct. Res. 96:114-124. Rapoport, S.I. (1973) The anisotropic elastic properties of the sarcolemma of the frog semitendinosus muscle fiber. Biophys. J. 13t14-36. Richmond, F.J.R., D.R.R. MacGillis,andD.A. Scott (1985) Muscle-fiber compartmentalization in cat splenius muscle. J. Neurophysiol. 533368-885. Schmidt-Nielsen, K. (1984) Scaling. Why Is Animal Size So Important. Cambridge: Cambridge University Press. Shear, C.R. and R.J. Bloch (1985) Vinculin in subsarcolemmal densities in chicken skeletal muscle: Localization and relationship to intracellular and extracellular structures.J . Cell Biol. 101:240-256. Street, S.F. (1983) Lateral transmission of tension in frog myofibers: A myofibrillar network and transvcrse cytoskeletal connections are possible transmitters. J. Cell. Physiol. 114:346364. Trotter, J.A. (1990) Interfiber tension transmission in scrics-fibcred muscles of the cat hindlimb. J. Morphol. 206(3) (in press). Zar, J.H. (1984) Biostatistical Analysis (2nd ed).Englewood Cliffs, NJ: Prentice-Hall.
Dynamic Shape of Tapered Skeletal Muscle Fibers J.A. TROTTER Department of Anatomy, School of Medicine, Albuquerque, New Mexico 87131
ABSTRACT The muscle fibers of the feline biceps femoris have tapered ends, across which tension is transmitted to the endomysium. The angle of taper of 11ends, measured on scanning electron micrographs, varied between 0.16" and 1.18'. The muscle fibers are highly variable in cross-sectional shape. The shape of the fibers has been quantified as the ratio (form factor [FF]) of the measured perimeter to the calculated circumference of a circle having an area equal to that contained by the fiber perimeter. The FF for 173 terminal portions of fibers varied between 1.06 and 1.85 and was found to have a highly significant negative correlation with sarcomere length. The slope of the regression line suggests that the fibers maintain both volume and surface area as they change length. These studies suggest that isovolumic muscle fibers maintain a constant surface area by changing shape as they change length. Many skeletal muscles, including the feline biceps femoris (Chanaud et al., '901, are composed of short in-series muscle fibers that terminate by tapering within fascicles (Bardeen, '03; Huber, '16; Adrian, '25; Barrett, '62; Richmond et al., '85; Loeb et al., '87; Gans et al., '89; Gaunt and Gans, '90). In such series-fibered muscles, adjacent fibers are not directly coupled by intercellular mechanical junctions, and fiber ends lack the folded morphology of muscletendon junctions (Trotter, '90). Tension, therefore, appears to be transmitted between fiber and endomysium across most or all of the sarcolemmal interface (Trotter, '90). The amount of interfacial area is thus an important mechanical parameter, and any changes in this parameter as fibers change length could have mechanical consequences. A skeletal muscle as a whole (Swammerdam, circa 1660, cited in Needham, '71; Abbott and Baskin, '62), the myofilament lattice (Huxley, ' 5 3 ; Elliott et al., '631, and individual fibers (April et al., '72; Edman and Hwang, '77) are all thought to remain nearly isovolumic a t all sarcomere lengths. An elongated cylinder] such as a skeletal muscle fiber, cannot change length while simultaneously maintaining constant volume, surface area, and shape (see Discussion). For the purposes of this paper, surface area refers to the interface between myofiber and endomysium and does not include caveolae or T tubules. Because the volume of muscle fibers is constant, t? 1991 WILEY-LISS,
INC.
