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Research Article

A Computational Analysis of Limb and Body Dimensions in Tyrannosaurus rex with Implications for Locomotion, Ontogeny, and Growth

  • John R. Hutchinson mail,

    jrhutch@rvc.ac.uk

    Affiliation: Structure and Motion Laboratory, Department of Veterinary Basic Sciences, The Royal Veterinary College, Hatfield, Hertfordshire, United Kingdom

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  • Karl T. Bates,

    Affiliation: Department of Musculoskeletal Biology, Institute of Aging and Chronic Disease, University of Liverpool, Liverpool, United Kingdom

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  • Julia Molnar,

    Affiliation: Structure and Motion Laboratory, Department of Veterinary Basic Sciences, The Royal Veterinary College, Hatfield, Hertfordshire, United Kingdom

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  • Vivian Allen,

    Affiliation: Structure and Motion Laboratory, Department of Veterinary Basic Sciences, The Royal Veterinary College, Hatfield, Hertfordshire, United Kingdom

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  • Peter J. Makovicky

    Affiliation: Department of Geology, Field Museum of Natural History, Chicago, Illinois, United States of America

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  • Published: October 12, 2011
  • DOI: 10.1371/journal.pone.0026037
  • Published in PLOS ONE

Reader Comments (6)

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Fast running T. rex can not be completely ruled out

Posted by Manuel_Garrido on 11 Jan 2012 at 20:37 GMT

I think nobody pointed this before.

Hutchinson´s 2004 paper says that using MOR 555, the extensor muscle mass in one hindlimb for Tyrannosaurus rex needed to get G=2.5 was 21% of body mass (20,7% exactly). G is the ratio of the Ground Reaction Force to body
weight and it was set at 2.5, considering it a reasonable value for a high-speed running T. rex. In Hutchinson and Garcia 2002 it is stated that if T. rex could reach G of 2.5 then it woud imply a Froude Nunber of 16 and an average forward speed of almost 20ms-1 (45 mph, 72 kmph).

Well, Hutchinson et al 2011 paper sets the extensor muscle mass in one hindlimb for MOR 555 in 20,1%, almost de same needed to get G of 2.5 and speed of almost 72 mph.

I thought that a Froude number of 16 was high for a G of 2.5 so I prefered to obtain an approximation of T. rex speed if it could reach 2.5 G using Alexander (1979 and 1983) equations relating Ground Reaction Force and Duty factor the former and Duty factor and Froud number (Fr) in mammals the last. I did not find equations for other animals than mammals and this should be considered in the interpretation of the results. I obtained a Froud number of 6,79 and a speed of 12.9 ms-1 (28.8 mph, 46.4 kmph). I repeat this is an approximation.

The main problem Hutchinson et al 2011 points about high speeds in T. rex is that ankle extensor muscle masses may be the critical limit on running capacity. In Hutchinson 2004 the ankle extensor muscle mass needed to reach G of 2.5 is 8,3% of body mass, while in Hutchinson et al 2011 the ankle
extensor muscle mass obtained in MOR was 2.42-2.99%. But there are some points that do reduce the ankle muscle mass needed for high speeds y T. rex.

1. Limb posture in Hutchinson 2004 is crouched and this reduces the effective mechanical advantage (EMA). With less bent limbs EMA would be larger and this would lower the ankle extensor muscle mass needed. A datum to test is an EMA of 1, that of more than 300 Kg running mammals.

2. In Hutchinson et al 2011 is said that T. rex probably had hip and thigh muscles relatively larger than any extant animal's and that the limb antigravity muscles may have been as large as or even larger than those of ratite birds, which themselves have the most muscular limbs of any living animal. Maybe it could be the same for ankle extensor
muscles. Perhaps some signs of this could be found in the bones.

3. Ankle extensor muscles have long tendons that let then contract close to isometry and enable then to be highly pennated. Pennation is a mecanism that reduces muscle mass mantaining physiological cross-sectional area. This is difficult to detect in extinct animals but it is something that can reduce the ankle extensor muscle mass needed and should be taken in consideration.

For these reasons I think a fast running T. rex can not be completely ruled out and more work is needed.

