Dust flux, Vostok ice core

Dust flux, Vostok ice core
Two dimensional phase space reconstruction of dust flux from the Vostok core over the period 186-4 ka using the time derivative method. Dust flux on the x-axis, rate of change is on the y-axis. From Gipp (2001).

Sunday, May 15, 2011

(Scaling) laws of life and death

Another blast from the past*

One of the enduring problems in historical geology is the relationship between speciation and extinction. Geological history is punctuated by episodes of mass extinction, when in response to tectonic, climatic, or even astronomic events, large numbers of species become extinct in a short period of time. What happens in the aftermath?

One idea is that with the Earth denuded of many lifeforms, there are a large number of ecological niches "up for grabs" by the first applicant. Natural selection may select for those organisms which have undergone morphological and/or behavioural changes that more efficiently exploit the opportunities of the vacant niche. The logical consequence is that speciation should increase rapidly after a mass extinction.

One problem with this idea is that natural selection is a mechanism for culling away the unfit--it is not a mechanism by which innovation occurs. Innovation has been linked to random mutation, although occasionally papers appear claiming that the mutations are not random. The Lamarckian idea of directed mutation in nature has been discredited.

We might say this is an existential problem, for just as one may contemplate one's own mortality, one may contemplate the mortality of the human race. Are there predictable laws of extinction, and are we governed by them?

In pondering this issue we are not helped by a belief in our own exceptionalism. So for the duration of this essay, let us consider merely the idea of extinction of any number of species, and only afterwards ponder its relevance for ourselves.

Extinction (for our purposes) represents termination of a species.

What can we say about the processes of speciation and extinction? Are they governed by the same dynamics?

Spectral power graphs for extinctions and originations of marine 
families over the last 500 million years. From Kirchner (2002).

The Fourier power spectra for rates of extinction and origination of fossil marine families provides insight into the dynamics of extinction and evolution. On the graph above, the extinction power spectrum is relatively constant, with undulations. The highest frequency undulation (the last wiggle on the left) is Raup and Sepkoski's (1984) "death star" peak--the extinction peak at 28 million year intervals attributed to the periodic approach of a neutron star to our solar system.

The graph for originations shows a steady decline with increasing frequency, so that at higher frequencies (shorter timescales) the rate of originations falls below the rate of extinction. Kirchner (2002) used this behaviour to infer that the rate of originations, especially at higher frequencies (periodicities less than 25 million years) was lower than the rates of extinctions; implying that there was a limit to the rate at which innovations could appear in the fossil record.

Although a correct observation, I believe the explanation for it is incorrect. The significance of the decline in rate of origination with frequency (note the nearly straight line of best fit on a log-scale) is that originations have a scale-invariant character, whereas extinctions (horizontal line of best fit) are random.

Over geologic time, rates of originations have to be at least as high as rates of extinction, or all life would be extinguished. We see that on long time intervals, rates of originations are higher than rates of extinction.

Scale invariance or random chance?

Scale invariance is a common characteristic of geological systems (Turcotte, 1997), and has been observed in such diverse phenomena as earthquakes, volcanic eruptions, and climate change. Such behaviour can be clearly demonstrated from the power spectra of some geological record. When projected on a log-log plot, the best fit of the power spectrum is a straight line of constant slope. The slope of this line is called the scaling exponent, and can be related to the fractal dimension which characterizes the size-time distribution of events.

Discrete scale invariance is a weaker form of invariance, where the scaling is not apparent at all frequencies, but only for a certain range of frequencies (Sornette, 1998). Discrete scale invariance is only described by a complex fractal dimension, the imaginary part of which is a simple function of the discrete scaling exponent (Saleur et al., 1996). Such behaviour is shown to exist in a wide variety of conditions, including diffusion, fracture propogation, fault rupture (in time), hydrodynamic cascades, turbulence, the Titus-Bode law and gravitational collapse of black holes (Sornette, 1998).

