4: Fitting Brownian Motion - Biology

4: Fitting Brownian Motion - Biology

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By fitting a Brownian motion model to phylogenetic comparative data, one can estimate the rate of evolution of a single character. In this chapter, I demonstrated three approaches to estimating that rate: PICs, maximum likelihood, and Bayesian MCMC. In the next chapter, we will discuss other models of evolution that can be fit to continuous characters on trees.

  • 4.1: Introduction
    Mammals come in a wide variety of shapes and sizes. Body size is important as a biological variable because it predicts so many other aspect of an animal’s life, from the physiology of heat exchange to the biomechanics of locomotion. Thus, the rate at which body size evolves is of great interest among mammalian biologists. Throughout this chapter, I will discuss the evolution of body size across different species of mammals. The data I will analyze is taken from Garland (1992).
  • 4.2: Estimating Rates using Independent Contrasts
    The information required to estimate evolutionary rates is efficiently summarized in the early (but still useful) phylogenetic comparative method of independent contrasts (Felsenstein 1985). Independent contrasts summarize the amount of character change across each node in the tree, and can be used to estimate the rate of character change across a phylogeny. There is also a simple mathematical relationship between contrasts and maximum-likelihood rate estimates that I will discuss below.
  • 4.3: Estimating rates using maximum likelihood
    We can also estimate the evolutionary rate by finding the maximum-likelihood parameter values for a Brownian motion model fit to our data. Recall that ML parameter values are those that maximize the likelihood of the data given our model (see Chapter 2).
  • 4.4: Bayesian approach to evolutionary rates
    Finally, we can also use a Bayesian approach to fit Brownian motion models to data and to estimate the rate of evolution. This approach differs from the ML approach in that we will use explicit priors for parameter values, and then run an MCMC to estimate posterior distributions of parameter estimates. To do this, we will modify the basic algorithm for Bayesian MCMC.
  • 4.5: Summary

4: Fitting Brownian Motion - Biology

Humans and other species continually perform microscopic eye movements, even when attending to a single point [1, 2, 3]. These movements, which include drifts and microsaccades, are under oculomotor control [2, 4, 5], elicit strong neural responses [6, 7, 8, 9, 10, 11], and have been thought to serve important functions [12, 13, 14, 15, 16]. The influence of these fixational eye movements on the acquisition and neural processing of visual information remains unclear. Here, we show that during viewing of natural scenes, microscopic eye movements carry out a crucial information-processing step: they remove predictable correlations in natural scenes by equalizing the spatial power of the retinal image within the frequency range of ganglion cells' peak sensitivity. This transformation, which had been attributed to center-surround receptive field organization [17, 18, 19], occurs prior to any neural processing and reveals a form of matching between the statistics of natural images and those of normal eye movements. We further show that the combined effect of microscopic eye movements and retinal receptive field organization is to convert spatial luminance discontinuities into synchronous firing events, beginning the process of edge detection. Thus, microscopic eye movements are fundamental to two goals of early visual processing: redundancy reduction [20, 21] and feature extraction.


► In natural viewing, microscopic eye movements whiten (decorrelate) visual stimuli ► Whitening is due to a match between eye movements and natural scene characteristics ► Elimination of input redundancy starts before any neural processing takes place ► In a continually moving eye, the process of edge extraction starts in the retina


Either a single rooted tree or a list of rooted trees, of class "phylo", corresponding to the first data set on which a BM model is to be fitted. Edge lengths are assumed to represent time intervals or a similarly interpretable phylogenetic distance.

Numeric state of each trait at each tip in each tree in the first data set. If trees1 is a single tree, then tip_states1 must either be a numeric vector of size Ntips or a 2D numeric matrix of size Ntips x Ntraits, listing the trait states for each tip in the tree. If trees1 is a list of Ntrees trees, then tip_states1 must be a list of length Ntrees, each element of which lists the trait states for the corresponding tree (as a vector or 2D matrix, similarly to the single-tree case).

Either a single rooted tree or a list of rooted trees, of class "phylo", corresponding to the second data set on which a BM model is to be fitted. Edge lengths are assumed to represent time intervals or a similarly interpretable phylogenetic distance.

Numeric state of each trait at each tip in each tree in the second data set, similarly to tip_states1 .

Integer, specifying the number of parametric bootstraps to perform for calculating the confidence intervals of BM diffusivities fitted to each data set. If <=0, no bootstrapping is performed.

