Chris Aldrich

Objectives

The objectives of the workshop are to bring mathematicians working in three key areas together to make progress in these problems. We will also invite several biologists who are keen to engage with mathematicians on the challenges posed by new data on evolutionary processes. Key challenges in the field at the moment are focused around the following emerging inter-related areas, each of which is raising mathematically interesting problems:

1. Inference with evolutionary trees and networks: Ultimately it is necessary not just to obtain evolutionary trees from data using standard methods, but to infer aspects of an underlying biological process. This requires understanding the likelihood of an evolutionary tree or network, or at least some of its informative features, using some stochastic process as the underlying ecological model. In principle, this approach allows simultaneous inference of both evolutionary trees and parameters of the ecological model. Coalescent theory has made considerable progress, for example, in obtaining tree likelihoods for sparsely sampled populations with geographical structure or with known past demographics (see for just one example [5]). In some simplified cases, epidemiological inference methods can estimate transmission trees [2], branching rates through time [5] and other aspects of epidemic spread [7]. However, none of these approaches is currently applicable if there is non-tree-like evolution, or where datasets are large. Furthermore, the range of models for which we can write down a tree likelihood is very limited. This raising nice new problems in probability, statistical inference and ecological modelling. Recently, more general processes (e.g. Lambda-coalescents, which allow multiple rather than strictly pairwise coalescent events) are beginning to be used to model populations with large offspring variance, or even to model selection in a non-parametric fashion [3]. This is potentially a powerful tool particularly for bacteria, which may acquire resistance to antibiotics and spread rapidly as a consequence, yielding both highly variable effective offspring numbers and a need to model selection carefully.

2. Understanding spaces of evolutionary trees: There are a large number of possible labelled, rooted binary trees for a given set of nn tips (ie for a given set of sequence data): (2n−3)!!=(2n−3)(2n−5)...(3)(1)(2n−3)!!=(2n−3)(2n−5)...(3)(1). This works out to 1018410184 trees on 100 tips; in contrast, current datasets for evolving bacteria contain thousands of tips. Not even the tools of Bayesian inference, the natural approach in such situations, can systematically explore spaces this big. This motivates the development of mathematical approaches for the exploration of tree space. These include new approaches to continuous tree spaces, including those from tropical geometry [8], and the use of tree metrics [1]. These in turn can lead to tools for averaging trees , and for navigating tree space in efficient ways [6] -- with profound applications in statistical inference from sequence data. Generalizing metrics to the case of evolutionary networks (for example tree-based networks) is another natural and important question.

3. Summarising trees and networks using combinatorial tools: Uncovering shape features, spectral features and other ways to describe trees using quantities that are mathematically tractable will be of considerable interest [4]. As one example, where likelihoods are truly intractable, rapid tools for likelihood-free inference can be used to infer evolutionary processes from sequence data, but only where there are informative ways to summarize key features of the data. Trees are natural combinatorial structures with connections to data; for example, a binary tree is a sequence of partitions of the set of tips (sequences in a dataset), where each partition is one block smaller than the previous one, moving back through time from the partition with each tip on its own to the partition with all tips in one block as we move from the tips of the tree to the root. If the tree is not binary (ie it allows multifurcations), more than two blocks can combine at a branching event. Because of the natural link to partitions, the study of tree shapes links to the enumeration of partitions and to lattice path combinatorics. These in turn allow the characterization and enumeration of possible tree shapes. Meanwhile the study of motifs in other biological networks has been fruitful, and could be extended to tree and evolutionary network shapes. Trees and evolutionary networks are of course also graphs (with an added time dimension); the tools of algebraic graph theory are now finding application in this area of mathematical biology.

The community's response to the idea for this workshop has been very positive. A * beside a participant's name indicates that they have expressed enthusiasm for the workshop, and plan to attend.

References [1] Louis J Billera, Susan P Holmes, and Karen Vogtmann. Geometry of the space of phylogenetic trees. Adv. Appl. Math., 27(4):733–767, November 2001. [2] Xavier Didelot, Jennifer Gardy, and Caroline Colijn. Bayesian inference of infectious disease transmission from whole-genome sequence data. Mol. Biol. Evol., 31(7):1869–1879, July 2014. [3] Alison M Etheridge, Robert C Griffiths, and Jesse E Taylor. A coalescent dual process in a moran model with genic selection, and the lambda coalescent limit. Theor. Popul. Biol., 78(2):77–92, September 2010. [4] Fanny Gascuel, Regis Ferriere, Robin Aguilee, and Amaury Lambert. How ecology and landscape dynamics shape phylogenetic trees. Syst. Biol., 64(4):590–607, July 2015. [5] Amaury Lambert and Tanja Stadler. Birth–death models and coalescent point processes: The shape and probability of reconstructed phylogenies. Theor. Popul. Biol., 90(0):113–128, December 2013. [6] Tom M W Nye. An algorithm for constructing principal geodesics in phylogenetic treespace. IEEE/ACM Trans. Comput. Biol. Bioinform., 11(2):304–315, March 2014. [7] David A Rasmussen, Erik M Volz, and Katia Koelle. Phylodynamic inference for structured epidemiological models. PLoS Comput. Biol., 10(4):e1003570, April 2014. 2 [8] David Speyer and Bernd Sturmfels. The tropical grassmannian. Adv. Geom., 4(3):389–411, 2004.

