GEOS 26400/36400/BIOS 23255/EVOL 32300 Principles of Paleontology

Evolutionary Rates, I

I. Importance of rates and trends in evolution
      -two principal questions in evolution of lineage: rate and direction
      -temporal component essential (importance of fossil record)
      -Why do some lineages or traits evolve at different rates?

II. Defining and Measuring Morphological rates
      -For mathematical purists: rate=derivative w/ respect to (continuous) time
      -Pragmatic: rate=(net change)/(elapsed time) (discrete time)
      -Relative vs. Absolute rates
        -Increments of size change depend on size, so measure relative change
        -Constant proportional (relative) rate yields log-linear change of morphology versus time.
           -Logarithmic scale natural for measuring proportional change
           -Haldane's (1949) formulation
              -rate (darwins, d) = ln(x2/x1)/(elapsed time)
	      -e.g., 10cm-->12cm in 2 m.y.: rate=ln(1.2)/2=0.09d
              *allows comparison of taxa with different body size
              *assumes proportional change (should be tested)
	-Rate of change in shape depends on whether change is isometric or allometric
	   -let s=y/x; then ln(s2/s1)=ln(y2/y1)-ln(x2/x1)
	-If change is small, linear and logarithmic change difficult  to distinguish 
     -Standardizing rates according to variability
            -rate of change within populations should increase with variance
                  -at least for traits that vary with fitness
            -Gingerich's (1993) formulation
                  -rate (haldanes, h) = change in standard deviations per generation
			-trait can be linear or logarithmic, whichever is more appropriate
                  -should allow comparison of size and shape measures
                  -difficult to apply to fossil data
                        -need to estimate variability within popultion
                              -potentially confounded by time-averaging (but probably not much)
                        -need to know generation time
                              -modern analogs and homologs

III. Dependence between net rates and time intervals over which they're measured
      -Longer intervals yield slower net rates.
        -Longer intervals incorporate more reversals and stasis.
	    -inverse correlation between net rate and interval of 
		measurement partly a result of comparing t (time) with 1/t 
      -How to make valid comparisons?
            -Compare rates calculated over the same interval length.
            -Compare rates of different lineages in the same strata.
            -Define rate quotient as magnitude of rate relative to statistical expectation
                  of rates for the corresponding interval length.
		-cf. encephalization quotient in allometry
		-potentially too much scatter in the data for this to be very useful

IV. Using rate-interval comparison be used to infer evolutionary patterns (Gingerich 1993)
    -plot log(interval length) vs. log(rate) for single evolutionary sequence
    -end-member expectations
            -extreme directionality (every step goes in the same direction)
                  *log(rate) independent of log(interval) (no stasis or reversal)
            -extreme stasis (fluctuation with no net change)
                  *log(rate) goes as -1 x log(interval)
            -symmetric random walk (equal chance of increase or decrease at each step)
                  -Random walks often yield what appear to be trends.
                  *log(rate) goes as -0.5 x log(interval)

V. Improved approach to inferring evolutionary patterns (Hunt 2007, Proc. Nat'l. Acad. Sci. USA)
   -Formal statistical models of directionality, stasis, and random walk yield probability of observed ancestor-descendant change under model, therefore likelihood of model given observed set of changes.
   -Survey of many time series shows little directionality, much stasis and random walk
   -Size shows greater tendency toward random walk, shape toward stasi