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First published online September 14, 2007
Journal of Experimental Biology 210, 3395-3406 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.007062

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Uniform strain in broad muscles: active and passive effects of the twisted tendon of the spotted ratfish Hydrolagus colliei

Mason N. Dean^1,*, Emanuel Azizi² and Adam P. Summers¹

¹ Ecology and Evolutionary Biology, University of California Irvine, 321 Steinhaus Hall, Irvine CA 92697-2525, USA
² Department of Ecology & Evolutionary Biology, 80 Waterman Street, Box G-B, Brown University, Providence, RI 02912, USA

View larger version (25K): [in this window] [in a new window]   Fig. 1. Hypothetical mechanism for reduction of strain in a broad muscle. The anterior (red) and posterior (blue) margins of the muscle exhibit the most extreme strains, so we will consider these linear tensile elements as the boundaries of the parameter space (A). In a typical `untwisted' morphology (B), the anterior face inserts furthest from the joint and therefore experiences greater strains, likely resulting in suboptimal force production (the region of optimal force production is indicated by the vertical gray bar in B and C). Conversely, in the `twisted' conditions (C), the two faces undergo similar strains and therefore their force production capability is similar and high. The anterior/posterior coloring scheme used here is employed throughout the remainder of the figures. MTC, musculotendon complex.

View larger version (71K): [in this window] [in a new window]   Fig. 2. Morphology of the cranial musculature in the head of the spotted ratfish Hydrolagus colliei. (A) The schematic on the right illustrates the musculature labeled on the left. (B) The twisted portion of the tendon (circled) expanded. Although all three adductors insert on the lower jaw, only the anterior adductor (AMA-) exhibits a pronounced twist in its tendon (its approximate middle indicated throughout the figure by a white arrow) where the anterior face (red arrow) inserts more posteriorly than the posterior face (blue arrow). The Hydrolagus schematic shown here is used in the remainder of the figures with the anterior jaw adductor isolated for clarity. AMA-ß, posterior division of the anterior adductor mandibula; AMP, adductor mandibula posterior.

View larger version (23K): [in this window] [in a new window]   Fig. 3. Geometric model of the anterior division of the adductor mandibula of the ratfish. The model can be easily adjusted to represent the twisted and untwisted conditions by changing the insertion points of the anterior and posterior faces (see Appendix). We examined the effect of gape angle on strain, in both twisted and untwisted conditions, by calculating the lengths of the anterior and posterior muscle faces at gape angles from 0–40°.

View larger version (29K): [in this window] [in a new window]   Fig. 4. (A) The effect of gape angle and tendon morphology on muscle strain in the anterior division of the adductor mandibula of the ratfish. In the untwisted condition (broken lines), strain increases much more rapidly in the anterior face (red line) than the posterior face (blue line) as the mouth is opened to maximum gape (40°). As a result, at maximum gape, there exists more than a 50% difference in strain between the faces of the muscle (relative to their respective resting lengths). This is an order of magnitude difference over what is seen in the twisted condition, where the strain differences in the two faces are never more than 5.5%. These results are represented schematically in (B) where strain differences (S) between the anterior and posterior faces of the muscle in each condition (twisted and untwisted) are colored in a gradient from yellow (equal strains) to green (an anterior face strain that is greater by 56% of resting length). AFS, anterior face strain; PFS, posterior face strain.

View larger version (10K): [in this window] [in a new window]   Fig. 5. The effect of insertion angle on the proportion of contractile force in the direction of bite force (effective force). Insertion angles closer to 90° exert 100% of their force normal to the jaw (in the same direction as bite force) and therefore have an effective force of 1.0. The twisted and untwisted conditions are modeled using the same insertion points and therefore have the same in-lever moment arms; however, the twisting of the tendon results in shallower insertion angles and therefore a lower resting effective force. As gape increases from 0 to 40°, insertion angle increases for the anterior face and decreases for the posterior face of both conditions. Because the anterior face of the twisted condition has a resting insertion angle of less than 90°, its effective force is highest at a gape of approximately 10°. Although the posterior face of the twisted system is always less efficient than its untwisted counterpart, its anterior face remains more efficient than the untwisted anterior face for larger gapes, resulting in an eventual equality of average effective force for both conditions. Ant., anterior; Post., posterior; Avg., average.

View larger version (92K): [in this window] [in a new window]   Fig. 6. The effect of gape angle and adductor mandibula geometry on regional strain variation (S). In all panels, gape angle is on the x-axis, theoretical manipulations of morphology are on the y-axis and S is on the z-axis. (A,B) Strain fields resulting from an MTC that is 24.5% muscle and (C,D) an MTC that is 100% muscle. (A,C) Modifications in the width of the muscle insertion; (B,D) the distance of the posterior insertion of muscle from the jaw joint. z-axis variables represent strain differences (S) between the anterior and posterior faces of the muscle, forming a topographic representation of strain heterogeneity. The shaded plateau region of each graph indicates morphologies where strains in the anterior and posterior muscle faces are nearly similar (±1.0% S in A,B; ±0.5% S in C,D). The broken green line represents natural morphology/geometry of the adductor mandibula of the ratfish. This natural morphology generally corresponds to regions of low strain variation across a broad range of gape angles.

View larger version (37K): [in this window] [in a new window]   Fig. 7. The effect of sarcomere length and tendon morphology on active and passive force production in the anterior division of the adductor mandibula of the ratfish. Sarcomere length is estimated from predicted muscle strains during jaw opening (see Fig. 5) and a resting length of 2.0 µm. In the untwisted condition (A,B), passive tension increases more rapidly in the anterior face than in the posterior face as the jaw is opened (A). This results in the two portions of the muscle beginning their active tension generation at disparate points on their active curves (B) and having heterogeneous force production capabilities, lowering the whole muscle force output at prey contact. In contrast, the faces of the muscle in the twisted condition (C,D) occupy similar portions on both their active (D) and passive (C) tension curves, allowing more optimal active tension generation and a wider gape without the detrimental effects of high passive tension forces.