References:

Hutchinson JR, Garcia M (2002) Tyrannosaurus was not a fast runner. Nature 415: 1018-1021
Hutchinson JR (2004a) Biomechanical modeling and sensitivity analysis of bipedal running ability. II. Extinct taxa. Journal of Morphology 262:441-461.
Hutchinson JR, Bates KT, Molnar J, Allen V, Makovicky PJ 2011) A Computational Analysis of Limb and Body Dimensions in Tyrannosaurus rex with Implications for Locomotion, Ontogeny, and Growth. PLoS ONE 6(10): e26037.doi:10.1371/journal.pone.0026037
Alexander (1979 and 1983). Jornal of Zoology. (sorry, I did not get the complete citations)

Manuel Garrido.
Madrid, Spain.

No competing interests declared.

RE: Fast running T. rex can not be completely ruled out

jrhutch replied to Manuel_Garrido on 12 Jan 2012 at 17:11 GMT

We thank the author for his comments, but we have considered all the points he raises in our current and previous work already (albeit some of them not explicitly in our papers) and still do not consider them to be sufficiently compelling to warrant more credulity for the fast-running hypothesis.

Garrido correctly emphasizes the ankle extensor muscle mass as the likely key limit, so our response focuses on that critical point and his three suggestions for means to reduce the ratio of muscle mass needed vs. muscle mass present and thereby increase the odds of T. rex being an adept runner:

1. We have dealt with this in numerous papers that estimated ankle muscle moment arms (r; ~0.09m) and ankle GRF moment arms (R; ~0.3 m). EMA is r/R, so it is around 0.3- there is no plausible way to get EMA near 1, which is the mean value for the whole limb in some large mammals. Using those values for dinosaurs is highly inappropriate because (1) mammalian ankle EMA is quite low relative to other joints (~50% the EMA of proximal joints), not around 1 even in large mammals (see data in Biewener, 1989; cited in our study), and (2) the long foot and cranially positioned centre of mass (and thus foot centre of pressure far from the ankle joint) mean that the value of R will always be much larger than the value of r. Gatesy et al. 2009 (cited in our study) dealt with this issue to some degree by analyzing all plausible limb poses and considering CoP/CoM interactions; their results strongly support this conclusion.

2. There is no evidence for the ankle extensors being unusually large in Tyrannosaurus, especially relative to large running birds such as ratites (extensor masses ~6% body mass/leg). In adults, the tibia is relatively short and not particularly stout for the size of the animal, showing no unusual features promoting an apomorphically large set of ankle extensors. We made the most plausible estimates for muscle mass based on leg proportions, analogies with living animals, and comparative anatomy, incorporating all relevant evidence. These estimates like any other involve some subjectivity but do not approach 6-8% body mass in any plausible models. We thus see no plausible argument for Tyrannosaurus adults having very large ankle extensor muscles. Juveniles may well have had relatively larger muscles, although as we noted in our paper this is somewhat ambiguous.

3. Our models have considered muscle pennation and varying fascicle lengths in great detail- Hutchinson (2004) in particular treats this issue. Furthermore, the work of Sellers and Manning (2007; cited in our study) provides somewhat independent bolstering of the plausibility of our past estimates, by using joint range of motion to estimate fascicle lengths and finding values (and running speeds) similar to our more crude estimates. There is no evidence that points in the direction of more extreme specialization of muscle architecture in tyrannosaurs than studies have already taken into account. Hence, scientifically, we cannot posit more speculative reconstructions that have no evidence.

As in any active field of science, but particularly in palaeontology, more evidence and more work is always needed, so we must agree with Garrido's final point. Any hypothesis can be refined or falsified by new evidence. We feel Hutchinson's and others' past studies have done this with the fast-running tyrannosaur hypothesis but new evidence could always shift that balance. Indeed, our modelling approaches provide several means to do so; new evidence can always be considered using them.

No competing interests declared.

RE: RE: Fast running T. rex can not be completely ruled out

Manuel_Garrido replied to jrhutch on 16 Jan 2012 at 06:55 GMT

First of all, thanks for your answer to my post.

I want to set that the fast running T. rex hypothesis is that in which it reaches a G of 2.5 as in Hutchinson and Garcia 2002. In my opinion it corresponds to a speed of about 12.9 ms-1 (28.8 mph, 46.6 kmph) and not to 20 ms-1 (45 mph, 72 kmph). In Sellers and Manning 2007, fig 4, the predicted T. rex speed, with a limb extensor muscle mass of 20.5% of body mass is 12-13 ms-1. This is consistent with my statements about extensor muscle mass and speed in an independent way. I proposed an EMA of 1 for the mean value of the whole limb, not only for ankle EMA.