Integer, specifying the number of simulations to perform for assessing the statistical significance of the log-transformed difference between the diffusivities fitted to the two data sets, i.e. of |log(D_1)-log(D_2)|. Set to 0 to not calculate the statistical significance. See below for additional details.

Logical, specifying whether to perform some basic checks on the validity of the input data. If you are certain that your input data are valid, you can set this to FALSE to reduce computation.

Logical, specifying whether to print progress report messages to the screen.

Character, specifying a prefix to include in front of progress report messages on each line. Only relevant if verbose==TRUE .

The Traditional Approach to Parameter Estimation

Traditionally, the parameters are estimated using the empirical mean and standard deviation of the daily logarithmic (or geometric) returns. The reasoning behind this can be seen by re-arranging the above equation for ( ilde) as follows,

From which it is possible to solve the implied simultaneous equations for (mu) and (sigma) , as functions of the mean and standard deviation of the geometric (i.e. logarithmic) returns. Once again, for a more in-depth discussion we refer the reader to ‘A Stochastic Processes Toolkit for Risk Management’, by Damiano Brigo et al..

Parameter Estimation when Data is Plentiful

An example computation, using the whole time-series generated above (364 observations of daily returns), is shown below. We start by taking a look at the distribution of daily returns.

The empirical distribution is relatively Normal in appearance, as expected. We now compute (mu) and (sigma) using the mean and standard deviation (or volatility) of this distribution.

We can see that the empirical estimate of (sigma) is close to the ex ante paramter value we chose, but that the estimate of (mu) is poor - estimating the drift of a stochastic process is notoriously hard.

Parameter Estimation when Data is Scarce

Very often data is scare - we may not have 364 observations of geometric returns. To demonstrate the impact this can have on parameter estimation, we sub-sample the distribution of geometric returns by picking 12 returns by random - e.g. to simulate the impact of having only 12 monthly returns to base the estimation on.

We now take a look at the distribution of returns.

And the corresponding empirical parameter estimates.

We can clearly see that now estimates of both (mu) and (sigma) are poor.

Create a bm Object

Create a univariate Brownian motion ( bm ) object to represent the model: d X t = 0 . 3 d W t .

bm objects display the parameter A as the more familiar Mu .

The bm class also provides an overloaded Euler simulation method that improves run-time performance in certain common situations. This specialized method is invoked automatically only if all the following conditions are met:

The expected drift, or trend, rate Mu is a column vector.

The volatility rate, Sigma , is a matrix.

No end-of-period adjustments and/or processes are made.

If specified, the random noise process Z is a three-dimensional array.

If Z is unspecified, the assumed Gaussian correlation structure is a double matrix.


Posted on October 29, 2012 by Edwin Grappin in R bloggers | 0 Comments

The Brownian motion is certainly the most famous stochastic process (a random variable evolving in the time). It has been the first way to model a stock option price (Louis Bachelier’s thesis in 1900).

The reason why is easy to understand, a Brownian motion is graphically very similar to the historical price of a stock option.

A Brownian motion generated with R (n = 1000)

The historical price of the index FTSE MID 250, source Yahoo Finance.

Even though it is not any more considered as the real distribution of stock options, it is still used in finance and methods to simulate a Brownian motion are really useful. As you will see the code is strikingly easy. We have used the Donsker’s principle.

If Y(i) is a serie of k variables normally distributed, then we can create X(i) = (Y(1) + … + Y(i))sqrt(i/k). Then X(i) is a proxy of the Brownian motion.

size = 1000
y = rnorm(size)
x = y

for (i in 1:size) <
x[i] = 1/sqrt(size)*(sum(y[1:i])*sqrt(i))


This first broad comparative study of environmental effects on encephalization within marine teleost fishes supports the prediction that encephalization correlates positively with BMR (using depth as a proxy) and hence the DMCH. However, we find little support for the hypothesis proposed by Gillooly et al. ( 2001 ) that a decrease in temperature, here indicated by latitude, correlates with smaller brains. We find no support for the hypothesis that increased energy intake due to higher trophic level increases encephalization, and therefore no support for the expensive tissue hypothesis, which predicts that increased investment in neural tissue correlates with diet-mediated decreases in gut mass. The only significant association between encephalization and trophic level was confined to species in the mesopelagic zone, and that correlation was opposite to the direction predicted. We discuss the implications of these results below.