either surface area or shape, or both, must vary with the length of the fiber. If the apparent surface area changes, then mechanisms for removing and adding surface to the muscle fiber must exist. I t has been suggested that transverse folding could serve this function within the physiological range of fiber lengths by storing excess membrane as fibers shorten (Dulhunty and Franzini-Armstrong, '75). That transverse folding actually serves this function has not been demonstrated quantitatively, however. Similarly, it has been argued that surface caveolae function as a reservoir of membrane that can be added to the fiber surface at very long sarcomere lengths (Dulhunty and Franzini-Armstrong, '751, although Poulos et al. ('86) have produced evidence to the contrary. In the studies on the reservoir roles of transverse folding and caveolae, it was assumed that fiber shape did not change as a function of sarcomere length. The latter assumption was apparently validated by the optical sectioning studies of Dulhunty and Franzini-Armstrong ('77), which indicated that the cross-sectional shape of isolated fibers deviated very slightly from a circle and was remarkably constant between fibers and a t all sarcomere lengths. In contrast, Blinks ('65), using similar techniques, observed a large variation in cross-sectional shape both between fibers and as a function of sarcomere length. The differences in the two studies might be due to the use of fibers from different muscles and dif-
212
J.A. TROTTER
ferent species of frogs. No general conclusion concerning whether fiber shape or surface area remain constant or change with sarcomere length could be drawn from these studies. The studies reported here were designed to estimate the amount of fiber surface associated with the tapered ends of cat skeletal muscle fibers in situ a n d to determine whether this parameter changes as the muscle changes length. The data are consistent with a constant surface area and a changing cross-sectional shape as fibers change length. MATERIALS AND METHODS
Three female cats weighing between 1.5 and 2.5 kg were anesthetized with ketamine given intramuscularly followed by Nembutal given intravenously. The skin of the right hind leg was removed, and the exposed biceps femoris muscle was kept moist with physiological saline during subsequent procedures. Longitudinal full thickness incisions were made in the anterior and posterior regions of the muscle to produce strips approximately 2 mm wide by 6 cm long. Sutures were tied to each end of a strip, and the limb was positioned to place the muscle in short, long, and intermediate lengths. At each length the sutures of one strip were tied to a wooden stick so that the muscle strip would remain approximately at that length as it was fixed. This procedure produced specimens from each cat in which the muscle fibers were at short, long, and intermediate sarcomere lengths within their normal anatomical range. Each strip was then excised from the muscle and placed in 2.5% glutaraldehyde, 0.2 M sodium cacodylate buffer, pH 7.2, at room temperature. This produced strips of anterior and posterior biceps femoris muscle fixed a t sarcomere lengths representing the full physiological range. After the tissues had been excised, the cats were sacrificed by an overdose of Nembutal. After approximately 30 min the strips were removed from the sticks and sliced with the aid of a dissectingmicroscope into strips about 1mm square by 3 mm long for transmission electron microscopy and about 1mm square by 2 cm long for scanning electron microscopy. Fixation was then continued overnight. Specimens for analysis by transmission electron microscopy were soaked for 1hr in fresh fixative of the same composition containing 0.2% tannic acid. They were then rinsed overnight in several changes of 0.2 M cacodylate buffer. Prior to further processing, the strips were sliced into two pieces.
One piece was later used for cross-sectional analyses; the other was used for longitudinal analyses, including sarcomere length determinations. These pieces were subsequently labeled and processed as matched pairs. The tissues were next postfixed in 1% osmium tetroxide, 0.1 M cacodylate buffer for 1 hr, and rinsed in water for 1 hr. After dehydration in a graded ethanol series the specimens were infiltrated with Spurr’s resin, which was polymerized for 24 h r at 65°C. Transverse and longitudinal ultrathin sections, approximately 100 nm thick, were produced using diamond knives and were imaged a t original magnifications of either ~ 1 , 0 0 0or x 5,000 in a n Hitachi H600 transmission electron microscope after staining with uranyl acetate and lead citrate. To prepare isolated muscle fibers for analysis in the scanning electron microscope, the strips of biceps femoris that had been fixed overnight were rinsed in several changes of cacodylate buffer for several hours and were then placed in ice cold 50% glycerol, 0.02% sodium azide, 0.1M sodium phosphate buffer, pH 7.0 (determined at room temperature). They were gently agitated for several days in the cold room and then stored in this solution at 4°C for up to 6 months. Single fiber segments were prepared from the strips by incubating them for 40-60 min at 60°C in 9.6 M HCl. When individual fibers were seen to have begun to separate from the strips the acid was rapidly diluted with a large volume of water a t room temperature. Fiber segments with tapered ends were selected with the aid of a dissecting microscope and were placed on freshly prepared, gelatin-coated, 12 mm round coverslips. The gelatin was hardened by exposure to 2.5% glutaraldehyde in water, after which the coverslips were passed sequentially through several changes of water and a graded ethanol series. After several changes of absolute ethanol, the fiber segments were dried from CO, by the critical point method. The coverslips were fixed onto aluminum stubs using silver paint, rendered conductive by a thin coating of goldipalladium deposited in a cold cathode sputter coater, and imaged in a n Hitachi S800 field emission electron microscope. The “taper angle” of a fiber end is the angle a t which the fiber surface intersects a line drawn parallel to the longitudinal axis of the fiber. To estimate this angle, scanning electron micrographs were made of the tapered portions of the fibers. Several widths
