Modelling Biological Evolution 2013: Conference Highlights

Over the last couple of days I have been attending the Modelling Biological Evolution conference at the University of Leicester organized by Andrew Morozov.

For me, the most interesting theme to have emerged is work on evolutionary branching: conditions under which polymorphisms (or even speciation) might arise. These were all talked about in the context of mathematical models (ODE-type formulations based on generalized Lotka-Volterra systems). The best talk I attended was by Andrew White (Heriot Watt University). He described various system of parasite-host co-evolution, the most interesting of which demonstrated increases in diversity: a new host could emerge that was resistant to current parasites, following which a new parasite could emerge that would infect that host. He rather nicely linked that work to experimental work from Mike Brockhurst (University of York) on phage infections of bacteria showing similar patterns. The results could of course be interpreted at a speciation level, or, probably more fairly, at the level of molecular diversification (e.g. of MHC types in an immune system). What I really appreciated about this resut is that it spoke to the idea that increased diversity can result through a positive feedback mechanism: diversification leads to new niches and thus the potential for further diversification. I have thought for some time that this is the most important mechanism that drives diversification / speciation in natural systems and it was nice to see an example of the mechanism in action.

The other talk I particularly appreciated on the subject was by Claus Rueffler (University of Vienna). He spoke about a result on complexity and diversity in Doebeli and Ispolatov 2010 that also contains this feedback idea. This paper relies on a specific model to obtain its result on conditions for evolutionary branching. Rueffler demonstrated general conditions under which branching might take place that depend only upon the properties of the Hessian matrix associated with key parameters in model space. The important point is that the analysis is model-independent: it only considers the properties of the model forms needed to obtain the result.

Similar ideas were presented by Eva Kisdi (University of Helsinki). She focussed on models that include evolutionary trade-offs (e.g. between virulence and transmissibility): her point was that instead of choosing a function and analyzing its consequences, one could consider desired properties of a model (e.g. branching or limit cycles) and then use “critical function analysis” to derive conditions for possible trade-off functions that would admit the desired behaviour. Eva made the important point that many models make ad hoc choices of functions and thus lead to ad hoc results of little predictive value.

I think Eva’s point really touched on some of the weaknesses that emerged in many of the talks that I attended: there was a great deal of theory (some of which was very good), but very little interface with real biological data. I find this somewhat surprising: modelling in ecology and evolution has been around for very much longer that modelling in say molecular biology (where I currently work), and yet seems to be less mature. I think that the field would really benefit from far greater interaction between theoretical and experimental researchers. Ideally, models should be looking to generate empirically falsifiable hypotheses.

Perhaps the most entertaining talks were given by Nadav Shnerb and David Kessler (both Bar Ilan University). Nadav’s first talk was about power-law-like distributions observed in genus/species distributions. Core to his work is Stephen Hubbell’s neutral theory of biodiversity.
Nadav showed that distributions of number of species within genera could be explained by a neutral model for radiation and the genus and species level coupled with extinction. Nadav’s most important point was that if you wish to make an argument that a certain observed trait is adaptive, then you have to rule out the null hypothesis that it could arise neutrally through mutation/drift. I hope that is something we addressed with regards global regulators in gene regulatory networks in Jenkins and Stekel 2010. David spoke about biodiversity distributions also, showing that adaptive forces could explain biodiversity data (they are generally poor at this due to competitive exclusion that occurs in many models) if the fitness trait is allowed a continuous rather than discrete distribution.

Nadav’s second talk was about first names of babies. This was very interesting – especially as I have a young family (and a daughter with a very old-fashioned name). He looked at the error distribution (easily shown to be binomial-like noise proportional to square root of mean) that is superimposed on a deterministic increase and decrease in popularity of a name over a 60 year period. His thesis was that the error distribution due to external events would be proportional to mean (not root mean), and, as only 5 names in his data set (Norwegian names in ~ 20th Century) did not fit binomial noise, he ruled out external events (e.g. celebrity) as being a major driver. The problem I have with this is that he didn’t rule out external events in the deterministic part of the data (e.g. initiating a rise in popularity of a name that then follows the deterministic feedback law he proposed).

Some thoughts on Thomas et al. 2012. Directional Migration of Recirculating Lymphocytes through Lymph Nodes via Random Walks.

This article [1] was published just over 4 months ago by Benny Chain’s laboratory in UCL, based on work carried out for Niclas Thomas’s PhD. It is a rare privilege to read an article that so clearly relates itself to work that I carried out – indeed work carried out during my own PhD. Therefore I have decided to post some thoughts on this (and related) articles, which I will also post on the PLoS ONE web site.

Thomas et al.’s work contains new data on the distribution of lymphocyte transit times, together with rigorous fitting of mathematical models to their data. Importantly, they show that their data can be fitted by a random walk model that allows for motion orthogonal to the main direction of motion (i.e. through the lymph node tissue). This random walk model, although implemented in one dimension, is intended to reflect a three dimensional motion in which the cells move either along the main axis of motion, or in dimensions perpendicular to it.