The influence of latitude on encephalization

Our results indicate that fishes at higher latitudes, and hence lower temperatures, are less encephalized. However, this relationship is only detected when all zones are analysed together. This suggests that the effect of lower ambient temperature on brain size, if it exists, may be too weak to be detected within smaller subsets of the data. Alternatively, the lack of an effect of latitude when depth zones are analysed separately may suggest either that cold compensation in higher latitude fishes (Torres & Somero, 1988a van Dijk et al., 1998 Hardewig et al., 1998 Brodeur et al., 2003 Portner et al., 2005 ) releases temperature constraints on encephalization or that temperature itself does not directly affect encephalization, but correlates with other factors that are responsible (Clarke, 1993 Clarke & Fraser, 2004 ).

The influence of temperature on the evolution of fish neural anatomy is poorly understood, although studies of Antarctic fishes suggest that the influence of temperature on brain metabolic activity may be mitigated by some combination of high concentrations of mitochondria, increased enzyme efficiency and/or increased enzyme concentrations (Hardewig et al., 1999 Kawall et al., 2002 Somero, 2004 ). Comparative analyses of Antarctic fish brains have revealed these to be more similar to other temperate teleosts than deep-sea fishes (Eastman & Lannoo, 1995 ), and given, even partial adaptation to permanently cold conditions such as those suggested above could provide an explanation for the lack of a strong trend between brain size and latitude in our analyses. However, the latitudinal temperature range within the epipelagic zone (36 to −2 °C) mirrors the temperature range for depth across all zones. If temperature were the main driver of encephalization, latitude should correlate with encephalization within the epipelagic zone. As we do not detect this pattern, we conclude that temperature does not directly affect encephalization.

The influence of feeding ecology on encephalization

There is a general correlation between shorter guts and higher quality diets in fishes (Kramer & Bryant, 1995 Elliott & Bellwood, 2003 Wagner et al., 2009 ), although there are exceptions (Day et al., 2011 Pogoreutz & Ahnelt, 2014 ). The expensive tissue hypothesis predicts that there is a trade-off between investing in guts and brains (Aiello & Wheeler, 1995 ) therefore, we would expect to detect an effect of trophic level on encephalization in our analyses. Although an intraspecific study in a marine fish did not show trade-offs between expensive tissues such as testes, liver and brain (Warren & Iglesias, 2012 ), we considered it possible that an indirect measure of a trade-off between gut length and brain mass across almost 500 species of fish would reveal such a pattern if it existed. However, our results offer no evidence that the expensive tissue hypothesis holds across marine fishes. The lack of any effect of trophic level on encephalization in the epipelagic zone is particularly striking, given that epipelagic zone species show the most variation in trophic level and composed over seventy per cent of the species in our data set.

Although encephalization in marine fishes of the mesopelagic was partially explained by trophic level (Tables 2 and 3), this finding disagrees with expectations under the expensive tissue hypothesis. Rather than finding an increase in encephalization at higher trophic positions, our analysis supported an inverse relationship. This trend of increased brain size relative to body size at lower trophic positions may be partially explained by the increased sensory needs of planktonic feeders at depths below 200 m. Upper mesopelagic fishes and mesopelagic fishes undergoing vertical migrations are characterized by extreme visual modifications that allow for detecting slight contrasts of light and making use of bioluminescence (Kotrschal et al., 1998 ). Plankton feeders in particular tend to have greater eye and lateral line modifications in order to detect more minute prey quantities (Bleckmann, 1986 Coombs et al., 1988 ). While changes in brain morphology have been associated with epipelagic fishes living in turbid water (Huber & Rylander, 1992 Kotrschal et al., 1998 ), it is unclear whether there are also systematic trends in the reduction or enlargement of specific brain regions in mesopelagic fishes at different trophic levels. Our analyses highlight this as a particularly fruitful area of future research.