In Gatesy et al. 2009 the highest result of G is 1,87, with an ankle extensor muscle mass of 5% of body mass. This implies an ankle EMA that is 24% higher than that of Hutchinson 2004 and leads to a decrease of 20% in the ankle extensor muscle mass needed to get a determined G. I agree that the new data of extensor muscle masses and CoM position of Hutchinson et al. 2011 applied to a model like that of Gatesy et al. 2009 may change these results.

Besides, muscle parameters can change the amount of extensor muscle mass needed. Modifying muscle parameters within the range of the sensitivity analysis realized in Bates et al. 2010 (in example: Force per unit of muscle area (FPUA) of 400000 Nm-2, cited in literature; or muscle fascicle length 15% lesser) would reduce the extensor muscle mass needed too. I think that varying unknown parameters of extinct animals within a justified and reasoned with extant animal data range to get an interval of possible results is not doing speculative reconstructions.

Taking in consideration these parameters (ankle EMA 24% higher, FPUA of 400000 Nm-2, ankle muscle fascicle length 15% lesser) and regarding that ankle extensor muscle masses are the critical limit on running capacity, the ankle extensor muscle mass needed to get a G of 2.5 decreases from 8.3 to 4.25% of body mass. This is under the ankle extensor mass of ratites. And mass needed to get G of 2.31, which corresponds to a speed of about 11 ms-1 (the upper limit of the speed range proposed in Hutchinson et al. 2011) is 3.94% of body mass, a quantity that is not far from 4.25% of the fast-running hypothesis.

I acknowledge these values are a combination that leads to the upper limit of the range of possible speeds and most of current evidence may point to more moderate velocity but I think the fast-running T. rex hypothesis, in which it reaches a G of 2.5 and about 12.9 ms-1 can not be ruled out completely.

Thank you again.

Manuel Garrido.
Madrid, Spain.

Reference:

Bates, Karl T. , Manning, Phillip L. , Margetts, Lee and Sellers, William I.(2010) 'Sensitivity Analysis in Evolutionary Robotic Simulations of Bipedal Dinosaur Running', Journal of Vertebrate Paleontology, 30: 2, 458 — 466.

No competing interests declared.

RE: RE: RE: Fast running T. rex can not be completely ruled out

Manuel_Garrido replied to Manuel_Garrido on 24 Jan 2012 at 23:58 GMT

I calculated maximal G using the estimates of ankle extensor muscle mass for MOR 555 from Hutchinson et al. 2011 and data fron Gatesy et al. 2009. I got paradoxical results.

For Min mass (5777 kg), ankle extensor muscle mass was 2,42% of body mass. Then G would be 0.91. This means that Min mass MOR did not got enough ankle extensor muscle mass even for walking.

For Max mass (10768 kg), ankle extensor muscle mass was 2,99% of body mass. Then G would be 1,08 (Df=0,72). Maximal speed would be of about 1,85 ms-1 (4,13 mph). Heavier MOR would be less slow but its maximal velocity was well under published estimates of T. rex speed [Hutchinson et al. 2011 (5-11 ms-1); Sellers and Manning 2007 (8 ms-1)].

After that I decided to use maximal ankle extensor muscle mass and minimal masses of the rest. For a body mass of 6141 kg, ankle extensor muscle mass would be 5,24%, G 1,96, Df 0,4 and speed about 8,17 ms-1 (18,24 mph). This is consistent with published estimates of T. rex speed.

Finally, this ankle extensor muscle mass of 5.24% exceeds the 4.25% that I calculated in my previous message to reach a G of 2,5 (about 12.9 ms-1), using the values of FPUA and muscle fascicle length from the sensitivity analysis of Bates et al. 2010. For this reason the fast running hypothesis can not be ruled out (for a six ton T. rex at least).

With this calculation, I only wanted to point an opinion about the upper limit of the range of speed of an extinct animal. I really did not want to criticize the model and the methods of John Hutchinson and collaborators. On the contrary, I think that over a large number of articles they have created a quantitative method to estimate and compare extinct animal locomotion, resolving how to gauge the unknown parameters like muscle fascicle length, limb posture, extensor muscle moment arms, body mass and extensor muscle mass. T. rex speed is just one outcome among all that their work will have and not the most important.