There were two models for lymphocyte recirculation that I proposed for my PhD, both of which were implemented in a one dimensional domain (along the lymph node). The first was a convection-diffusion model [2], which could be thought of as a biased random walk, although could be sufficiently general to include other mechanisms. The second was a model that proposed that lymphocyte migration was halted due to encounters with dendritic or other cell types [3, 4]. Both models could explain the available data on lymphocyte recirculation transit times.

Following my own PhD, two photon microscopy technology developed to the point where the motion of individual lymphocytes could be tracked [5, 6, 7]. This has led to the discovery that lymphocyte migration is essentially random, and that the hypothesis I set forward in [3, 4] is false. Thomas et al.’s work, following work by Textor et al. [7], has shown that indeed the distribution of lymphocyte recirculation times can be explained well by a three-dimensional diffusion model.

So, how do I feel to have had some of my research empirically falsified? Well, actually, it makes me rather happy! Now don’t get me wrong: I would much rather that the two photon microscopy had shown T cells moving along the lymph nodes, stopping at dendritic cells for a while, and then moving along again, as my work of [3, 4] proposed. That is not the case. I take solace in two things. First, the convection-diffusion model of [2] is essentially correct in its most naive form. But more significantly, I can hear Karl Popper cheering me on from the side-lines: my hypotheses were good science, even if incorrect. The important point to learn from [2, 3, 4] is that the observed distribution of lymphocyte transit times is non-trivial: it demands a mechanistic explanation, and, at that time, no such explanations had been proposed. The dendritic cell hypothesis was perfectly plausible (in fact, I came up with it through discussions with Benny Chain and David Katz, with whom Benny Chain shared a lab) – and worthy of experimental testing. (Parenthetically, this was in the bad old days of “mathematical biology”, when theoreticians generally worked independently of experimentalists. I remember once, at that time, a theoretician proudly stating at a conference that no experimental data could falsify their model. Thankfully things are much better today under the “systems biology” paradigm, and Thomas et al’s article is an excellent example of experiment and theory working so well together).

To be honest, the thing that annoys me a little is that I didn’t think to check myself, during my PhD, whether three dimensional diffusion could explain the distribution of recirculation times (and full credit to Niclas Thomas and others for investigating this!). At the time, I was unhappy with the value of the one-dimensional diffusion coefficient that my first model needed to fit the data: based on a Brownian motion calculation for T-cell diffusivity, I thought that it was two orders too fast to be realistic as random motion, and so looked to other explanations. Indeed, this was discussed at my PhD viva, and my examiners (an eminent modelling-friendly immunologist and an eminent mathematical biologist) made me correct my thesis to include the Brownian motion calculation. But the assumptions behind the calculation were completely wrong, for reasons that my examiners and I should have known. First, on biological grounds, the calculations were based on Brownian motion of T-cells – when of course we knew that T-cells move actively – and even then there was some data available on speed of such movement (e.g. from Tim Springer’s lab). Second, the calculations were based on a 1-D diffusion – when of course we know that 3D diffusion is qualitatively and quantitatively different (e.g. the recurrence / transience of 1D / 3D random walks taught in undergraduate Markov chain courses). I had even written a 3D diffusion simulator (for another part of my PhD) which could have easily been used to test the hypothesis. Hind-sight is a wonderful thing!

All-in-all, I congratulate Niclas Thomas on this work. I think it is wonderful and enhances our knowledge of this extremely important field.

1. Thomas N, Matejovicova L, Srikusalanukul W, Shawe-Taylor J, Chain B. 2012. Directional Migration of Recirculating Lymphocytes through Lymph Nodes via Random Walks. PLoS ONE 7(9): e45262.

2. Stekel, D.J., Parker, C.E. and Nowak, M.A. 1997. A model of lymphocyte recirculation. Immunology Today, 18:216-21.

3. Stekel, D.J. 1997. The role of inter-cellular adhesion in the recirculation of T lymphocytes. Journal of Theoretical Biology, 186:491-501.

4. Stekel, D.J. 1998. The simulation of density-dependent effects in the recirculation of T lymphocytes. Scandinavian journal of immunology47:426-30.

5. Miller MJ, Hejazi AS, Wei SH, Cahalan MD, Parker I. 2004. T cell repertoire scanning is promoted by dynamic dendritic cell behavior and random t cell motility in the lymph node. Proceedings of the National Academy of Sciences of the United States of America 101: 998-1003.

6. Beltman JB, Mare AFM, Lynch JN, Miller MJ, de Boer RJ. 2007. Lymph node topology dictates t cell migration behavior. The Journal of Experimental Medicine 204: 771–780.

7. Textor J, Peixoto A, Henrickson SE, Sinn M, von Andrian UH, Westermann, J. 2011. Defining the quantitative limits of intravital two-photon lymphocyte tracking. Proceedings of the National Academy of Sciences of the United States of America 108: 12401–6.