Drift of continuous traits.

Wikipedia claims that there are no formal models of blending inheritance, although I would be inclined to disagree with this, as Fisher 1930 discusses blending inheritance in enough detail to call it a formal model.  Further, one older web site defines blending inheritance as “A discredited model of inheritance suggesting that the characteristics of an individual result from the smooth blending of fluid like influences from its parents.” (http://groups.molbiosci.northwestern.edu/holmgren/Glossary/Definitions/Def-B/blending_inheritance.html).  Nevertheless, it should be clear that any phenotypic view of evolution will have to include some continuously inherited aspects.  As I have indicated, in the phenotypic view it is phenotypes that create new phenotypes, and that new phenotypes are defined by a transition equation that is determined by the patterning node.  The patterning node, as I have described it, is specifically those influences on a phenotype that are “heritable” in the sense that a trait value in the parent/teacher phenotype in some manner defines the trait value in the offspring/student phenotype.  By the way, it seems to me that “parent” and “offspring” are loaded enough terms that I am looking for a term that would include both parents and non-parent mentors, both of which can contribute to non-genetical inheritance, and the phenotype to phenotype transition equation.

For genetically transmitted traits the rules are quite clear:  for some aspects of the transition equation the simple rules of Mendelian inheritance will suffice, for others some modification of the quantitative genetic concept of heritability will be needed.  The beauty of genetic traits is that because they have particulate inheritance, and we are diploid, there is “hidden” variation, the “segregation variance”, not to mention variance that is tied up in epistatic and dominance relationships, but can feed into the heritability (see last weeks post).  The result is that with quantitative genetics we have the best of all possible worlds.  Indeed, you can basically think of Fisher as having made the world safe for the Biometicians by giving them an explanation for why they can ignore the associated loss of heritable variance.  That is, quantitative genetics assumes “blending inheritance” with no loss of heritable variance.

The problem comes when we have traits that truly follow blending inheritance.  Mendel is not going to save us from the fact that culture and language are continuously inherited.  Fortunately a basic model was provided for us by Fisher.  Of course, Fisher was dealing only with bi-parental inheritance, but the logic still works.  The idea is that if the offspring/student learns from exactly two individuals (i.e., their parents), and those two individuals each contribute equally then the variance among the offspring will be the variance of the average of the two parents.  If we assume random mating we know that the variance of an average is simply the variance of an individual divided by the sample size, n.  Thus the variance of the offspring is equal to V(parents)/2.  Obviously, in the absence of mutational input the variance will go down by one half per generation until it asymptotically approaches zero.

Of course learned traits are never learned perfectly, thus, it is natural to assume that there will be some level of mutation in the transmission process.  In this case we can solve for the steady state where

Cultural drift eq 1

It should be apparent that this balance occurs when the variance in the trait is equal to 2*V(mutation).  Thus, in this simple case of bi-parental inheritance we can say that the heritable variance will be equal to twice the mutational variance for the trait.

More generally, an offspring/student may have multiple individuals that contribute to their patterning node.  Thus, the transition equation for the trait may be a weighted average of a larger set of parents and teachers.  In that case:

Cultural drift equation 2

where wti is the weight given to the teachings of the ith parent/teacher.  I assume that the sum of the weights is equal to one.  Note that with equal weighting as the number of parent/teachers go up the equilibrium variance goes down rapidly, and quickly becomes nearly equal to the mutation rate.

This raises the interesting point that the effects of drift on a continuously inherited trait is a function of not only the population size, but also the details of how it is inherited.  If an new phenotype learns its trait value by sampling all or most of the community, and copies it as closely as possible there were be essentially no heritable variance.  In contrast, if the trait is acquired by copying one model, and then modifying it substantially (i.e., learning it badly, and making it up so it is similar) there will be substantial heritable variance.  This is something that we rarely think about – the mating and interaction structure of a population can radically influence the amount of heritable variance.  For genetic systems this manipulation primarily takes the form of changes in mating structure, and manipulation of the degree of inbreeding (more on this later).  With blending inheritance there are many more ways that the degree of heritable variance can be manipulated.  For some traits there may be tight formal teaching by numerous teaching, with “mutations” carefully controlled.  These traits become very stable, and can change very little over long periods of time.

Balanese danceBalinese dance 2 

(Left: http://lauradavis.net/roadmap/virtual-vacation-dancing-our-way-through-bali-day-25  Right http://www.coloradocollege.edu/events/2013-05-04-a-celebration-of-balinese-performing-arts) Balinese dance is carefully taught, and has become a very stable part of the culture of Bali.

On the other hand there are traits that are very malleable, fashion comes to mind, in which a premium is placed on innovation and being unique, but not too unique.  In this case a premium may be placed on “mutation”, and individuals may choose to heavily weight some “teachers” over others.

cnn pll-fashion-four-girls-story-top

“(Here) you have Aria’s (from left) vintage rock-and-roll … Hanna’s high-end glam with her famous pops of color, you have the softness of Spencer … and my sexy, tough and modern Emily,” costume designer Mandi Line said. “Aria is my fantasy doll, Hanna is my high school me, Spencer is who I learn from the most, and Emily comes the most natural to me.” (http://www.cnn.com/2012/09/05/showbiz/tv/pretty-little-liars-fall-fashion-tv)

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