Manuel Garrido.
Madrid, Spain.

No competing interests declared.

RE: RE: RE: Fast running T. rex can not be completely ruled out

Manuel_Garrido replied to Manuel_Garrido on 25 Jan 2012 at 00:00 GMT

I calculated maximal G using the estimates of ankle extensor muscle mass for MOR 555 from Hutchinson et al. 2011 and data fron Gatesy et al. 2009. I got paradoxical results.

For Min mass (5777 kg), ankle extensor muscle mass was 2,42% of body mass. Then G would be 0.91. This means that Min mass MOR did not get enough ankle extensor muscle mass even for walking.

For Max mass (10768 kg), ankle extensor muscle mass was 2,99% of body mass. Then G would be 1,08 (Df=0,72). Maximal speed would be of about 1,85 ms-1 (4,13 mph). Heavier MOR would be less slow but its maximal velocity was well under published estimates of T. rex speed [Hutchinson et al. 2011 (5-11 ms-1); Sellers and Manning 2007 (8 ms-1)].

After that I decided to use maximal ankle extensor muscle mass and minimal masses of the rest. For a body mass of 6141 kg, ankle extensor muscle mass would be 5,24%, G 1,96, Df 0,4 and speed about 8,17 ms-1 (18,24 mph). This is consistent with published estimates of T. rex speed.

Finally, this ankle extensor muscle mass of 5.24% exceeds the 4.25% that I calculated in my previous message to reach a G of 2,5 (about 12.9 ms-1), using the values of FPUA and muscle fascicle length from the sensitivity analysis of Bates et al. 2010. For this reason the fast running hypothesis can not be ruled out (for a six ton T. rex at least).

With this calculation, I only wanted to point an opinion about the upper limit of the range of speed of an extinct animal. I really did not want to criticize the model and the methods of John Hutchinson and collaborators. On the contrary, I think that over a large number of articles they have created a quantitative method to estimate and compare extinct animal locomotion, resolving how to gauge the unknown parameters like muscle fascicle length, limb posture, extensor muscle moment arms, body mass and extensor muscle mass. T. rex speed is just one outcome among all that their work will have and not the most important.

Manuel Garrido.
Madrid, Spain.

No competing interests declared.

RE: RE: RE: RE: Fast running T. rex can not be completely ruled out

jrhutch replied to Manuel_Garrido on 31 Jan 2012 at 11:06 GMT

We thank Manuel Garrido for his careful analysis and response. While we agree that there might be solutions that could enable a fast-running tyrannosaur, they are less plausible solutions because they push assumptions close to our past their limits. In particular we find the usage of 400 kNm^2 muscle stress as excessively speculative. The most cautiously executed physiological experiments on maximal stress routinely obtain values around 225 kN/m^2 (see review by Lieber and Ward 2011, Phil Trans Roy Soc B 1570:1466-1476, "Skeletal muscle design to meet functional demands"). But we welcome anyone to submit a peer-reviewed paper making their case for how a fast-running tyrannosaur is the best hypothesis, considering all the factors we have cited and appropriate methods.

No competing interests declared.