2 13
TAPERED MUSCLE FIBER SHAPE
were measured along the tapered regions, and the longitudinal distances between the points of measurement were also determined. The slope of the regression line relating the half-widths to the cumulative length is the tangent of the taper angle. The angle is therefore found by taking the arctangent of the regression slope, as illustrated in Figure 1. The ratio between the measured perimeter of a transversely sectioned fiber and the circumference of a circle having the same area as that of the fiber cross section is a quantitative expression of the shape of the crosssectional profile of the fiber. This measure, designated a form factor (FF), estimates the extent to which the profile deviates from circularity. Because a circle has the minimum perimeter possible for a given area, FF values are always 1 or greater. Because all the fibers analyzed had FF > 1,the use of a single diameter measurement to express fiber size was impossible. The cross-sectional area could have been used as an accurate way to express fiber size, but the values for area are not commonly used for this purpose. Therefare, the length of the diameter of the circle with the same area as the fiber profile was calculated and designated the equivalent diameter (ED). Sarcomere lengths (SL) were determined from longitudinal sections by taking the average distance between at least 10 Z-line pairs in a myofibril. SL were determined from electron micrographs of longitudinal sections; FF and ED were determined from measurements of transverse sections of blocks that were paired with those from which the longi-
1
L4
tudinal sections were taken, as described above. All length and area measurements were made using Sigmascan software and digitizing tablet (Jandel Scientific, Corte Madera, CA) and an IBM PS/2 Model 50 microcomputer. The resolution of the digitizing tablet was adjusted with the software to be either 0.76 or 6.1 mm. Statistical analyses followed standard parametric procedures for linear regression and comparison of slopes by analysis of covariance (Zar, '84). Unless otherwise specified, statements of statistical significance are based o n P I0.05. RESULTS
Taper angles of isolated fibers The tapered ends of isolated anterior and posterior biceps femoris fibers analyzed by SEM showed no prominent surface folds or irregularities (Fig. 2). The tapering was not uniform, and some regions of the tapering portion had no obvious taper. Considering only the tapered portions of the fibers, the range of values for taper angle of the 11 fibers analyzed was 0.16" to 1.18";the mean value was 0.72". The range of taper angle values is shown in Figure 3. Shape of fiber cross sections The cross-sectional shapes of both large and small fiber profiles varied between round and angular, and the outlines varied between smooth and rough. Fibers with all degrees of these characteristics were seen at all sarco-
t
Fig. 1. Method of estimating taper angle from scanning electron micrographs. Arctan I(W,/Z)/LJ = 8, for i = 1 , 2 , 3 , . . . =.
214
J.A. TROTTER
Fig. 2. Scanning electron micrographs of the tapered end of an isolated biceps femoris muscle fiber. Note the absence of surface specializations and the gradual taper
toward the end. Bar = 50 pm in upper panel and 10 p m in lower panel.
TAPERED MUSCLE FIBER SHAPE 1.6
I
1.4
-t
1.2
.)
b 1.0
3
Y
2
3-
' LL: W
0.a 0.6
0.4 0.2
0.0 1
2
3
1
1 ii
4
5
6
7
8
9 1 0 1 1
FIEER NUMBER
Fig. 3. Bargraph of the taper angles of 11 fibers similar to that shown in Figure 1. Error bars show the standard error nf the angle measurements made on each
fiber.
mere lengths. Figures 4 and 5 are tracings of smooth, rough, round, and angular profiles of fibers with short (Fig. 4) and longer (Fig. 5) sarcomere lengths. The profiles from fibers with short sarcomere lengths appeared to be both rougher and more angular than those from longer fibers (compare profiles on the left sides of Figs. 4 and 5). The smaller profiles also appeared to be slightly rougher and more angular than the larger ones (compare the upper left panels of Figs. 4 and 5). The shape of the fibers was quantified using the FF (described in Materials and Methods). The resolution of the digitizing tablet was set at 0.76 mm, and the micrographs analyzed were printed at a final magnification of x 12,700. Hence the measurement resolution was 60 nm. The distribution of FF for the tapered ends (ED between 2.7 and 10.1 +m) of 173 fibers is shown as a histogram in Figure 6a. The distribution is prominently skewed, indicating that a large number of fibers have very large FF. In contrast, a histogram of the form factors of 247 fibers with ED greater than 10 pm shows a distribution that is much less skewed (Fig. 6b). There is thus a tendency for the fibers to assume less circular profiles as they taper toward their ends.
