Previously, I explained in general terms what Alexander’s formula is and how you can (it was claimed) use it to calculate the speed of a dinosaur. Today, I’ll talk about the biomechanical modeling approach, then discuss some of the problems and uncertainties of both methods, and especially the issue that struck me as “just wrong” and illogical. And then we get to the meat: I’ll explain what the underlying assumptions of both methods are, why this basic tenet is wrong.
Dinosaur speeds – how do we determine them, part 2?
A smart and very detailed approach to estimating how fast a dinosaur could move is biomechanical modeling. Where Alexander’s formula (and, btw, it derivate, Thulborn’s formula for running gaits. More on that later.) uses tracks and extrapolation from living animals, biomechanical modeling looks at the skeletons and compares to extrapolations from extant animals.
A prime example, and the most detailed application I have so far seen, is the work done on Tyrannosaurus rex by John Hutchinson and colleagues using the modeling program SIMM by Musculographics. You can get an excellent introduction and summary of their first (Hutchinson and Garcia 2002) paper here in a popular article by A.A. Biewener. Simply said, John and his co-authors looked at the amount of extensor muscles in the leg, which counter the ground reaction force, and the moment arms of the muscles. There’s a lot of uncertainty involved, especially in the moment arms, but it was quite clear that a T. rex has way too little ability to create torques (force times moment arm) in the limb joints to run like a scaled-up chicken. The big problem for Rexy is that when you get bigger, volume (thus weight) goes up by the cube, whereas forces go up by the square (muscle cross section). So you need more than proportional muscle masses and/or moment arms. At the same time, your bones also grow in strength by the square, while weight goes up by the cube, so you need much stronger bones to support your much bigger weight, which adds even more weight…. it is a runaway thing, and large animals to a certain degree counter it by a more straight limb posture. But that has limits, and then you’re forced to abandon gaits that create very high loads on the limbs. Elephants don’t trot and gallop for a reason.
John and colleagues later followed up the short Nature article with a detailed and long paper in 2005, which took the model from 2D to 3D (Hutchinson et a. 2005). It was decidedly more accurate, and found even smaller moment arms. Oops!
Now, it must be said that potentially, the SIMM models used for these studies had too little cartilage in the joints, but correcting that can’t change the results significantly.
Fast dinosaurs – an open and shut case?
Overall, it looked like dinosaurs were pretty slow. But then I kept finding contradicting evidence. Aside from the “why didn’t they ever ADAPT for higher speeds?” question, what got me wondering was the huge moment arm of the caudofemoralis longus muscle (CFL). I don’t mean “huge”, I mean huge as in
In sauropods and hadrosaurs, the fourth trochanter, where the CFL attaches to the femur, is close to or halfway down the femur shaft. Add to that the fact that the CFL pulls in a nearly orthogonal direction, thus creating very little compression in the hip, and you have a system that can pull the entire limb back with a tremendous amount of power – for relatively little muscle mass. Remember, torque = force * moment arm! Double the latter, which requires a tiny shift in where you attach the muscle, and PRESTO you gain as much as you would from doubling the muscle cross section! That’s why mammals have the large cnemial crests and calcaneal tubers, they increase the forces for extending knee and ankle……. hm, isn’t that odd? I means, just plain weird? If it can be done so cheaply, why don’t dinosaurs have such a nifty tuber on the ankle? Why do most dinosaurs have small to inexistent cnemial crests on the fronts of their tibiae? And having a large bony patella can help as well – and ossifying a part of a large tendon, well, where’s the problem?
(image w/o text from wikipedia, public domain delcaration here)
A horse hind limb. There is quite a moment arm for the hip extensors; some come from, others wrap around the posterior process on the hip. But check that knee, and especially that ankle! WOW!
Now, the tuber calcanei is present in basal archosaurs, and although initially it did not function as a moment arm increase for the extensor muscles, it did take that role with the evolution of erect gaits (simplifying matters a bit here). Why oh why would dinosaurs do away with it, if that meant slowing themselves down?
Here’s a pic to illustrate things again. Plateosaurus engelhardti and Equus ferus caballus limbs side by side, with muscle paths and moment arms sketched in.I stole the Plateosaurus from Scott Hartmann (the go-to guy for accurate skeletal drawings). Much simpified, too.
Note that this is scaled to roughly same limb length. The horse is standing, thus the hip is higher up.
Now, that’s a clear pattern. I simplified that for my talk in a table, in which I gave numbers for “torque producing potential”. I arbitrarily chose the value 2 for the mammalian hip (no I didn’t, I used it so that I would only get integers for the others).
You can quibble if it should be 3-2-2 for mammals or what. No matter – the point is that dinosaurs had WAY stronger hips, but much weaker ankles.