RE: RE: RE: RE: RE: Fast running T. rex can not be completely ruled out

Manuel_Garrido replied to jrhutch on 04 Feb 2012 at 00:25 GMT

Thanks for your answer.
There really is evidence of higher muscle stress values, about 400-500 kN/m^2 (O'Brien et al. 2010; Zheng et al. 1998; and references therein). Heinrich Mallison regarded 390 kN/m^2 as the likely upper end of the best estimate range of muscle stress (Mallison 2011). Finally 400 kN/m^2 was in the range of the sensitivity analysis done in Bates et al. (2010). I think sensitivity analysis of muscle stress is justified and 400 kN/m^2 as upper limit is supported.
O'Brien, T.D., Reeves, N.D., Baltzopoulos, V., Jones, D.A., and Maganaris, C.N. 2010. In vivo measurement of muscle specific tension in adults and children. Experimental Physiology, 95:202-210.
Zheng, N., G. S. Fleisig, R. F. Escamila, and S. W. Barrentine. 1998. An analytical model of the knee for estimation of internal forces during exercise. Journal of Biomechanics 31:963–967.
Mallison, Heinrich. 2011. Defense capabilities of Kentrosaurus aethiopicus Hennig, 1915. Palaeontologia Electronica Vol. 14, Issue 2; 10A:25p.
Bates, Karl T. , Manning, Phillip L. , Margetts, Lee and Sellers, William I.(2010) 'Sensitivity Analysis in Evolutionary Robotic Simulations of Bipedal Dinosaur Running', Journal of Vertebrate Paleontology, 30: 2, 458 — 466.
And finally I got Alexander´s papers references.
Alexander, R. M., Maloiy, G. M. O., Hunter, B., Jayes, A. S. and Nturibi, J. (1979), Mechanical stresses in fast locomotion of buffalo (Syncerus caffer) and elephant (Loxodonta africana). Journal of Zoology, 189: 135–144.
Alexander, R. M. and Jayes, A. S. (1983), A dynamic similarity hypothesis for the gaits of quadrupedal mammals. Journal of Zoology, 201: 135–152.
Manuel Garrido.
Madrid, Spain.

No competing interests declared.

RE: RE: RE: RE: RE: RE: Fast running T. rex can not be completely ruled out

jrhutch replied to Manuel_Garrido on 04 Feb 2012 at 08:30 GMT

We are aware that higher muscle stresses have been reported in the literature but it is highly likely that these are due to experimental errors; not all experiments are done equivalently well. As we noted above, the best mechanical measurements done by the most experienced muscle physiologists (e.g. Lieber) routinely obtain almost 50% lower values. But we agree sensitivity analysis of quantitative parameters is always important-- it merely needs to be executed with plausibility in mind.

No competing interests declared.

RE: RE: RE: RE: RE: RE: Fast running T. rex can not be completely ruled out

Celestist replied to Manuel_Garrido on 09 Feb 2012 at 23:36 GMT

Seller increased the upper limit to 450kN/m^2, their allosaurus' top speed increased from ~=9.5m/s to 10.7m/s, the increase doesn't seem to be significent enough to push T-rex speed from 8~11m/s to 20m/s.

[1]"Sensitivity analysis in evolutionary robotic simulations of bipedal dinosaur running"

panthera leo, panthera tigris or large terror bird's top speed usually around 55-60km/hr, It is rather extreme that T-rex be able to run at a faster speed. Not to mention the potentially fatal danger of falling down.

[2]Body mass, bone" strength indicator," and cursorial potential of Tyrannosaurus rex
[3]Terror birds on the run: a mechanical model to estimate its maximum running speed

T-rex's prey iterms are not particularly fast either, it will quite biazze to think T-rex will take such a extreme adoptive measure that is rather unnecessary.

off course, Fast running T. rex can not be completely ruled out, there is off course the probability that T-rex can run faster than 8~11m/s.

Sinerely
Celestis
sinerely

No competing interests declared.

RE: RE: RE: RE: RE: RE: RE: Fast running T. rex can not be completely ruled out

Manuel_Garrido replied to Celestist on 31 Mar 2012 at 13:58 GMT

First of all, thanks for your comment and sorry for my delayed answer.

I pointed the fast-running hypothesis can not be ruled out in a six ton T. rex. Of course this implies to set the physiological variables near their upper limits and I agree that the more far from the mean values, the less probable the results are, but they still are within the extant animal range and therefore they are possible.

With respect the lower limits, if we set the physiological variables to get a six ton T. rex with a maximum speed of about 5 ms-1 and use them to obtain the maximum speed of a nine-ten ton T, rex (Sue), we get that it did not have enough force to walk (an absurd result), so the maximum speed of a six ton T. rex should be higher.

Recently, a paper pointed that hadrosaurs (less dangerous preys than ceratopsians or ankylosaurs), could have been quite fast (14-15.7 ms-1, Sellers et al. 2009). They could be a reason for fast running tyrannosaurs.

A detailed explanation of my calculations can be read at

http://dml.cmnh.org/2012F...

Thanks.
Manuel Garrido.


REFERENCE

Sellers WI, Manning PL, Lyson T, et al. (2009) Virtual palaeontology: gait reconstruction of extinct vertebrates using high performance computing. Palaeontologia Electronica 12, 11A: 26.

No competing interests declared.