My Erdős Number below 6 degrees of separation

Paul Erdős was a Hungarian mathematician who authored or co-authored hundreds of publications across a wide variety of disciplines. He was so prolific that a certain level of prestige is associated to having co-authored a paper with him. The idea for the Erdős Number came from another mathemetician, Casper Goffman. This has now turned into a kind of science cult, in the same way that the Bacon number (actually a newer concept stemming from the Erdős Number) has in acting (see here). Those who have co-authored with Erdős are given an Erdős number of 1. Someone who has never written a paper with Erdős, but who has co-authored a paper with one of his co-authors are given an Erdős number of 2, and so on with increasing degrees of separation. As noted by Lenski, while the ranks of those with 6 degrees of separation from Erdős and Bacon are quite high, very few people have an Erdős-Bacon Number. 

For a long time, I've been pestering my mathatically inclined collaborators to find out what their Erdős number is so that I can plot my own. It turns out that I need not have bothered as my line is more direct than I had thought. 

My first attempt equalled six degrees of separation:

Daniel Kleitman published many papers with Erdős, and has an Erdős number of 1

Lior Pachter has an Erdős number of 2, having co-authored a publication with Daniel Kleitman.

Richard Lenski has an Erdős number of 3, having co-authored a publication with Lior Pachter.

Jonathan Losos has an Erdős number of 4, having co-authored a publication with Richard Lenski.

Anthony Herrel has an Erdős number of 5, having co-authored a publication with Jonathan Losos.

I therefore have an Erdős number of 6, having co-authored a publication with Anthony Herrel

Thus my collaborators have either an Erdős number of 7 or lower (if they've got a more direct line to Erdős). It could be that my Erdős number is lower, or will get lower. Stay tuned for any updates!

Here are the relevant publications:

Beerenwinkel, N., Pachter, L., Sturmfels, B., Elena, S.F. and Lenski, R.E., 2007. Analysis of epistatic interactions and fitness landscapes using a new geometric approach. BMC Evolutionary Biology, 7(1), 1-12.

Blount, Z.D., Lenski, R.E. and Losos, J.B., 2018. Contingency and determinism in evolution: Replaying life’s tape. Science, 362  (6415).

Erdös, P. and Kleitman, D.J., 1968. On coloring graphs to maximize the proportion of multicolored k-edges. Journal of Combinatorial Theory, 5(2), pp.164-169.

Kleitman, D. and Pachter, L., 1998. Finding convex sets among points in the plane. Discrete & Computational Geometry, 19(3), pp.405-410.

Measey, G.J., & Herrel, A. 2006. Rotational feeding in caecilians: putting a spin on the evolution of cranial design. Biology Letters 2, 485-487.

But then my friend [and collaborator] Ben Stevenson [Erdős number 5] pointed out that the American Mathematical Society has their own Erdős calculator: 

David [who was the first person I asked many years back] it turns out has an Erdős number of 4. As David and I published a few times, my Erdős number is actually 5.

Here are the relevant publications

Antille, A., Kersting, G., Zucchini, W. (1982) Testing symmetry. J. Amer. Statist. Assoc. 77, 639–646.

Borchers, D. L., Zucchini, W., Heide-Jørgensen, M. P., Cañadas, A., Langrock, R. 2013 Using hidden Markov models to deal with availability bias on line transect surveys. Biometrics 69 (3), 703–713.

Erdős, P., Janson, S., Łuczak, T., Spencer, J. (1996) A note on triangle-free graphs. Random discrete structures (Minneapolis, MN, 1993),  117–119, IMA Vol. Math. Appl., 76, Springer, New York, 

Janson, S., Kersting, G. 2011 On the total external length of the Kingman coalescent. Electron. J. Probab. 80, 2203–2218.

Measey, G.J., Stevenson, B., Scott, T, Altwegg, R and Borchers, D. (2017) Counting chirps: acoustic monitoring of cryptic frogs. Journal of Applied Ecology 54: 894–902 https://doi.org/10.1111/1365-2664.12810 pdf

Two very different talks for Society for Integrative and Comparative Biology (SICB) 2021

I’m co-author on two very different talks at this years’ SICB conference, held online. 

The first talk by Carla Madelaire takes the work she and Adriana did on CORT back in 2019 to another level by asking whether saliva assays work as well as blood. This work has particular relevance on repeating CORT levels on individuals. Great to see this being presented!

The second study uses some data collected by myself and Anthony Herrel many years back. We collected some burrowing data on various caecilian species. Now Aurélien Lowie, from Ghent University, has taken this and other data to look at how a number of different live caecilians perform in relation to their skull shape. 

