- Marina Dimitrov
- Further understand the mechanism of energy transfer between the tuna’s internal locomotor structure and the surrounding fluid medium.
- Explore the interesting phenomenon that tuna control their swimming speed predominantly through tailbeat frequency rather than tailbeat amplitude or “stride length” – unlike kangaroos, for example.
- Cost of transport remains fairly constant over a range of speeds (1-2 bodylengths/second), while tailbeat frequency changes significantly (almost doubles) – inconsistent with theory of forcing an oscillator at close to its natural frequency for minimum energy expenditure.
- What is the “natural frequency” of the tuna locomotor system – specifically, the arrangement of red muscle myomeres, ribs, posterior oblique tendons (POTs), and great lateral tendon (GLT) that power cruising (Figure 3)?
- [later, summer research maybe] Is a more sophisticated model of “natural frequency” necessary, to incorporate interaction with the surrounding fluid?
- Is this frequency comparable to the tailbeat frequencies and swimming speeds most commonly observed in wild tuna?
- [later, summer research maybe] Is this frequency comparable to that of vortex shedding from a tuna’s body at cruising speeds?
- [another relevant bio study to do] Is a tuna muscle fiber’s force-velocity curve more forgiving than its force-length curve (hence the constant amplitude, variable frequency swimming style)?
Plan for Success
- I am already well-informed in tuna biomechanics (literature and personal experience).
- Start with the simplest possible model and increase complexity until it adequately represents the situation.
- One pin joint for tail fin, pulled by one tendon on either side (GLT equivalent, also representative of force generated by muscles earlier in body).
- Maybe need to represent bending of spine (although it is not as significant in tuna as it is in other fish) – progressively add more degrees of freedom and representative actuators as needed.
- If interested in complete POT system of anatomical complexity, can extend Matt Millard’s kinematic MATLAB model by adding tendon properties Melinda Cromie had measured, etc.
- Maybe measuring impulse response of passive system is not enough – then need to include muscle actuation.
- Compare frequency of model/simulation to tailbeat frequencies and swimming speeds most commonly observed in wild tuna – presumably similar. If not, reevaluate model assumptions.
- Support from experienced collaborators (Melinda Cromie, Matt Millard, John Dabiri).
I am currently working on building the following simple model in OpenSim:
This represents the last set of musculotendon actuators in the tuna locomotor system (see Figure 5, later, for actual anatomy).
- keel (ellipse) is currently fixed to the ground - in reality it moves back and forth driven by the rest of the fish (Figure 3, later), but I am focusing on the tail flick
- tail is pin joint - in reality a set of several vertebrae with higher flexibility than the fairly rigid keel made up of several almost-fused vertebrae
- two linear musculotendon actuators inserting on either side of the tail - in reality, some parts of the tendon also attach to the skin
- origin of actuators is fixed - in reality, the muscles anterior pull the others forward as they contract
Below is the OpenSim model so far.
Things still to do:
- add wrapping geometry for the keel (ellipse), so the muscles don't pass through it when the tail angle changes
- fill in correct mass and inertial properties for tail
- maybe get a cool shape
- correct coordinate limits and coordinate limit force values incorporating vertebrae bending research
- correct muscle parameters to represent tuna
- select values to feed into existing musculotendon model (some research indicates that shark red muscle operates similar to mammalian, and that tuna tendons are mammal-like)
- do some geometry for initial muscle conditions so not pre-stretched (or are they pre-stretched in tuna? hm)
Once I have constructed and verified the model, I will use it to determine the natural frequency of this simplified system. Validation will be necessary to determine whether this represents reality well enough.
Lingering roadblocks and ideas to overcome them:
- Is this simplified system representative of the real tuna locomotor system? Sure, I can find the natural frequency of the model, but how do I relate that back to the much more complicated biological system that includes many more actuators upstream?
- How to find "natural frequency"?
- Incorporate fluid forces as well? Or just biomechanics in isolation?
- Impulse response with no activation, just passive properties?
- Frequency sweep of muscle activations and measure tail amplitude? Would I then make the amplitude range higher than physiologically possible not to rail?
- Measure stiffness at many static angles with appropriate muscle activations by applying force and measuring deflection? (a Carmichael thought)
Extra Background for Reference
Tuna, one of the top predators in the ocean, swim with an incredible mix of power, speed, and efficiency. In this “thunniform” swimming, the front part of the body remains fairly straight. Bending the last part of the spine swings the stiff tail region right before the fin, called the peduncle, back and forth. The tail fin, in turn, bends relative to this peduncle, adding a characteristic flick to the tail strokes. Figure 1 shows this combined mode of oscillation.
(sorry, not published yet, but here is a video where you can get the idea)
Figure 1 – Tail motion of a swimming tuna (Dimitrov et al. in preparation). Note the two primary motions of interest: that of the peduncle at the end of the spine, and that of the tail fin relative to the peduncle.
This gives the tuna fine control over the interaction between vortices coming off the body and those generated by the tail, as modeled with computational fluid dynamics (CFD).1,2 Vortices are rotational patterns of fluid motion that a fish creates while swimming, generating force and leaving the equivalent of footprints in the water (Fig. 2).
Figure 2 – 3D fluid structure behind an oscillating tuna tail fin modeled using CFD (left), and a 2D slice of that (right), showing how the vortices form almost like “footprints”.2
During cruising, tuna swim by almost exclusively using a system of red muscle myomeres, ribs, and posterior oblique tendons (POTs), shown in Figure 3, which transmit muscle forces to the spine.3 The horizontal ribs coming out of each vertebra act as struts, allowing the muscle attached to each tendon to pull on a vertebra farther down the spine. Figure 4 shows how these muscles contract along the body throughout the swimming “stride.”
(sorry, not published yet, but here is a figure mashup from another paper that illustrates it)3
Figure 3 – A 2D engineering model (left) of the red muscle and posterior oblique tendon (POT) system in tuna (right), with the normally stacked muscles separated and folded out (Cromie et al. in preparation).
Figure 4 – Muscle activation patterns of a swimming yellowfin tuna. From the head towards the tail, the muscles on one side start contracting, until they are all contracted. This then repeats on the other side of the body.4
Posterior to this POT system, a bony keel jutting out horizontally from the vertebrae acts to increase the moment arm of the great lateral tendon (GLT), which attaches to the tail fin (Fig. 5). Passing over the keel rather than lying close to the backbone provides the GLT and associated muscle myomeres with greater mechanical advantage as they pull on the tail. The GLT also attaches to the skin at various points on the way back to the tail.
(sorry, not published yet, but here is a photo from another paper illustrating the anatomy)3
Figure 5 – The great lateral tendon (GLT, blue) and associated muscle myomeres (red) of a yellowfin tuna (Dimitrov et al. in preparation).
- Yang, L. & Su, Y. CFD simulation of flow features and vorticity structures in tuna-like swimming. China Ocean Eng. 25, 73–82 (2011).
- Zhou, K., Liu, J. & Chen, W. Numerical Study on Hydrodynamic Performance of Bionic Caudal Fin. Appl. Sci. 6, 15 (2016).
- Westneat, M. W., Hoese, W., Pell, C. A. & Wainwright, S. A. The horizontal septum: Mechanisms of force transfer in locomotion of scombrid fishes (Scombridae, Perciformes). J. Morphol. 217, 183–204 (1993).
- Altringham, J. D. & Shadwick, R. E. in Fish Physiology (ed. Stevens, B. B. and E.) 19, 313–344 (Academic Press, 2001).