Thus, spectral power (P) varies as a function of frequency (f) so that for an arbitrary change f → λf there exists a number μ such that P(f) → μP(λf). This is a homogeneous function encountered in the theory of critical phenomena (Bak et al., 1988), and is solved by a power law P(f) = Af**(-α), where α = -log(μ)/log(λ). Power laws are the “fingerprint” of scale invariance, as the ratio of P(f):P(λf) is independent of f. Thus the relative spectral density is a function of the ratio of the two frequencies, and this property is the fundamental one which associates power laws to scale invariance, self-similarity, and self-organized criticality (Bak et al., 1988).

Fractal analyses of the scaling behaviour can be used to provide more information about the dynamics of speciation and diversification. The scaling exponent is the slope of the best fit line fitted to the log-log power spectrum. There is a correlation between the size-time distribution of events and the scaling exponent. If this slope is ~0, then the the distribution of events through time resembles white (i.e., random) noise.

Importantly, there is no trend observed in the power spectrum for extinctions. The implication to be drawn from this observation is that extinctions are randomly distributed, both in size and in time. They are not governed by processes at all like those that control diversification. The apparently random nature of extinctions is something of a mystery, as it would suggest that extinctions are not related to originations (new critters outcompeting the old). Nor does it fit with extinctions being related to large tectonic, climatic, or external events (bolides), all of which are believed to be systems at a state of self-organized criticality (Bak et al., 1988), and show a measured increase in power with decreasing frequency. Randomness in extinctions may imply that they are dominated by gambler's ruin.

By contrast, the power spectrum for the originations undulates about a line with a constant negative slope on the log-log graph. The slope of the eyeballed line of best fit (don't have access to the real data) is about -0.9 (i.e. a = 0.9). This is consistent with a system at a state of self-organized criticality.

Natural systems displaying self-organized criticality (SOC) are known throughout the geological realm. Tectonic and volcanic activity shows such a distribution, as does the distribution of large climatic disturbances. From the data analysis of Kirchner (2002), it is unclear whether the fingerprint of SOC arises from external influences or is an internal character of the evolutionary process.

From a geological perspective, it is natural to assume that SOC is imprinted on evolution by environmental processes. But scaling laws are observed at the level of proteins (Unger et al., 2003) as well as at the gene- and species level (Harrada et al., 2011), suggesting that SOC is inherent in life itself.

Bak, P., Tang, C., and Wiensenfeld, K., 1988. Self-organized criticality. Physical Review A, 38: 364-374.

Bonnet, E., Bour, O., Odling, N. E., Davy, P., Main, I., Cowie, P., and Berkowitz, B., 2001. Scaling of fracture systems in geological media. Reviews of Geophysics, 39: 347-383.

Erwin, D. H., 1998. The end and the beginning: recoveries from mass extinctions. Trends in Ecology and Evolution, 13: 344-349.

Herrada, E. A., et al., 2011. Scaling laws of protein family phylogenies. http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.4540v2.pdf

Kirchner, J. W., 2002. Evolutionary speed limits inferred from the fossil record. Nature, 415: 65-68.

Kirchner, J. W. and Weil, A., 1997. No fractals in fossil extinction statistics. Nature, 395: 337-338.

Raup, D. M., and Sepkoski, J. Jr., 1984. Periodicity of extinctions in the geological past. Proceedings of the National Academy of Sciences 81 (3): 801–805.

Saleur, H., Sammis, C. G., and Sornette, D., 1996. Discrete scale invariance, complex fractal dimensions, and log-periodic fluctuations in seismicity. Journal of Geophysical Research, B101: 17,661-17,677.

Sepkoski, J. J. Jr., 1993. Ten years in the library: new data confirm paleontological patterns. Paleobiology, 19: 43-51.

Sornette, D., 1998. Discrete-scale invariance and complex dimensions. Physics Reports, 297: 239-270.

Unger, R., Ulierl, S., and Havlin, S., 2003. Scaling law in sizes of protein sequence families: from super-families to orphan genes. Proteins, 51 (4): 567-576. doi:10.1002/prot.10347.

* I wrote this article nearly ten years ago, which is why the references are so dated.Updated recently just for you.

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