See more on the SICB 2021 website (links below!).

Corticosterone levels in the saliva as a measure of stress in toads

CB Madelaire, D Dillon, AMG Barsotti, J Measey, FR Gomes, CL Buck

Glucocorticoids have been widely used as a physiological marker of stress, and elevated baseline glucocorticoids levels in vertebrates have been associated with environmental changes. The use of minimally invasive sampling techniques and analysis of non-traditional sample types to monitor stress in wild populations has increased due to the importance of understanding how animals respond to environmental disturbances. The use of saliva samples can be a powerful tool to monitor both endocrine shifts and responses to stressors in wild populations. This sampling method does not require a large amount of manipulation and it can be used to sample smaller species, contributing to an increase of studies in environmental endocrinology and conservation efforts of understudied species. This study validated corticosterone (CORT) measurements in the saliva of the guttural toad (Sclerophrys gutturalis) using samples collected in the field and after a standardized stress protocol. We show that small amounts of saliva (0.018±0.028 g) are sufficient to quantify CORT. Salivary CORT levels were higher after exposure to a standardized stress protocol when compared to field levels of CORT, indicating that saliva samples can reflect biologically meaningful levels of CORT in the guttural toad. Because levels of salivary and plasma CORT were not correlated in either the field sampled animals or following exposure to acute stress, we conclude that CORT in the saliva and plasma might show different response dynamics to stimuli.

Keywords

conservation physiology, glucocorticoid, acute stress, anura, salivary glucocorticoids

9:47 AM - 9:48 AM SAST on Saturday, January 2

https://sicbannualmeeting.pathable.co/meetings/virtual/cHp2e4tMMLd5d5xih

Under pressure: the relationship between cranial shape and in vivo maximal burrowing force in caecilians (Gymnophiona)

Co-Author(s): A Lowie, A Herrel, B De Kegel, M Wilkinson, GJ Measey, JC O'Reilly, N Kley, P Gaucher, J Brecko, T Kleinteich, D Adriaens

Caecilians are elongate and limbless amphibians. Except one aquatic family, they all have an at least partially fossorial lifestyle. It has been suggested that they evolved sturdy compact skulls with fusion of ancestrally separate bones and tight sutures as an adaptation for head-first burrowing. Although their cranial osteology is well described, relationships between form and function remain poorly understood. In this study, we report data on in vivo burrowing forces for more than 120 specimens belonging to 13 different species. Over 80 caecilians were µCT-scanned and their skulls segmented. Using fixed and semi-sliding anatomical landmarks, we performed 3D geometric morphometrics to quantify skull variability across species. Finally, using correlation tests, linear models and two-blocks partial least squares, we investigated the relationships between the overall cranial shape and in vivo burrowing force in caecilians. Surprisingly, results show that despite differences in the head morphology across species, there is no relation between overall skull shape and this particular measure of burrowing performance. Although a phylogenetic signal may partly obscure the results, our conclusions join previous studies using biomechanical models and suggest that any differences in their degree of fossoriality are not driving the correlated adaptive evolution of head shape and maximal burrowing force. As the cranium has multiple functions such as feeding, and houses major sensory organs, or respiratory systems, further studies are needed to fully understand the selective pressures shaping the evolution of skull form.

Keywords

amphibian, burrowing, geometric morphometrics, gymnophiona, skull, performance, head-first burrowers, head shape

10:05 AM - 10:06 AM SAST on Saturday, January 2

https://sicbannualmeeting.pathable.co/meetings/virtual/E7JfQGzh9PTq2oiFe

Does spatial sorting act on Xenopus tadpoles in the same way it does adults?

Natasha Kruger asked this simple question during her PhD studies on the invasive population of Xenopus laevis  in France. Although the question appears simple, the experiment to investigate whether the phenomenon of spatial sorting occurs in tadpoles was complex and substantial. 

Natasha sampled adults from the core and periphery of the French invasion of Xenopus laevis, then bred couples from different ponds in mesocosms within their invasion range in France. In addition, she conducted a simultaneous experiment in the laboratory with one individual per plastic cup. During the development in both experiments, she monitored their morphology, developmental rate, time to metamorphosis and survival. She found that unlike adults, the tadpoles did not show many differing traits between core and periphery. One significant result was that tadpoles from the periphery raised in the laboratory reached a greater size at metamorphosis, somewhat similar to results in Cane Toads. But no significant size increase was seen in the mesocosm experiments. 