Change of fiber shape with sarcornere length The FF data shown in Figure 6 came from three cats. To determine whether there was a
215
correlation between sarcomere length SL and FF, regression analysis was applied to these data from each cat (Table 1). SL values were obtained from electron micrographs of fibers in the same region as those used for crosssectional analysis, but not on the identical fibers, as explained in Materials and Methods. For reasons given in the Discussion, the data were analyzed in logarithmic form. For each cat a significant correlation was seen between FF and SL. Analysis of covariance indicated that the slopes of the three regressions were not statistically different (Table 1).The slope ofthe common data was -0.268. Graphs of the data for the three cats and for the pooled data are shown in Figure 7. There is a large scatter in the data for all sarcomere lengths because of both an inherent variability of profile shape and the tendency of small fiber profiles to have less circular shapes. Nevertheless, there is a highly significant negative correlation between FF and SL (P < 0.005) for both the individual cats and for the pooled data, and the R2 values indicate that between 12% and 24% of the variance in FF can be attributed to the change in SL. The equation of the line calculated from the common data is lnFF = 0.54 - 0.2681nSL. A change in SL from 3 to 2 pm would thus produce a n 11% increase in FF. A similar analysis of the nontapered portions (i.e., profiles with ED > 12 pm) of fibers in the same specimens that were used for the above measurements yielded a common slope of -0.21, with an R2 of 0.25 (N = 143; P < 0.001). Both tapered and nontapered regions thus change shape as a function of SL. Source of shape change with change in SL The higher F F values associated with shorter sarcomere lengths could have been caused by largc-scale or small-scale shape changes. Examples of large-scale changes are increases in the ellipticity or angularity of the profile. A small-scale shape change would be seen as increased roughness in the fiber profile. To discriminate between these possibilities, the FF of the 173 fiber profiles discussed above were determined at a resolution of 480 nm, and the results were compared with the 60 nm determinations described above. An example of the tracings produced by these two settings of the digitizing tablet is shown in Figure 8. Large-scale shape changes are detected at both resolutions, but only the 60 nm tracing detects roughness changes. The relative contribution of shape
216
J.A. TROTTER
FF = 1.6 3D = 5 . 4
?F = 1.1
PD =
4.4
?F = 1.5
Fig. 4. Cross-sectional profiles of muscle fibers fixed at short sarcomere lengths (average SL = 1.9 pm). These profiles were chosen to represent the range of shapes of both small and large profiles. The profiles on the left are
more angular and have rougher surfaces; those on the right are smoother and more nearly round. The form factor (FF) and equivalent diameter (ED) values are indicated in each panel. Bars = 1pm.
217
TAPERED MUSCLE FIBER SHAPE
FF ED
FF = 1.4 ED = 2 . 8
= 1.2 = 3.6
r I
'F = 1.1 1D = 9.9
~
~~~
Fig. 5 . As in Figure 4,except that these fibcrs were fixed at longer sarcomere lengths (average SL = 2.5 km). Bars = 1 km.
218
j!
J.A. TROTTER
,
25
I
1
I
n
a
20 rn w K
15 0
K
w
I
1
I
z 3 1
LL
3z
I
10
20 5 10
0
I]
1 .o
0
1.2
1.4
1.6
1 .o
1.8
FORM FACTOR
FORM FACTOR
Fig. 6. Histograms of the distributions of form factor values for the tapered ends of muscle fibers (a)and the middle regions of fibers (b).Data for a were taken from profiles with equivalent diameters (ED) 5 10.1 IJ-m;fibers used in b had ED 2 10.2 pm.
changes a t these two scales was estimated by comparing the slopes of the regression lines through the lnFF vs. lnSL data for each cat (Table 2). I t can be seen from Table 2 that between 66% (cats 1and 2) and 72% (cat 3) of the change in FF is accounted for by largescale shape changes, and the rest is produced by increases in roughness.