A caveat to these results is that the accuracy of trophic level calculated by FishBase (Froese & Pauly, 2014 ) is unclear. Fish species on FishBase for which diet data are lacking can still be assigned a trophic level calculated based on what is known about the closest relative in the database with available diet information (Froese & Pauly, 2000 ). This is an assumption that has unknown reliability, as patterns of trophic disparity and missing data will vary nonrandomly among taxa. Several studies have attempted to verify FishBase's assignment of trophic level, with some showing good agreement between empirically derived measures and FishBase trophic levels (Kline & Pauly, 1998 Mancinelli et al., 2013 ) and others showing weak or variable agreement (Faye et al., 2011 Carscallen et al., 2012 ). It is unknown how many of the species included in our analyses had trophic levels that were calculated from other species’ data, a caveat that emphasizes the need for natural history studies to aid our understanding of macroevolutionary trends (McCallum & McCallum, 2006 ). Given this, and that we tested the expensive tissue hypothesis indirectly, our conclusions regarding the validity of the expensive tissue hypothesis should be interpreted with caution.

The influence of depth on encephalization

Our analyses support a strong correlation between depth and encephalization across marine teleosts (Tables 2 and 3). Basal metabolic rate is known to decrease significantly with depth, and temperature only accounts for approximately 2% of that change (Torres et al., 1979 ). Further, the decrease in BMR experienced by deep-water fishes surpasses that experienced by high-latitude fishes at the same temperature (Torres & Somero, 1988a ). Therefore, our results support the DMCH, which states that BMR is a primary physiological factor limiting encephalization.

The DMCH is supported by patterns we detect across depth zones analysed together and independently. The transition from the mesopelagic to the bathypelagic zone has been linked with an abrupt shift towards brain size reduction in teleosts (Fine et al., 1987 Kotrschal et al., 1998 Lannoo & Eastman, 2000 ), and there is also evidence of a similar large-scale pattern of brain reduction in sharks (Yopak & Montgomery, 2008 Yopak, 2012 ). Our results are consistent with these observed shifts (Tables 2 and 3), but it is possible this trend may be explained by greater depths promoting the evolution of alternate sensory systems that are more suited to conditions at depth (Deng et al., 2013 ), which require less brain mass and are therefore less costly to maintain.

Basal metabolic rate is predicted to continue to decrease in bathypelagic teleosts with increasing depth. As brain size is tightly correlated to BMR, the DMCH predicts that encephalization should also continue to decrease with depth. However, is there a lower limit to encephalization in fishes? There is an abundance of slow-moving ambush predators at extreme depths (Koslow, 1996 ), which suggests the possibility that energetic trade-offs might reduce activity levels instead of tissue mass. Further analysis of the relationship between BMR, encephalization and energetic budget trade-offs is currently not possible, as studies of brain morphology and metabolic rates have only been conducted on a few species of deep-sea fishes. As we continue investigating the ecology and evolution of deep-sea fishes, understanding the drivers of brain size reduction and the factors limiting this reduction will significantly increase our understanding of sensory biology.

Broad comparative studies show that encephalization correlates with higher BMR in primates and bats, but not rodents and carnivores (Isler & van Schaik, 2006b ). Avian studies also report a lack of correlation between BMR and encephalization instead, there appears to be a trade-off between pectoral muscle mass and brain mass (Isler & van Schaik, 2006a ). Taken together, these results emphasize that, regardless of the energetic budget, allocation to encephalization is not the only successful strategy towards maximizing fitness (Isler & van Schaik, 2009 ). Given the breadth of ecologies and life histories of the epipelagic fishes in our data, it is possible that life history differences may be counteracting the influence of depth, latitude and trophic level in a variety of ways habitat (reef versus pelagic), feeding mode (ram suction versus sweeping) and social systems have all been linked to brain size (Bauchot et al., 1988 van Staaden et al., 1995 Huber et al., 1997 Dunbar, 1998 Kotrschal et al., 1998 ). This suggests that a broader hypothesis, such as the energy trade-off hypothesis, may generate more specific predictions that can facilitate identification of common trade-offs that must be controlled for when testing encephalization hypotheses within a group of interest. Finally, given our results, we propose that in addition to controlling for body size and phylogeny, comparative encephalization studies examining functional explanations for encephalization should control for BMR where appropriate.


