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Papers in computational evolutionary biology

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Internal coarse-graining of molecular systems

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ResearchBlogging.org
Feret J, Danos V, Krivine J, Harmer R, & Fontana W (2009). Internal coarse-graining of molecular systems. Proceedings of the National Academy of Sciences of the United States of America, 106 (16), 6453-8 PMID: 19346467, PNAS page, Supporting Information.

 

Models of molecular dynamics suffer from combinatorial explosion: the phenomenon of an exponential number of combinations arising from  a small set of basic entities. A protein with 10 phosphorylation sites, for example, can exists in 2^10 = 1024 distinct forms (states); if any two of these can form a complex, then the number of distinct molecular species rises to 525312. For a modeller tasked with building a mathematical description of such a system combinatorial explosion is a major problem, for it prohibits explicit representation of every species, and—more importantly—makes straightforward models (i.e. one equation per species) computationally intractable. On the other hand, a simple system like the one described above can reasonably be expected to admit a simple model capturing its essential features. How to build it, then?

One solution is to use rule-based languages, where instead of modelling molecular species, one builds parametrised models of  the biochemical reactions the species engage in. The key idea is that most of the technical differences between species do not matter for their ability to take part in a particular interaction, and hence there are substantially less interaction patterns (a.k.a. rules) than there are species, each pattern being applicable in a large chunk of the species space. In this way rule-based modelling avoids the combinatorial explosion as far as specification of the system is concerned. The execution cost, however, is often still prohibitive.

Feret et. al. offer an ingenious method of reducing the computational cost of the analysis of rule-based models. It is based on the simple observation that while an external human observer may distinguish between two different species, the dynamical system itself may be unable to do so. To quote from the paper (emphasis added):

…an experimental technique might differentiate between SOS recruited to the membrane via GRB2 bound to SHC bound to the EGF receptor and SOS recruited via GRB2 bound to the EGF receptor directly. However, from the perspective of the EGF signalling system, such a difference might not be observable for lack of an endogenous interaction through which it could become consequential. The endogenous units of the dynamics may differ from the exogenous units of the analysis.

The natural consequence of this observation is that one can use the information contained in the rules to infer what species are indistinguishable in the above sense and provide just one equation per cluster of indistinguishable species (called a fragment in the paper). This is exactly what authors do, and the results for their benchmark model of the EGFR pathway are very encouraging. In the case of a simpler model (39 rules), there are 10 times less fragments than species; in the case of the bigger model (71 rules), the methods yields a staggering million million-fold (10^12) dimensional reduction.

It is important to realise that the notion of dynamical indistinguishability of species is not merely a technical device for model reduction. It captures a property that is essential to evolution and dynamical stability of molecular systems, and does it from the semantic rather than syntactic perspective (i.e. by focussing on the equivalence of dynamics rather than equivalence of model descriptions). As such, it is worth investigating in much greater detail. Another important point is that the method is not a statistical heuristic that may fail for special cases. All species lumped together in a fragment are provably indistinguishable from each other. The only sub-optimality is the possibility that two species are in fact dynamically indistinguishable, but the method separates them anyway. These issues are discussed at length in the supporting information, linked above.

Finally, a word of warning: the authors use and develop sophisticated mathematics and computer science, not molecular (nor even theoretical) biology. Readers without quantitative background may struggle to follow the paper.

(Full disclosure: one of the authors is going to act as an examiner of my Ph.D. thesis.)

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October 18, 2010 at 16:23

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Curvature in Metabolic Scaling

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Tom Kolokotrones, Van M. Savage, Eric J. Deeds and Walter Fontana Curvature in Metabolic Scaling Nature 464:753-756, 2010. Nature page

This paper is not about evolution, but it is short, recent, published in Nature and comes from Fontana Lab, so there is definitely no harm in reviewing it. It deals with metabolic scaling, that is the relationship between an organism’s metabolic rate and its body mass. Experimental measurements seem to indicate that the metabolic rate is proportional to the body mass raised to a fixed power. The actual value of the exponent was first thought to be 2/3, and then 3/4; the latter was also derived by West et. al. from an involved theoretical model of vascular system [1].

Kolokotrones et. al. took a large dataset and showed that instead of a simple power law a more complex expression involving two exponents is a much better fit. When plotted on a log-log scale, the graph of this function is a slightly convex curve, rather than the straight line resulting from a pure power law; hence the title of the paper. Of course by introducing a new degree of freedom you will always get a better fit, but the improvement in this case is considerable, and, crucially, the curve can be approximated in different regions by pure power laws with the well-established exponents. This shows that essentially both the 2/3 and 3/4 hypotheses were correct.

A mechanistic explanation for the 3/4 theory was provided by West’s model, and so the authors set out to modify it to get a two-exponent formula instead. Apparently it is possible by postulating a different moment of transition between the pulsatile and smooth blood flow dynamics. More details can be found in Supplementary Information, if you’re interested (I am not).

Now, it is possible that the curved fit does not represent any underlying biological principle. As mentioned above, the curve can be approximated by two or more power laws acting on different parts of the data. It is conceivable that the relationship is in fact a pure power law, but evolutionary distant families of mammals (the study is on mammals) evolved—for whatever reasons—different exponents. Through phylogenetic analysis, Kolokotrones et.al. show that this is not the case, and that curvature is observed in subsets of data corresponding to closely related species. Other factors, such as habitat and food type were also excluded, suggesting that there is an underlying mechanistic principle at work.

[1] West, G. B., Brown, J. H. & Enquist, B. J. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126 (1997).

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April 7, 2010 at 17:44

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