Contribution of longitudinal ruffling The above measurements consider only the cross-sectional shape of the fibers. To determine whether longitudinal ruffling, i.e., accordion-like folding of the fiber surface, also varies with sarcomere length, 68 fibers were examined in longitudinal sections, representing SLs between 1.75 and 3.58 pm. Two points on the fiber periphery were chosen, and the TABLE 1. Regression Cat
N
length of the sarcolemma between the two points was measured, as was the straight line between the two points. The ratio between these two measurements quantifies the ruffling of the fiber surface and is thus a ruffling ratio (RR). RR as a function of SL is shown graphically in Figure 9. A highly significant correlation exists between these two parameters (P < 0.001). The equation of the regression line is RR = 1.03 - 0.0089SL. From this equation it can be calculated that a change in SL from 3 to 2 pm increases the RR, and consequently the surface area, by 0.89%. In contrast, as noted above, transverse shape changes over this same range of SL increases surface area by 11%over that minimally required to enclose the fiber volume. of
M F F ) on ln(SL)
R
R*
b
SE ofb
P
0.144 0.239 0.119
-0.356 -0.273 -0.229
0.119 0.075 0.073
< 0.005
1
55
0.380
2 3
44 74
0.489 0.345
< 0.001 < 0.005
Analysis of covariance Source Cat 1 Cat 2 Cat 3 Pooled Common
Sum (X') 1.0138 2.5842 2.5603
Sum (XY) -0.362 -0.705 -0.568
6.1584
-1.654
Sum (P) 0.8943 0.8057 1.1238 2.8238
N 55 74
h -0.356 -0.273 -0.229
173
-0.268
44
Res.SS 0.7650 0.6130 0.9893 2.3673 2.3793
Res.DF 53 42
72 167 169
F = 0.4224; P > 0.5. Common slope = -0.268. N, number of fibers analyzed; R, correlation coefiicient;K2,coefficientof determination; b, slope of regression line; SE, standard error of b; P, probability. For an explanation of the terms used in the analysis of covariance, see pp. 300-302 in Zar ('84).
219
TAPERED MUSCLE FIBER SHAPE
2.0 1.8
k
l
1 I
1.6
l i
2
I
f
1. 1.4
1.2
OO 0
@8
-
8
0
0 0
0 0
x%
3
-
0
1.2
O :
-
U 0
0 0
1.0
'
I
I
0
0
I
1
"1 0
-
:
I
SARCOMERE LENGTH bm)
SARCOMERE LENGTH (pm) 1 .a
80
I
3
0 0
0
1.6
0 1
0
e
0
0
008
0 0
K
E E z
1.8
0 0
1.4
1.6 00
0
0
2
0 0
. -
0-0
0
0
0 0
8 8 O
:%O
0
I
5
e
1.4 .
1.2 1.2 0
1 .c
I
I
2.0
2.5
O
'
I
0
0
I 3.0
3.5
SARCOMERE LENGTH (pn)
Fig. 7. Scatter plots of FF values in relationship to sarcomere lengths. 1-3: Data from individual cats; c: combined data from the three cats. The least-squares
linear regression line, together with the 958 confidence interval for the slope of the line, is shown for the combined data.
DISCUSSION
amount of surface area amplification is approximately equal to lisin0, where 0 is the taper angle.' Therefore, the value of this ratio for the 11ends analyzed varies between 49 (for 0 = 1.18") and 358 (for 0 = 0.16"). The tension produced by a muscle fiber is proportional to its cross-sectional area. If the tension is transmitted across the tapered sur-
Functional consequences of smooth tapered ends The absence of any junctional specializations in the cat muscle fibers used in these studies requires that tension be transmitted across the smooth surface membrane of the tapered ends (Trotter, '90). The small taper angle of the ends produces an interface between muscle and connective tissue that is loaded almost completely in shear and also greatly amplifies the surface area for tension transmission relative to the cross-sectional area of the fiber (Trotter, '90). This is because the area of the curved surface of a cone is greater than the area of its base, and the
'Given a right cone with b,aseradius R, and height H, the area of the curved surface is 7iR )/(R'+H2), and the area of the base i s 7iR2.The wrfacebase ratio is then .;iK ~(R2+I12))l.iiR7, which Because d(R2+H') is the hypotenuse of simplifies to b/(R2+HZMR. a right triangle, it can also be expressed as RisinO, where O i s the angle bctween the tapered side and a line parallel to the height. Thus the surfaceibase ratio can be expressed as (RlsinOYR, or l/sinfi.