Widespread environmental change requires an understanding of how animals modify movements and space use to accommodate changes in habitat or barriers to movement (Sawyer et al., 2013 ). Behavioural constraints such as site fidelity can limit the ability of individuals to select the highest quality habitats within a landscape (Merkle et al., 2015 ), and these constraints can be detrimental if behaviours persist in spite of decreasing benefits in the face of rapid environmental change (Sih et al., 2011 ). Our study demonstrates that site fidelity is associated with a number of environmental drivers and that, at least in elk, site fidelity is conditional on experiences at sites in the previous year, suggesting that ungulates are equipped with some flexibility to respond spatially to environmental change (Riotte-Lambert & Matthiopoulos, 2020 ). Nonetheless, a number of studies have observed animals returning to sites even after changes to their environment (Wiens et al., 1986 Wyckoff et al., 2018 ), suggesting that their attraction to known sites overrides responses to their environment, at least in the short term. For species with weak fidelity, there is a need to conserve landscapes in ways that ensure animals retain access to large areas that contain ephemeral resources that can vary substantially in space from year to year (Nandintsetseg et al., 2019 Teitelbaum & Mueller, 2019 ). Future studies that quantify behavioural flexibility across lifetimes of individuals and that experimentally manipulate key resources would allow insight into the magnitude and relative strength of environmental and memory effects (Van Moorter et al., 2013 ).


Fluorescence Correlation Spectroscopy (FCS) is a well-established technique that allows measuring diffusion coefficients and chemical reaction rates in solutions [1]. The technique is based on the measurements of the fluctuations of fluorescence intensity emitted from the defined volume of the sample. These fluctuations may be produced by fluorescent molecules diffusing in and out of a sampling volume, which is defined by laser focal volume, and/or photochemical reaction involving fluorescent molecules within the focal volume [2]. The focal volume is therefore a central technical parameter of the FCS method. To evaluate correctly the dynamics of the sample it is critical to measure accurately the size of the focal volume. There are three approaches used to determine the size of the focal volume using samples containing molecules at known concentration, measuring solution with molecules of predetermined diffusion coefficient or scanning the static conformational feature of the sample [3]. All three methods require independent calibration system, which physical, chemical or optical properties may differ from that of the sample. For example the water soluble fluorescence probes (Alexa or Rhodamine) are typically used for the calibration purposes [4]. The diffusion coefficients of such probes, assume to be known in advance, may very due to the nature of the solution containing the fluorescent probe and/or measuring technique used. This in turn will affect the determined value of the focal volume. The other difficulty may result from the differences between a sample and the calibration system with the respect to their optical properties. Typically, the focal volume is determined experimentally using diluted solution of hydrophilic fluorescent dye. The resulting value of the focal volume is derived from experimental data using a series of methodological and technical assumptions regarding mainly the geometry of the optical system and illumination beam, probe stability and negligible influence of other optical effects [5]. Whereas the determination of the focal volume, based on the diffusion of the small fluorescent dye, can be readily used for the characterization of processes taking place in solutions they may not suffice when suspensions of particulate systems are studied [6–9]. The quantitative analysis of processes taking place in a suspension may be affected by slow diffusion time of particulate, photo-physical properties of a fluorescence dye and scattered light. In the paper a methods for the determination of the focal volume and the evaluation of the photo-bleaching and/or blinking effect on FCS measurements are presented. The presented approach relies on the combination of the two autocorrelation techniques dynamic light scattering (DSC) and fluorescence correlation spectroscopy (FCS). The method makes possible the measurement of the correct value of the diffusion coefficient and the determination of the focal volume “in situ”, in the suspension of particulates labelled with fluorescent probe.

Alternative pore hindrance factors: What one should be used for nanofiltration modelization?

In nanofiltration it is important for predictive purposes to obtain retentions and/or reflection coefficients from known sizes of the pores and the molecules of uncharged solutes. This correlation is also needed in order to model the mass transport of salts or other charged species. To complete these model and predictive needs, the hindrance factors have to be correlated with the ratio between the pore and the molecule sizes, λ. There are several correlations proposed in the literature. Moreover, the effect of the applied pressure was not accounted for in these correlations until recent revisions of the transport model. In some cases the action of the pore-wall friction has been also neglected.

Here we make a revision of these different assumptions on the hindrance factors, we discuss their effect on the transport and we show some conditions that a correct correlation should accomplish. It is shown that it is important to consider both the pressure and the pore-wall friction because the corresponding terms have important contributions to both retention and reflection. It is, nevertheless, less relevant an accurate choice of a relationship for the pore hindrance factors in terms of λ, as far as, both retention and reflection are mainly controlled by partitioning in the ranges where the different proposed correlations differ, what leads to the same transport predictions. In any case a theoretically correct correlation can be chosen attending to the conditions that the pore reflection must accomplish.

Watch the video: Βιολογία #4: Φυσική Επιλογή (January 2023).