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The Structure, Function, and Evolution of Biological Systems Instructor: Van Savage Spring 2010 Quarter 4/6/2010 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. 2. 3. 4. 5. 6. 7. 8. Individuals Genes Phenotypes Behaviors Strategies (economic, cultural, games, etc) Tumor cells and tumor treatment Antibiotic resistance Language Special case of Price’s Theorem We will learn full version in much greater detail soon. Cov(w,g) p g w 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 Can understand all of this again in terms of covariance. Covariance of A and B implies effect of recombination. Zero covariance implies no recombination Cov(A,B) E(AB) E(A)E(B) x11 p1q1 D D is the measure of gametic disequilibrium and time evolution can be expressed in terms of this and the recombination rate x’ij=xij+(-1)i+jrD D’=D(1-r) Recombination with selection Must assign fitness and then use formulas and do algebra similar to what we have been doing. 1 x ij [Cov(w,gij ) rDw1122] w 1 [Cov(w,gij ) rw1122Cov(A1,B1 )] w 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 Cov(wx ,wy ) w xy w x w y Can now see covariance plays central role in all of evolution. In fact, it is as central as fitness itself. If no interaction, then the covariance is 0. w xy w x w y 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 wWT 1 e 0 Still assuming a mutation is deleterious, we look at combined effects of two mutations w x 1 s ~ e x sx sx sy wx wy e e w y 1 sy ~ e and e (sx sy ) ~ 1 (sx sy ) sy Non-Additive w xy w x w y w xy w x w y w xy w x w y 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 ,g gA w,g gB A B ,g gA w,g gB ,g A B A gB gA gB Modeling more than two mutations If all mutations have the same deleterious effect, and k mutations are lethal, then ks we k ~ 1 s ~ 1 ks 1 kL k How can we modify this for epistasis? wepi 1 sk 1 ~ (1 s) k1 sk1 ~e What about these forms for epistasis? 1 wepi (1 ks) or Lethal number of mutations 1 k wepi 1 kL Modeling more than two mutations wepi 1 sk 1 ~ (1 s) k1 sk1 ~e How do we interpret synergy and antagonism? Mutations to steps in sequence are antagonistic (green) Mutations to steps in parallel can be synthetically lethal if it knocks out a loop, which is extreme synergy (red), or multiplicative (black). Recent papers using models of epistasis: Lenski,Ofria, Collier, Adami Definition of digital organism Carrying capacity of 3600 individuals Probability of point mutation was 0.0075 per instruction copied and probability of insertion and deletion is 0.05 per division Each generation is 5-10 updates and each update is execution of on average 30 instructions per individual Start with genome length of 20 instructions 28 different types of instructions (like amino acids) Phenotypic rewards are multiplicative Instructions are mathematical operations Advantages of digital organisms 1. Allows us to choose condition and seek generalizations beyond organic life forms 2. Allow us to perform experiments, in terms of time scales and numbers, that are unattainable with real systems 3. Use evolving programs to solve computational problems Complex organisms Selection criteria: 1. Baseline allocation of CPU time is proportional to genome size Why? Larger genomes does not necessarily imply more complex or better at “solving problems” or getting resources. See next plot. 2. Certain mathematical operations, which require novel combinations of instructions (e.g., performing an XOR operation using NAND operations), are rewarded with additional CPU time. This is a type of selection. Solving computational problem is solving fitness problem or getting more resources. Simple organisms Selection criteria starting with complex organisms: 1. Baseline allocation of CPU time is independent of genome size 2. mathematical operations are not rewarded with additional CPU time Removes a type of selection. Does not seem biological. But, there is still selection for shorter replication time, so some biological analogy. Complex vs. Simple organisms Complex organisms average genome size=91.3 instructions (Because of assumption 1 or 2?) Simple organisms average genome size=19.8 instructions Simple organisms have more lethal mutations Tests for epistasis Decay test—see whether successive mutations (1-10) become increasingly worse (synergy), better (antagonistic), or are multiplicative. wepi 1 sk 1 ~ (1 s) k1 sk1 ~e Both show an average type of antagonistic epistasis, but complex organisms show it more strongly. Why do you think? Tests for epistasis Pair test—Explicitly calculate and compare double mutant fitness (Wxy) to the product of each single mutant fitness (WxWy) for each fitness for all pairs (double mutants). Cov(w x ,w y ) w xy w x w y Simple organisms have more lethals. Both have more antagonistic than synergistic interactions Simple organisms actually have a higher proportion of non-lethal, antagonistic interactions. Recent papers using models of epistasis: Segre, DeLuna, Church, Kishony Quantitative Epistatic Interactions Perturbation X Phenotype (Growth Rate) Perturbation Y Synergy Antagonism Suppression See also: Boone, Science (2004) Weissman, Cell (2006) Boeke, Nature Genetics (2003); Cell (2006) Giaever, Nature Genetics (2007) Quantitative genetic interactions Proliferation Rates How much information is concealed within the remaining 99.5% of gene interactions??? Gene X Gene Y + + 1 ~0.5% Aggravating 0.9 0 0.72 0.8 0 Alleviating 0.72 ? 0.8 # Cells WT X Y XY Functional Association XY XY Time 0. 8 Interactions between mutations in yeast metabolism using Flux Balance Analysis Varma and Palsson, 1994 Famili et al, 2003 Rate of biomass production (growth rate) • 829 - metabolic reactions • 343206 gene pairs Flux Balance Analysis • Computational model of metabolism • Growth rate predictions for wild-type and deletion mutants • Main Assumptions: – Steady-state – Mass-conservation – Optimality Nutrients B A C • Developed and experimentally verified in E. coli and yeast by Palsson et al: Nature 2002; PNAS 2003; Nat. Genetics 2004 Biomass (new cells) D Measures of epistasis Since covariance is as fundamental as fitness, why not define relative covariance instead of relative fitness. We define it relative to tri-modally binned covariance that itself varies, so relative to a shifting baseline. Absolute covariance Cov(wx ,wy ) wxy wx wy Relative covariance wxy wx w y ˜ BinnedCov (wx ,w y ) w˜ xy wx w y Cov(wx ,wy ) Measures of epistasis Additive BinnedCov(wx ,wy ) wxy wx wy 0 Antagonistic is binned as synthetic lethal BinnedCov(wx ,wy ) 0 wx wy wx wy Synergistic is binned as buffering (wxy=wx<wy) BinnedCov(wx ,wy ) wx wx wy Can think of this as relative fitness being relative to product of fitnesses and again a shifting baseline Additive BinnedCov(wx ,wy ) 0 Antagonistic is binned as synthetic lethal ˜ w 1 ~ 1 r xy Synergistic is binned as buffering (wxy=wx<wy) w 1 ˜ ~1 w 1 r xy r xy Measures of epistasis What do we expect for distribution of new measure of epistasis? If multiplicative, then relative covariance should still be 0 For antagonistic, since ε is unimodal, if wxwy is constant, then relative covariance is unimodal. If wxwy is unimodal in same form as ε , then relative covariance should be 1. For synergistic, since ε is unimodal, if wx-wxwy is constant, then relative covariance is unimodal. If wx-wxwy is unimodal in same form as ε, then relative covariance should be -1. Measures of epistasis How does ε vary with relative covariance? ˜ ~ 2 Will accentuate differences in distribution Measures of epistasis—based onFBA predictions in yeast Sort of unimodal distribution goes to trimodal distribution Opposite of Lenki et al. because synergy is enriched. Why? Measures of epistasis—RNA viruses A bit more continuous for real data. We will see more real data later on. Higher level epistasis—interactions among functional groups rather than loci Interactions are mostly monochromatic. No reason a priori that this should be, except it signifies functional organization. Cov(WModA,WModB ) WModA,ModB WModAWModB Can we do reverse and cluster monochromatically to find functional groups? Construct network for all pairwise interactions, Start with each gene in its own group. Cluster by pairs if they interact with other genes in same way. Require monochromaticity, each group must interact with all other groups in same way Within a group there is no requirement for monochromaticity Make cluster sizes as large as possible Next class we will move onto more theory for evolution of epistasis, synergy, and antagonism as well as evolution of resistance in antibiotics. First Homework set is due in two weeks (April 20, 2010).