-
220
J.A.TROTTER
Fig. 8. Profile of fiber from Figure 4, traced at ti0 nm (left)and 480 nm (right)resolution. The major shape features are present in the 480 nm tracing, but the roughness of the periphery is smoothed out.
face, the stress in this surface must vary as the inverse of the surface area, since stress = forcelarea. The tapering of the end is therefore an effective mechanism of stress reduction at the interface between muscle fiber and connective tissue. Because of the majpitude of the stress reduction produced by the tapered end, the load applied to the connective tissue is small in comparison with what it would be if it were concentrated in a smaller area. This reduction in load could make it possible for the collagen content of the endomysium to be lower, and for the fibril orientation to be more nearly isotropic, than would otherwise be the case.
TABLE 2. Regression slopes at two resolutions Source
60 nm
480 nm
480160
Cat 1 Cat 2 Cat 3
-0.356
-0.235
-0.273
-0.181 -0.164
0.66 0.66
-0.229
0.72
Slopes of regression line at 60 and 480 nm resolution are shown. Portion of slope at 60 nm present in slope at 480 nm is also gven.
0 O
O
0
1.01
1.00 1.5
1 4.0
2.0 2.5 3.0 SARCOMERE LENGTH (M , )I
3.5
Fig. 9. Scatter plot of the relationship between the longitudinal ruffling ratio and sarcomere length. The least-squares linear regression line is shown, together with the 95% confidence interval for the slope. The R' value for this regression is 0.204, N is 68, and P < 0.001.
221
TAPERED MUSCLE FIBER SHAPE
Shape change and constant surface area The data in this paper suggest that the surface area of a skeletal muscle fiber remains constant a t all SL and that the fiber maintains a constant surface area by varying its shape. The necessity of a shape change to keep both surface area and volume constant in a cylinder that varies in length can be appreciated by considering that, in geometrically similar structures (i.e., structures of the same shape), surface area (S) varies as the second power of the length (L), while volume (V) varies as the third power of I,. The ratio S/V therefore varies as (see SchmidtNielsen, ’84). To maintain both S and V constant while L changes, geometric similarity cannot be maintained: The shape has to change. The amount of shape change expected for a cylinder that remains isovolumic and isoareal as it changes length can be predicted from the following argument. ‘FF, the quantitative measure of shape change in a cylinder that changes length, was defined in Materials and Methods as the ratio of the perimeter of the cross section of a cylinder to the circumference of a circle with the same area. The FF expected for a shortening cylinder that maintains constant volume and surface area is derived as follows. Suppose a muscle fiber a t its maximum length is a uniform cylinder with L, = length (=l), A, = cross-sectional area, Po = perimeter, and LOP, = surface area. The fiber shortens to a new length, X (X < Lo), maintaining constant area and volume. At length X, the cross-sectional area = A,, and the perimeter of the cross section =$ P,. Then, because surface area and volume are constant: lA,
= XA,
LoPo= lP,,
XP,.
L,A,
=
Therefore Ax = A,/X
and Px = P,iX.
FF has been defined previously as P/C, where C is the circumference of the circle with area A. At L = LO= 1,P = Po = ~TR,,,and C = C, = ~ T R , where , R, is the radius, because at ‘This derivation of’FF for a shortening cylinder was provided by Dr. Peter Purslow, AFRC Institute nf Food Research (Bristol Laboratory, Langford, Bristol, U.K.).
L, the cross section is circular. Therefore, FF,=P,/C,= 1 . A t L = X , P = P x = 2 n R , , / X . Cross-sectional area at length X, A,, is equal to the area of a circle, radius R,, such that A, = IT^. The initial area = A, = T R ~ . Since A, = AJX, T~R;= vR;/X
and R,
=
&/Xo’.
Therefore, the circumference, C,, of the circle of area A, is C,
=
‘.
2vRx = XTR,/X~
At L,, then, FF, = P,/C, = (2~rR,/X)/(2aR,/ X”5 ) . Simplification of the last equation yields FF,
=
X-”.
The derivation of FF for a right cone, assuming that it is also circular in cross section at its maximum length, Lo, differs only in the beginning steps. Surface area is equal to aR,,(R~+ L;)”, where R,, is the radius of the base of the cone; for large ratios of L,/R,, such as those that are characteristic of the muscle fibers used in these studies, this expression is approximately equal to TR,L,, or to P,L,/2. Volume is equal to nR;L,,/3, OF A&,,/3. Then, because volume is constant, L,A,/3
=
1&/3
=
XAJ3
and thus L,A, = lA,
= XA,.
