December 21, 2013

Dragons, Sandpiles, and Cavefish: an evolutionary inquiry

One area of evolutionary science that has always fascinated me involve subtle evolutionary mechanisms. Having an interest in evolutionary modeling and theoretical biology, I am particularly interested in evolutionary mechanisms that are nonlinear, and provide a path towards complex evolutionary dynamics. It is somewhat different from a traditional phylogenetic model, and requires a significant departure from standard population genetics thinking as well.

Whether this belongs to the extended evolutionary synthesis or not is not clear, although a mechanism-first approach is inclusive of development and other life-history considerations. We will begin by looking at a new paper [1] on the evolution of Mexican cavefish (Astyanax mexicanus) populations. A. mexicanus had been previously identified as a prime example of developmental processes playing a role in morphological divergence between species [2]. Namely, the cave-dwelling morph has lost its eyes, which are not needed in the cave environment. Figure 1 shows the latest version of this story.

Overview of Hsp90 phenotypic capacitance mechanism, based on account from [1].

In the latest paper, an inducible system is tested which depletes the available amount of Hsp90 (a chaperrone molecule which aids in protein folding). A few notes on the changes that have been linked to the absence of Hsp90:

1) the relationship between Hsp90 (chaperone) and proteins is one of a metastable signal transducer [3]. For example, one folding state results when chaperrone is present, while another state results when the chaperrone is not. This results in a sigmoidal response function. As the chaperrone is depleted, some deleterious traits become unmasked. But for large-scale changes to occur, a complete depletion of the
chaperrone is required.

Schematic demonstrating the shape of a sigmoidal function.

2) Hsp90 is intentionally overproduced in the sense that enough of the chaperone is available when unpredictable environmental stresses occur, requiring a greater amount of chaperrone to achieve the proper folding. This baseline is a conserved mechanism for morphological robustness (sometimes called phenotypic buffering).

3) in general, the more environmental stress that exists during development, the more Hsp90 is needed and used. When Hsp90 is exhausted, deleterious and large-scale changes can be unmasked (sometimes called phenotypic capacitance).

4) since selection does not act directly upon masked variation, multiple variants can be unmasked at the same time, revealing large changes in phenotype (similar to the notion of hopeful monster).

A hopeful monster represented using the Fisherian model of evolutionary mutation. Taken from [4].

The bottom line is that while an eyeless phenotype would not have a high fitness in an above-ground environment, having eyes would be quite costly in a cave environment. Thus, eyeless phenotypes would suddenly have a high fitness, but only in the context of this niche. But how do you get from point A to point B, particularly when most existing theoretical models assume gradual and/or genotypic-driven change?


This is where statistical models of extreme events comes into play. While the biological model suggests that eyeless phenotypes is the consequence of a failed protective mechanism, we can also understand these changes as extreme events that have a statistical distribution in any evolutionary context. Fortunately, we can turn to statistical physics for two candidate models: the Abelian sandpile model [5], and the Dragon King model [6].

Conceptual model and agent-based Java simulation of an Abelian Sandpile Model (a.k.a. the Bak-Weisenfeld-Tang model). Adding sand grains to the pile results in avalanches distributed according to a power law distribution.

In both cases, we must make the assumption that extreme events are not only possible but inevitable as evolutionary outcomes. In both "sandpile" and "dragon" evolution, extreme events can drive processes like speciation, niche specialization, and evolutionary diversification. The difference between sandpile evolution and dragon evolution involves whether or not extreme events are due to the same process as other evolutionary outcomes which are smaller in magnitude. This should not be interpreted as a verdict on so-called "gene-centered" evolution [7] -- while sandpile evolution is more likely to be dominated by changes in the genetic architecture, dragon evolution simply provides ways to organize the expression of these genotypic changes.

Distribution in time (top) and probability distribution (bottom) of Dragon King events (notice that they deviates from a conventional power law in the tail region). Images taken from [8].

The sandpile model demonstrates that the same underlying process (in this case, the growth and avalanche dynamics of a sandpile) is responsible for observed events of every magnitude. While this process is stochastic and unpredictable, it can be characterized using a power law distribution [5]. While you can predict the existence of avalanches (and perhaps at a certain frequency), you cannot predict when they will happen or the chain of events that lead up to them. In "sandpile evolution", the mutational structure serves as the driving force for evolutionary change. Even when the genotypic mutation rate is constant, cumulative changes (driven by delayed feedback) could sometimes lead to large-scale and sudden changes in phenotype.

