Some comments about Tyrannosaurus rex speed.

["Manuel Garrido" <bagaraatan@gmail.com>]

I tried to do an exercise and get an upper limit of the speed range of 
Tyrannosaurus rex using published data. After analyzing the literature on 
the speed of this dinosaur I put the more favorable to fast running T. rex 
values on the parameters of the equation 1 of  Hutchinson (2004a) to 
calculate the highest possible G (the ratio of Ground Reaction Force to body 
weight) and therefore the highest possible speed.

mi=(100·G·g·R·L·d)/(cos p·FPUA·c·r)

See below for significances.

Most of published data are about MOR 555, so I used this T. rex to begin. 
First I assumed that ankle extensor muscle masses are the critical limit on 
running capacity (Hutchinson et al. 2011).

I took from Gatesy et al. (2009) the highest result of G (1.87), with an 
ankle extensor muscle mass (mi) of 5% of body mass. This implies the best 
limb posture for fast running I found published. Other results can be found 
in Hutchinson and Garcia (2002) and Hutchinson (2004b).

I took two values of muscle parameters within the range of the sensitivity 
analysis did in Bates et al. (2010). These were Force per unit of muscle 
area (FPUA) of 400000 Nm-2 (instead of 300000 Nm-2), cited in literature and 
muscle fascicle length (L) 15% lesser.

Finally, for getting the largest possible ankle extensor muscle mass I 
decided to use the maximal estimation of shank mass with the minimal 
estimated masses of the other parts of MOR 555 from  Hutchinson et al. 
(2011). Body mass 6553 kg, ankle extensor muscle mass relative to body mass 
(mi) 4.91%.

Then I calculated the maximum G using these values. I got, for a 6553 kg T. 
rex, with a mi of 4.91%, a maximum G of 2.89. This implies a Duty Factor 
(Df) of about 0.27, a Froude number (Fr) of about 10.88 and a speed of about 
16.34 ms-1 (36,47 mph, 58.82 kmph). G of 2.89 is over the value of 2.5, 
considered the limit for a high-speed running T. rex (Hutchinson and Garcia 
2002). This maximum theoretical speed is consistent with that of Paul 
(1998), but it still below 20 ms-1 and it is only the upper limit of the 
possible range of maximum speed and should be considered that actual maximum 
speed may well be lower.

I acknowledge that I have taken the parameters to their maximum limits and 
that it is possible that the actual values were lower and therefore the 
speed were lower, but I think the fast-running hypothesis can not be ruled 
out in a six ton T. rex (see below for a heavier one). Besides these 
calculations must be validated using the models and taking into account COM 
position and should be considered preliminary. However, one or more of these 
high values must be accurate because if we apply none of them and use the 
ankle extensor muscle values from Hutchinson et al. (2011) we get a value of 
G lesser than 1 and this would imply that T. rex could not even walk.

There are much less published information about FMNH PR 2081 (Sue) so I 
assumed that the ankle effective mechanical advantage (EMA=r/R) was the same 
as MOR 555 and adjusted ankle extensor muscle fascicle length (L) 
proportional to limb length. It was a 9-10 tons T. rex with only slightly 
longer limbs than MOR. Equation 1 is Body mass independent but greater body 
mass and nearly equal limbs implies a lower mi (that is a percentage of body 
mass) and therefore reduced G and speed.

In fact, the maximum mi I got for Sue using the maximal estimated shank mass 
with the minimal estimated masses of the other parts from Hutchinson et al. 
(2011) was 3.58% of a body mass of 10346 kg. This was below the mi of  about 
5% that I calculated to get a G of 2.5 and so this did not support a 
fast-running ability. Applying the maximum values to the equation 
parameters, with a mi of 3.58%, I obtained a speed about 8.41 ms-1 (18.77 
mph,30.28 kmph)[in the middle of the range published in Hutchinson et al. 
2011(5-11 ms-1), but still a running gate (Hutchinson and Garcia 2002)] and 
not applying these values I obtained G lesser than 1 (no walking ability) 
again.

Conclusions.
Assuming these body masses of MOR and Sue are correct
1- Fast-running hypothesis can not be ruled out in a 6 ton tyrannosaur, but 
should be discarded in a 10 ton T. rex, taking into account the data we have 
(of course, it must be said that data do not rule out a slower maximum speed 
in MOR either).

2- Results show there could be two T. rex morphotypes differentiated by the 
ability of locomotion, despite that the difference may be due to ontogeny. 
This may have paleobiological implications that could be discussed 
elsewhere.

3-This model seems to be very good to compare locomotion abilities of 
extinct animals. It could be used not only in intraspecific but also 
interspecific comparisons (i. e. predator-prey or two different predators).

4- To reconstruct very large tyrannosaur locomotion may be necessary to use 
high values (within the range known in extant animals) in one or more of the 
parameters of the model. Using only conservative values can lead to results 
such as T. rex could not walk.

I want to say I only changed some values in the model of of John Hutchinson 
and collaborators and commented the results. I think that they have created 
a solid quantitative method to estimate and compare extinct animal 
locomotion with their papers. T. rex speed is just one outcome among all 
that their work have and not the most important.

Thanks and excuse me for my bad English.
Manuel Garrido.
Madrid, Spain.


Methods:
mi=(100·G·g·R·L·d)/(cos p·FPUA·c·r)
mi is the extensor muscle mass, expressed as percentage of body mass,G is 
the ratio of Ground Reaction Force to body weight, g is the acceleration due 
to gravity, R is the moment arm of the Ground Reaction Force, L is the mean 
muscle fascicle length, d is the muscle density, cos p is the cosine of the 
mean angle of muscle fascicle pennation, FPUA is the Force per unit of 
muscle area,c is the fraction of maximum exertion by the muscles and r is 
the mean moment arm of the muscles.

To estimate speed from G, I used these equations:

To get the Duty Factor we can solve from Alexander et al. (1979).

Df=Pi/(4*G)

To get the Froude Number and speed we solve from equations from Alexander 
and Jayes (1983).

Fr=(Df/0,53)^(-1/0,28)

s=(Fr*g*hh)^0,5

hh is hip height. I used 2.5 m for MOR (Hutchinson and Garcia 2002) and 
estimated it proportional to limb length for Sue (2,7 m).

Note these two papers are about mammal locomotion. It should be taken into 
account and speed values should be considered only indicative.

References:

Hutchinson JR (2004a) Biomechanical modeling and sensitivity analysis of 
bipedal running ability. I. Extant taxa. Journal of Morphology 262: 421-440.

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

Gatesy SM, Baeker M, Hutchinson JR (2009) Constraint-based exclusion of limb 
poses for reconstructing theropod dinosaur locomotion. Journal of Vertebrate 
Paleontology 29: 535-544.

Hutchinson JR, Garcia M (2002) Tyrannosaurus was not a fast runner. Nature 
415: 1018-1021.

Hutchinson JR (2004b) Biomechanical modeling and sensitivity analysis of 
bipedal running ability. I. Extant taxa. Journal of Morphology 262: 421-440.

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.

Paul GS (1998) Limb design, function and running performance in 
ostrichmimics and tyrannosaurs. Gaia 15: 257-270.

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.