Similarly, because surface area is constant LOPOI:!= 1P0/2 = XP,i2
and thus L”P, = lPo = XP,
Therefore, A, =.AJX, and P, = P,/X, and the derivation continues as above for cylinders. F F for a cone is thus identical to that for a cylinder. It can be shown that these relationships hold for the volume and surface area of cones of any cross-sectional shape. For both cylinders and cones, then, the relationship between FF and X is a power function, the slope of which is -0.5. In logarithmic notation: lnFF = 0.51nX. Hence, as the muscle fiber shortens, FF must become greater. For example, if the cylinder were to shorten from 3 units to 2 units, X would be 0.67 and FF would be 1.22. That is, F F would increase by 22%. The lnFF vs. lnSL regression of the common data in the present study yielded a slope of -0.268. Both the sign of this slope and its
222
J.A. TROTTER
approximate magnitude are consistent with a change in shape in order to maintain constant surface area. The discrepancy between the calculated slope and the theoretical one may be accounted for in part by the large scatter in the FF data caused by the wide variation in fiber shape at all sarcomere lengths and by the inclusion in the FF vs. SL graph of all fibers with equivalent diameters I 10.1 p,m, although there was a n increased FF associated with smaller ED. It could also be true that constancy of surface area is only approximately maintained. The contribution of ruffling to a constant surface area is very small. Previous investigators have described "festoons" along the fiber surface of much greater magnitude than those seen in the present study (Street, '83; Shear and Bloch, '85). In both of the cited papers, however, the large festoons were seen in fibers with sarcomere lengths of about 1 km and were not seen in fibers within the physiological range of lengths. In the present study there was a n order of magnitude greater change in transverse fiber shape than in longitudinal surface ruffling as fibers changed length. Approximately two thirds of the shape change appears to be due to large-scale adjustments in shape. How this comes about is unknown. Although it was not quantified, there did not seem to be much myofibril-free sarcoplasm beneath the sarcolemma, and shape changes would therefore seem to involve myofibrils as well as the nonfibrillar sarcoplasm. That is, the sarcolemma appears to be a tight-fitting skin that conforms to the shape of the underlying contractile structures rather than a loose bag that changes shape independently of the underlying structures. Thus there may be a mechanism by which the packing of myofibrils, or even the myofilament lattice, changes as a function of fiber length. It is also possible that the magnitude of the shape changes seen in these studies is partly due to constraints resulting from the packing together of the fibers in a tissue. There are presently no data on these questions.
Significance of a constant surface area The tendency to maintain a constant surface area is important because tension is transmitted across the surface of the tapered end to the endomysium (Trotter, '90). The perifiber matrix (endomysium) is a sheath composed of small collagen fibrils, and other connective tissue macromolecules, that sur-
rounds the muscle fibers (see Mayne and Sanderson, '85). If the endomysium is tightly adherent to the surface of the fiber, as appears to be the case (see Fields, '70, and Rapoport, '73, and references therein), it is to be expected that it will conform to the fiber shape. Furthermore, if the endomysial volume remains constant at all SL, as seems likely, then its thickness will also remain nearly constant, because thickness of the endomysial sheath is equal to (constant) volume of endomysium divided by (constant) area of fiber surface. If the endomysial thickness remains constant, then the separation between muscle fibers should also remain constant, a t all SL. Considering series-fibered muscles as fiber-composite materials, with muscle fibers representing the fibrous (or anisotropic) phase and the endomysium representing the amorphous (or isotropic) phase (Piggott, '80), the constancy of the distance between fibers may make important contributions to the mechanical stability of the muscle. ACKNOWLEDGMENTS