A dynamical phase space representation of the Dragon King event, taken from [8]. In this case, the dynamical behavior of a coupled chaotic oscillator sporadically wanders far outside of its attractor orbit, resulting in an extreme event.

The Dragon King model, by contrast, assumes that events of large magnitude are not due to the same processes as events of small magnitude [6]. Dragon King events, such as financial crises [9], coherent structures in turbulent fluids [6], and the behavior of coupled chaotic oscillators [8], cannot be characterized well by a typical power law distribution, with exceptional differences in the tail region [6]. While "dragon
evolution" relies upon two or more concurrent processes, there is a historical contingency that allows for one of these processes to be sporadically amplified. This amplificion is accomplished through cumulative negative feedback from some mediating factor (perhaps chaperrones). Much as in the case of sandpile evolution, this generates large-magnitude events as a low frequency.

From what I can tell, the model of phenotypic capacitance for the cavefish matches the Dragon King criteria quite well. In this case, you have a dynamical system -- a variable concentration of Hsp90 that changes deterministically with respect to stochastic environmental fluctuations. When the Hsp90 concentration reaches zero (which happens rarely and represents a lower-bound), the phenotypic system sojourns far from equilibrium. Crucially, the depletion of Hsp90 and the absence of Hsp90 behave as independent systems: the depeletion of Hsp90 merely allows for deleterious phenotypes to be expressed.

It is of note that the original Hsp90 experiments in Drosophila [10], most of these phenotypes turned out to be embryonic lethal. But, using a different mechanism, the absence of Hsp90 allows for suites of mutations (representing latent variation) to be expressed, and resulting in a coherent, non-lethal embryonic phenotype that can have high fitness in a narrow range of environmental contexts. Perhaps this is the beginnings of a mathematical model for evo-devo!

Reconciling the Dragon King in phase space with phenotypic evolution. Images taken from [8] and [4].

NOTES:

[1] Rohner, N., Jarosz, D.F., Kowalko, J.E., Yoshizawa, M., Jeffery, W.R., Borowsky, R.L., Lindquist, S., and Tabin, C.J.   Cryptic Variation in Morphological Evolution: Hsp90 as a Capacitor for Loss of Eyes in Cavefish. Science, 342, 1372-1375 (2013). Associated Phenomenon blog article can be found here.

[2] Jeffery, W.R.   Evolution and development in the cavefish Astyanax. Current Topics in Developmental Biology, 86, 191-221 (2009).

[3] Pratt, W.B., Morishima, Y., and Osawa, Y.   The Hsp90 Chaperone Machinery Regulates Signaling by Modulating Ligand Binding Clefts. Journal of Biological Chemistry, 283, 22885-22889 (2008).

[4] Chouard, T.   Revenge of the Hopeful Monster. Nature, 463, 864-867 (2010).

[5] Bak, P., Tang, C., and Wiesenfeld, K.   Self-organized criticality: an explanation of 1/ƒ noise. Physical Review Letters, 59(4), 381–384 (1987).

[6] Sornetts, D. and Ouillon, G.   Dragon-kings: mechnisms, statistical methods, and empirical evidence. European Physical Journal, 205, 1-26 (2012).

[7] For the latest shots (and resulting kerfuffle) in this debate, please see: Dobbs, D. "Die, Selfish Gene, Die" has evolved. David Dobbs' Neuron Culture blog, December 13 (2013).

[8] de S. Cavalcante, H.L.D., Oria, M., Sornette, D., Ott, E., and Gauthier, D.J.   Predictability and Suppression of Extreme Events in a Chaotic System. Physical Review Letters, 111, 198701 (2013).

[9] Sornette, D.   Dragon-kinds, black swans, and the prediction of crises. International Journal of Terraspace Science and Engineering, 2(1), 1-18 (2009). Associated TED talk can be found here.

[10] Lindquist, S.L. and Rutherford, S.   Hsp90 as a capacitor for morphological evolution. Nature, 396, 336-342 (1998).

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