The Structure, Function, and Evolution of Biological Systems Instructor: Van Savage Spring 2010 Quarter 4/1/2010

Crash Course in Evolutionary Theory

What is fitness and what does it describe? Ability of an entity to survive and propagate forward in time. It is inherently a dynamic (time evolving property). Can assign fitness to 1.Individuals 2.Genes 3.Phenotypes 4.Behaviors 5.Strategies (economic, cultural, games, etc) 6.Tumor cells and tumor treatment 7.Antibiotic resistance 8.Language

Evolution of allele frequency and Wright’s equations Conclusions 1.Increases in direction of slope of fitness function 2.Allele frequency climbs peak until maximal fitness and this derivative or slope is zero 3.Peak occurs when marginal fitness for A 1 and A 2 are equal, implying relative fitness of heterozygote 4.Prefactor is actually a variance, so strength of selection depends on variance. No variance implies no selection.

How do we maintain variance? Mutation and migration What is typical effect of a mutation? Wild Type fitness=1 (relative fitness) Hetero. Mutant fitness=1-hs Deleterious double mutant=1-s Genetic Load=

Mutation-selection balance Given a forward mutation rate, μ, and backward mutation rate, ν Special case that h=0, we have and Genetic Load A1A1 A2A2 μ(1-p) νp

How good are these approximations?

Other important factors 1.Density dependence 2.Multiple alleles (more then two) 3.Multiple Loci (more than one) 4.Fertility selection is pair specific

Do better for finite-size populations with conditional probabilities Fundamental formula in statistics is Note that P(A 1 )=p and we define So the marginal fitness is

Do better for finite-size populations with conditional probabilities Definition of average fitness is now Measure, g ij, is the proportion of A 1 alleles within a genotype, so mean value of g is p

Special case of Price’s Theorem We will learn full version in much greater detail soon.

Additive Genetic Variance From statistics Least-squares regression of w on g Known as additive genetic variance and used by breeders Variance in fitness is square of deviations in fitness, s

Special case of Fisher’s Fundamental Theorem of Natural selection This term captures selection favoring the most fit. Need variance for selection to act. Small values of fitness lead to rapid changes to increase it. Large value lead to small changes because we are near the peak. Fitness is always increasing More general form of Theorem is Extra term captures effects of density dependence. Also, need to account for fluctuating environments

Additional effects for more than two loci 1.Recombination—breaking, rejoining, and rearranging of genetic material. Major extra source of variation. 2.Epistasis—interactions between loci (i.e., non-independence). Fitness effects of alleles affect each other in non-additive way.

Recombination Why do we need two loci for re-arrangements to matter? A1A2A1A2 A2A1A2A1 up versus down makes no difference in our model A1B1A2B2A1B1A2B2 up and down are now differentiated by the B alleles A2B1A1B2A2B1A1B2 A1B1A2B1A1B1A2B1 Does this re-arrangement make a difference? A2B1A1B1A2B1A1B1

Recombination Now need four frequencies for each possible pairing of A and B alleles? A1B1A1B1 Freq of =x 11 A2B1A2B1 A1B2A1B2 A2B2A2B2 Freq of =x 21 Freq of =x 12 Freq of =x 22 A1A1 Freq of =p 1 =x 11 +x 12 A2A2 Freq of =p 2 =x 21 +x 22 Freq of B i =q i = Freq of A i =p i =

Recombination For which genotypes with will recombination have an effect A 1 B 1 ? Take all possible genotypes with an A 1 or B 1 A1B1A2B1A1B1A2B1 A1B1A1B1A1B1A1B1 A1B2A2B1A1B2A2B1 A1B1A2B2A1B1A2B2 A1B1A1B2A1B1A1B2 A1B2A2B1A1B2A2B1 A1B1A2B2A1B1A2B2 r 1-r

Recombination Can understand all of this again in terms of covariance. Covariance of A and B implies effect of recombination. Zero covariance implies no recombination D is the measure of gametic disequilibrium and time evolution can be expressed in terms of this and the recombination rate x’ ij =x ij +(-1) i+j rD D’=D(1-r)

Recombination with selection Must assign fitness and then use formulas and do algebra similar to what we have been doing. Additional term captures effects of recombination and whether it slows or speeds up evolution. “-” if i=j and “+” is I does not equal j

Epistasis Interactions among fitness effects for different alleles If no interaction, then the covariance is 0. This is know as additive (or sometimes multiplicative.

Additive Choose relative fitness so that the wild type fitness is 1, and look at exponential (continuous) versions Still assuming a mutation is deleterious, we look at combined effects of two mutations and

Non-Additive Synergistic (negative epistasis) Antagonistic (positive epistasis) What is the distribution of these effects? What fraction of mutation pairs are antagonistic? What fraction of mutation pairs are synergistic?

Graphical representation

Modeling more than two mutations If all mutations have the same deleterious effect, and k mutations are lethal, then How can we modify this for epistasis? What about these forms for epistasis? or Lethal number of mutations

Next class we will move onto interactions between loci and genes and possible touch on drift and coalescence. Some material is in Chapter 2 of Sean Rice’s book, but you don’t need to know more beyond what was covered in class Read papers for next week on distribution of epistatic interactions, modeling epistasis, the evolution of sex, and the evolution of antibiotic resistance.