The technical aspects of these studies were all expertly performed by Rayanne Ozbaysal. Conversations with Carl Gans were very helpful in clarifying the arguments concerning constant surface area. This paper was greatly improved by the suggestions made by Peter Purslow and by four anonymous reviewers. Statistical aspects of this work were discussed with Dr. Betty Skipper, Professor of Biostatistics (University of New Mexico School of Medicine). This work was supported by NIH grant AR39922. LITERATURE CITED Abbott, H.C., and R.J. Baskin (1962) Volume changes in frog muscle during contraction. J. Physiol. (Land.) 161:379-391. Adrian, E.D. (1925) The spread ofactivity in the tenuissimus musclc of the cat and in other complex muscles. J. Physiol. (Lond.) 60:301-315. ADril. E.W., P.W. Brandt. and G.F. Elliott (1972) The myofilament lattice: Studies on isolated fibers. 11. The effects of osmotic strength, ionic concentration, and pH upon the unit cell volume. J. Cell Biol. 53r5345. Bardeen, C.R. (1903) Variations in the internal architecture of the m. obliquus abdominis externus in certain animals. Anat. A n z . 23t241-249. Barrett, B. (1962) The length and mode of termination of individual muscle fibres in the human sartorius and posterior femoral muscles. Acta Anat. 48,242-257. Blinks, J.R. (1965) Influence of osmotic strength on cross-section and volume of isolated single muscle fibers. J. Physiol. (Lond.) 177r42-57. Chanaud, C.M., C.A. Pratt, and G.E. Loeb (1990) Functionally complex muscles of the cat hindlimb. 11. Me-
TAPERED MUSCLE FIBER SHAPE chanical and architectural heterogeneity in the biceps femoris. Exp. Brain Res. (in press). Dulhunty, A.F., and C. Frdnzini-Armstrong (1975) The relative contributions of the folds and caveolae to the surface membrane of frog skeletal muscle fibres at different sarcomere lengths. J. Physiol. (Lond.) 250: 513-539. Dulhunty, A.F., and C. Franzini-Armstrong (1977) The passive electrical properties of frog skeletal muscle fibres at different sarcomere lengths J. Physiol. 266: 687-711. Edman, K.A.P., and J.C. Hwang (1977) The forcevelocity relationship in vertebrate muscle fibers at varied tonicity of the extracellular medium. J. Physiol. (Lond.) 269;255-272. Elliott, G.F.. J. L o w . and C.R. Worthineton (1963) An X-ray and light-diffraction study of the filament lattice of striated muscle in the living state and in rigor. J. Mol. Biol. 63295-305. Fields, R.W. (1970) Mechanical properties of the frog sarcolemma. Biophys. J. 10:462-480. Gans, C., G.E. Loeb, and F. De Vr6.e (1989) Architecture and consequent physiologicalproperties of the semitendinosus muscle in domestic goats. J. Morphol. 199:287297. Gaunt, A.S., and C. Gans (1990) Architecture of chicken muscles: Short fiber patterns and their ontogeny. Trans. R. SOC. Lond. [Biol.]2401351-362. Huher, G.C. (1916) On the form and arrangement in fasciculi of striated voluntary muscle fibers. Anat. Rec. 11:149-168. Huxley, H.E. (1953) X-ray analysis and the problem of muscle. Proc. R. Soc. Lond. TBiol.1 141159-62. Loeb, G.E., C.A. Pratt, C.M. Chanaud, and F.J.R. Richmond (1987) Distribution and innervation of short,
223
interdigitated muscle fibers in parallel-fibered musclcs ofthe cat hindlimb. J. Morphol. 19111-15. Mayne, R., and R.D. Sanderson (1985) The extracellular matrix of skeletal muscle. Collagen Rel. Res. 51449468. Needham, D.M. (1971) Machina Carnis. Cambridge: Cambridge University Press. Piggott, M.R. (1980) Load Bearing Fiber Composites. Oxford: Pergamon Press. Poulos, A.C., J.E. Rash, and J.K. Elmund (1986) Ultrarapid freezing reveals that skeletal muscle caveolae are semipermanent structures. J. Ultrastruct. Mol. Struct. Res. 96:114-124. Rapoport, S.I. (1973) The anisotropic elastic properties of the sarcolemma of the frog semitendinosus muscle fiber. Biophys. J. 13t14-36. Richmond, F.J.R., D.R.R. MacGillis,andD.A. Scott (1985) Muscle-fiber compartmentalization in cat splenius muscle. J. Neurophysiol. 533368-885. Schmidt-Nielsen, K. (1984) Scaling. Why Is Animal Size So Important. Cambridge: Cambridge University Press. Shear, C.R. and R.J. Bloch (1985) Vinculin in subsarcolemmal densities in chicken skeletal muscle: Localization and relationship to intracellular and extracellular structures.J . Cell Biol. 101:240-256. Street, S.F. (1983) Lateral transmission of tension in frog myofibers: A myofibrillar network and transvcrse cytoskeletal connections are possible transmitters. J. Cell. Physiol. 114:346364. Trotter, J.A. (1990) Interfiber tension transmission in scrics-fibcred muscles of the cat hindlimb. J. Morphol. 206(3) (in press). Zar, J.H. (1984) Biostatistical Analysis (2nd ed).Englewood Cliffs, NJ: Prentice-Hall.