Spatial sorting is all about optimising traits for dispersal. As the tadpole life-history stage in Xenopus laevis  is not configured to disperse, it's not surprising that we didn't see the effects of spatial sorting on these populations. However, we did see the suggestion that animals at the periphery metamorphose at a larger size - larger metamorphs would facilitate dispersal.

As it's only ~40 years since the start of this invasion, it may be that we are now only seeing the start of the influence of spatial sorting on the larval stage of X. laevis. It would be great to do this same experiment in another 20 years to see whether there's been any changes in the distribution. According to the latest paper by Philipp Ginal et al (2020), there'll be no stopping this species playing all over Europe.

Read more about Natasha, who recently graduated her PhD from Stellenbosch University (see here). Natasha made this great graphical abstract to explain the paper:

Read more about the work on invasive Xenopus here:

Kruger, N., Measey, J., Vimercati, G., Herrel, A., Secondi, J. (in press) Does the spatial sorting of dispersal traits affect the phenotype of the non-dispersing stages of the invasive frog Xenopus laevis through coupling? Biological Journal of the Linnean Society https://doi.org/10.1093/biolinnean/blaa191 pdf

Eagle eyed blog readers will see the name of Giovanni Vimercati popping up here on this paper with Natasha. After graduating his PhD from the MeaseyLab, Gio went to work on this project for Jean Secondi (Natasha's French supervisor) at the University of Angers (near the Xenopus invasion site). Since then, Gio has gone to work at the Basher lab in Switzerland, but we haven't seen the last of him on this blog!

And don't forget to look at the modelling paper by Philipp!

Ginal, P., Mokhatla, M., Kruger, N, Secondi, J., Herrel, A., Measey, J. & Rödder, D. (in press)  Ecophysiological models for global invaders: Is Europe a big playground for the African clawed frog?   Journal of Experimental Zoology Part A     http://dx.doi.org/10.1002/jez.  2432     pdf

Hormones tell us more about rapid adaptation of invasive Guttural Toads

Back in January 2019, the wonderful PhD student Adriana Barsotti visited Stellenbosch together with Dr Carla Madelaire (both from Fernando Ribeiro Gomes' lab in USP), to add some endocrinological studies to our ongoing work on invasive Guttural Toads in Cape Town (see blog post here). Within no time at all, the fab duo set to work on sampling invasive toads in Cape Town's exclusive Constantia suburb, and from there they travelled to Durban to sample native toads with MeaseyLab MSc student, Carla Wagener (see blog post here). Along the way, the two Braziliaras taught us some some useful caipirinha skills (see here). 

Now Adriana has published the paper resulting from all of her work in South Africa:

In this great paper, we learn:

Invasive toads have lower body condition and tend to become dehydrated in the more arid conditions of Cape Town;
Invasive toads have the highest BKA and N:L ratio which could favor dispersal in their novel environment;
Dehydration is a stressor for native and invasive toads;
BKA remains comparatively higher in invasive toads even after acute stressors.

...and all this after only 20 years of invasion!

Carla published her findings earlier this year, showing us that invasive Guttural Toads have changed the way that they think about water (see blog post here).

The expertise from Fernando's lab is a great example of how our understanding grows through collaborations. Not only that, but it was great fun to host Adriana and Carla in the MeaseyLab, and we look forward to hosting more of Fernando's brilliant lab members in the future.

Read the papers in full here:

Barsotti, A.M.G., Madelaire, C.B., Wagener, C., Titon Jr, B., Measey, J., Ribeiro Gomes, F. (in press) Challenges of a novel range: water balance, stress, and immunity in an invasive toad. Comparative Biochemistry and Physiology,https://doi.org/10.1016/j.cbpa.2020.110870 pdf

Madelaire, C.B., Barsotti, A.M.G., Wagener, C., Sugano, Y., Baxter-Gilbert, J., Ribeiro Gomes, F., Measey, J. (2020) Challenges of dehydration result in a behavioral shift in invasive toads. Behavioral Ecology and Sociobiology74: 83. https://doi.org/10.1007/s00265-020-02866-5 pdf

Natasha graduates at small ceremony

The global pandemic has meant that Stellenbosch University cannot hold the large graduation ceremonies that we've seen in the past (e.g. see here). Instead, small ceremonies of a handful of graduands have taken place this week, allowing PhD student to graduate in person. 

One MeaseyLab member graduating this week is Dr Natasha Kruger. Read more about Natasha in this blog post written for her defense in Lyon, France.

Also graduating this week were the MeaseyLab's two honours students: 

Dan van Blerk & Emily Rippon

Congratulations to you all! You make us very proud.