OK, halfway there to my crazy theory. I kept banging my head against that “concrete wall across the autobahn”: what was WRONG? I had all this solid evidence from all the mentioned and many other publications, but the odd moment arm pattern, and the apparently “evolutionary failure” dinosaurs that couldn’t speed up. Oh, and then there’s the exceptions, smaller theropods on the line leading to birds, who throw away the CFL over time and suddenly produce tracks indicating high speeds…
I kept building models, running them, varying factors, variables, gaits – all to no avail. Things simply didn’t match!
From bad to worse
Over time, additional things emerged that didn’t add up either. I got interested in dinosaur tails not just because of the CFL, but also because of defense behavior. That led, in the end, to my paper on how Kentrosaurus might have wagged its tail. Baseball batter from hell is quite an appropriate summary. And in the course of it I realized that dinosaur tails were always drawn undermuscled. Anorexic. Dinos from the Sahel. Check Fig. 4.5 in that paper for a quick summary of paradigm compared to reality. Twice the cross section for the CFL means double the force! So dinosaurs were not just massively over-engineered in the hip, they were in fact over-engineered to the fare thee well and back again, twice over! And then Vivian Allen and colleagues came up with the same thing (Allen et al. 2009) in a different context, just (*blush*) based on WAY more data than my measly X-rays of a handful of reptiles and the Alligator sections I used in the Kentrosaurus paper (muchas gracias here, for the umpteenth time, to D. Ray Wilhite for them! You deserve a few gazillion more Thank Yous!). And Scott Persons’s thesis ended in a paper (Persons and Currie 2010) that focused on the CFL and found the exact same thing. Thus, I was NOT mad.
RTFP! stands for “read the f-cking paper!” I’d read them, many times. Some I had understood (I believed) right away, others were harder to digest. But one day I realized that I would have to keep digging away until I found out why things didn’t make sense. And I came to realize that Alexander’s formula, while troublesome in detail, was sound in principle. And the SIMM models were fine, too. So I began to look for underlying assumptions that could skew results. And it dawned on me that both lines of reasoning had one thing in common. They assumed that mammalian and avian limb kinematics (the way the limbs move) are similar to those of dinosaurs. In the words of A.A. Biewener from the pop artcile on Hutchinson and Garcia (2002):
“Their analysis rests on assumptions about the limb posture and the magnitude of reaction forces exerted by the ground on the limbs of Tyrannosaurus, and about the kinematic similarity between dinosaurs and living birds and mammals.“
Alexander (1976: p. 129) just says, as quoted before:
“The faster an animal walks or runs, the longer, in general, are its strides. [...] I have now obtained a relationship between speed, stride length and body size from observations of living animals and applied this to dinosaurs to achieve estimates of their speeds.”
In 2006, he clarified (and I like to think he did so because people used his formula unquestioning):
” The method cannot claim to be accurate. We cannot be certain that the relationship between relative stride length and Froude number was the same for dinosaurs as for mammals.“
He then proceeds to list other problems, but that’s beside my point. Basically, “The faster an animal walks or runs, the longer, in general, are its strides.” may not be true for dinosaurs, we just don’t know. And if not, then we can’t use the formula.
I mean, wow!
So if (a big “if” at that time to me) “The faster an animal walks or runs, the longer, in general, are its strides.” was not true, the required moments in the distal joints, knee and ankle, might not be as large as allometric scaling suggests……
More concrete wall – and a tiny breakthrough
So there I was trying to figure out if there really was a relationship between stride length and stride frequency in dinosaurs, and I made no progress whatsoever. It took me ages to come up with the simple expedient of creating a 3D walking dinosaur model (thus with known hip height), make it use maximum possible (reasonable) stride length, then see what kind of top speed Alexander’s formula would give. The result made it clear to me that no, we can’t use Alexander’s formula for typical dinosaurs. And now I’ll turn this into a cliffhanger: I’ll tell you what exactly the model told me – in the next post!
Alexander, R.M. 1976. Estimates of speeds of dinosaurs. Nature 261:129-130.
Allen, V., Paxton, H., and Hutchsinson, J.R. 2009. Variation in center of mass estimates for extant sauropsids and its importance for reconstructing inertial properties of extinct archosaurs. Anatomical Record, 292:1442-1461.
Hutchinson, J.R. amd Garcia, M. 2002. Tyrannosaurus was not a fast runner. Nature 415:1018-1021.
Hutchinson, J.R., Anderson, F.C., Blemker, S., and Delp, S.L. 2005. Analysis of hindlimb muscle moment arms in Tyrannosaurus rex using a three-dimensional musculoskeletal computer model: implications for stance, gait, and speed. Paleobiology, 31:676-701.
Persons, W.S. and Currie P.J. 2010. The tail of Tyrannosaurus: reassessing the size and locomotive importance of the m. caudofemoralis in non-avian theropods. The Anatomical Record: Advances in Integrative Anatomy and Evolutionary Biology 294(1):119–131.