Homework problem
Consider the following data for Beak Length in Buntings (a bird):
 

Midparent	Offspring
12.61	12.36
12.66	12.49
12.77	12.59
13.26	12.79
13.41	12.83
13.41	13.02
14.16	13.54
14.41	13.49
15.04	13.78

1) Plot this data.
2) Calculate: The mean values for midparent and offspring Beak length.
3) Calculate the offspring-midparent covariance, and the variance in midparent values. (see Handout)

4) Find the heritability of beak length in these birds.

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Quantitative genetics (cont.)

Recall the matrix equation for the joint evolution of two traits:

Or, simply:

Here, the vector of partial regressions of fitness on the phenotypes:

is the Fitness Gradient; a vector pointing in the direction of maximum increase in fitness.

This form of the equation is useful both because it applies to any number of characters (just make the matrix and vectors bigger) and because it illustrates that with covariation between the characters, selection does not necessarily move the population along the quickest rout uphill  (which would be the gradient).

The figure on the left below illustrates the fact that if there is no genetic covariance between two traits, then the direction of evolution is the same as the direction in which fitness would increase most quickly.

Furthermore, selection can cause a character to change in a maladaptive way if it covaries with another character that is under strong selection.

Consider the illustration below, illustrating the joint evolution of two characters that are correlated (correlation = 0.6):

In this case, the quickest way to increase fitness would be to increase ₁ and decrease ₂ (i.e. following the fitness gradient, ).
However, because the traits are positively correlated, ₂ increases.

Example: Galapagos Finches
Studied by P. and R. Grant

Birds followed (by marking individuals) for many years.

In 1977, a drought meant that there were far fewer seeds than usual, and those that remained were large. This imposed selection for birds with deep beaks.

By contrast, having a long beak was detrimental because of leverage problems.

Based just on observed changes:
S_depth = 0.3
S_length = 0.21

So both length and depth increased.

However, calculating partial regressions (which hold other factors constant) of fitness on these traits produced:
_w,depth = 0.43
_w,length = -0.17

So long beaks are selected against. Length increased because of high covariance with depth. Jul 8, 2021

Soon after this episode, a particularly wet year resulted in selection for smaller beaks, so much of the change was undone.

This may well be a common phenomenon: temporarily strong selection in one direction followed by similarly strong selection in another direction.
The result, if averaged over many generations, would appear to be very slow evolution.

Recall that the rate of morphological evolution is sometimes measured in "darwins"; one "darwin" being a change by a factor of e (2.718..) per million years.
Rates of evolution estimated from the fossil record are on the order of one to a few darwins.
The change observed on the Galapagos example discussed above was around 20,000 darwins.

It is common (in fact, the norm) for phenotypic traits to be genetically correlated with one another.
In particular, many traits are correlated with body size.

Allometry:
describes relation between shape or other characters and body size.

For two structures on an organism, with sizes x₁ and x₂, It is often the case that:

    x₂ = bx₁^a

Where a and b are constants. Alternately, taking the natural log of both sides of the equation, we get:

    ln(x₂) = a ln(x₁) + ln(b)

Thus, the logs of the two sizes show a linear relationship.

This is a very general (though not universal) relation.

Since shape is determined by the ratio of different size measures, this means that shape will generally change as size changes.  Strictly, this is Allometry,  if shape stays the same as size varies, it is properly called Isometry.  However, "allometry" is the word generally used to refer to any such relationship.

The significance of this for evolution is in the fact that most populations show more variation in overall rate and duration of body growth than in specific growth parameters of particular characters.

Thus, populations evolve along allometric trajectories quite quickly, only deviating from them under strong selection.

Example: Irish Elk:

This extinct deer had extremely large antlers, both in absolute size and relative to its body size.

Irish Elk lie close to the allometric line for all deer; closer than some smaller species.

Thus, the antlers of Irish Elk are just about the size that you would expect them to be for a deer of that size.

Note: Moose lie below the line, so their antlers are relatively smaller.

In most organisms, there is a lot of variation along growth trajectories, with some individuals growing faster and others slower.
As a consequence, it is generally quite easy to evolve along allometric trajectories, but more difficult to evolve off of them (meaning changing the allometric relationship between two traits).

Heterochrony

Heterochrony broadly refers to changes in the rate or timing of developmental events.

Of course, any change in morphology must involve change in the rate or timing of something, so we restrict the term heterochrony to cases in which an entire developmental process changes as a unit.

For morphological traits, we can characterize a growth process as an " Ontogenetic Trajectory".

There is a whole nomenclature for different ways that an ontogenetic trajectory can change.

In the case of the Irish Elk, A new morphology resulted simply from extending the growth trajectory of the ancestor.  This is an example of Hypermorphosis.

The opposite of hypermorphosis, shortening the growth trajectory, is called Progenesis.

Progenesis is quite common. It occurs when there is selection for a juvenile stage to become reproductive.

Example:Tunicates

In most Tunicates, the adult is sessile -- remaining attached to a rock while filtering plankton out of the water.
There is a brief larval stage that can swim, but these quickly settle and metamorphose into an adult.

In one group, the "Larvatians", the gonads mature in the larval stage, so it becomes sexually reproductive, and the ancestral "adult" stage is lost completely.