Brief reply to: Is majoring in liberal arts a mistake for students? https://medium.com/@vkhosla/is-majoring-in-liberal-arts-a-mistake-for-students-fd9d20c8532e#.d64awbm87

What magisterial sounding pontification! Sadly, it’s not much different than the early philosophies of Socrates and Plato or many of the other early progenitors of the humanities and liberal arts. I get the impression that the author hasn’t read much philosophy and has not much grounding in the liberal arts. While I agree with the spirit in which the piece is written, I find it deplorable that there aren’t what should be obligatory mentions of words like trivium, quadrivium, or philosophy, but rather the corpus of work in which the author seems steeped is that of only modern day authors of popular science (Pinker, Gladwell, Kahneman, et. al.) who have some interesting viewpoints, but ones which require at least a grounding in the liberal arts to pick apart. Several times Khosla demeans the liberal arts and uses the repeated example that a reader should be able to pick apart and think critically about articles in The Economist. To do this requires a knowledge of logic and rhetoric which are two of the pillars of what? — yes, the liberal arts!

He also seems unaware of big movements within the humanities and sciences like Bill Gates and David Christian’s Big History Project which are going a long way towards providing a more balanced education in history, economics, physics, chemistry, biology and evolution. I find here, no prima facie evidence of his knowledge of Thomas Kuhn or Karl Popper, which might help win me to his argument. In all, aside from the passing references to one or two recent works, this entire argument is not much different from many that could have been written at the beginning of the industrial revolution. How blind so many must be to seemingly think there’s something new here.

Most appalling to me here is that the author doesn’t seem to give even a passing nod or small wink to C.P. Snow or “The Two Cultures” [http://boffosocko.com/2013/11/28/two-cultures/]which, at heart, is really the substance of his entire argument, he’s just blind to it’s existence.

Yes, we certainly need more emphasis on the quadrivium portion of the liberal arts, and in particular mathematics and critical thinking which seem to have been left by the wayside. It is deplorable that the highest extent of mathematics that 99% of college students are exposed to terminates in the 17th century for the most part. Sadly, many college students are left without the ability to think critically and deeply, not to mention the hordes of students in America who barely make it through high school and don’t attend college. One also only needs to skim through recent issues of Nature [http://www.nature.com/news/reproducibility-a-tragedy-of-errors-1.19264], one of the world’s most pre-eminent scientific journals to discover that a multitude of advanced researchers with Ph.D.s lack the ability to properly design scientific experiments or evaluate the simple statistical analyses to reach the correct conclusions. What does this mean for readers of The Economist who aren’t even presented with any actual data and are supposed to be able to think critically about a writer’s hidden assumptions.

Yes, we need far, far more, but alas, this poor article only touches the tip of the issue and it sadly only does so with less than half of the picture.

@fadesingh, While I do agree in part with your comment that the technical side of the math could have been explored a bit more, to me, part of what this story is highlighting (and to great benefit as well) is the "other" personal side of mathematics which is rarely seen by the broader public. Most of math and math history is full of seemingly brilliant solo (read: lone wolf) researchers developing mathematics de novo in dark, smoke and caffeine-filled rooms and emerging with iron clad proofs. This particular story shows the growing more collaborative and friendly side of math in addition to the years of slow development of friendships and theory which have culminated into something potentially interesting. I would suggest that in this case you not take them too hard to task on the subject, particularly as the article was being written contemporaneously with the publication of the journal article itself. (The article here was published <i>just</i> in advance of the arXiv post, such that it didn't even include the link to the paper itself, though it was added the following day.)

I love that Quanta is continually exploring the areas of math and science at the depth and level which they've become accustomed. They're filling a very important gap in science communication between technical journal articles and somewhat sophisticated outlets like National Geographic, Scientific American, and Wired (in which they are also distributed, but yet are still an editorial cut above comparatively. [Cross reference: <a href="http://stream.boffosocko.com/2015/evolution-of-a-scientific-journal-article-title-from-nature-to">Evolution of a Scientific Journal Article Title (from Nature to TMZ)</a>] I'm sure that C.P. Snow himself would praise them for helping to close the gap between the "Two Cultures."

Chris Aldrich

17w5104: Mathematical Approaches to Evolutionary Trees and Networks Workshop @BIRS_math 2/12/17 #ITBio

Organizers

Objectives