computational insights and the theory of...
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Computational Insights and the Theory of Evolution
Christos H. Papadimitriou UC Berkeley
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Evolution Before Darwin
• Erasmus Darwin
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Before Darwin
• J.-B. Lamarck
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Before Darwin
• Charles Babbage
[Paraphrased] “God created not species, but the Algorithm for creating species”
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Darwin, 1858
• Common Ancestry • Natural Selection
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The Origin of Species
• Possibly the world’s most masterfully compelling scientific argument
• The six editions 1859, 1860,
1861, 1866, 1869, 1872
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The Wallace-Darwin papers
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Brilliant argument, and yet many questions left unasked, e.g.:
• How does novelty arise?
• What is the role of sex?
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After Darwin
• A. Weismann
[Paraphrased] “The mapping from genotype to phenotype is one-way”
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Genetics
• Gregor Mendel [1866] • Number of citations between 1866 and 1901:
3
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The crisis in Evolution 1900 - 1920
• Mendelians vs. Darwinians • Geneticists vs. Biometricists/
Gradualists • Population genetics
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The “Modern Synthesis” 1920 - 1950
Fisher – Wright - Haldane
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Big questions remain e.g.:
• How does novelty arise?
• What is the role of sex?
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Evolution and Computer Science
• “ How do you find a 3-billion long string in 3 billion years?” L. G.Valiant
At the Wistar conference (1967), Schutzenberger asked virtually the same question
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Valiant’s Theory of the Evolvable
• Which functions (traits of an organism) can evolve by natural selection?
• Properly formalized, this question leads to identifying obstacles to evolution
• For example, the function has to be learnable (actually, statistically so)
• Evolvability is a (quite) restricted form of learnability
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Evolution and CS Practice: Genetic Algorithms [ca. 1980s]
• To solve an optimization problem… • …create a population of solutions/genotypes • …who procreate through sex/genotype
recombination… • …with success proportional to their objective
function value • Eventually, some very good solutions are
bound to arise in the soup…
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And in this Corner… Simulated Annealing
• Inspired by asexual reproduction • Mutations are adopted with probability
increasing with fitness/objective differential • …(and decreasing with time)
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The Mystery of Sex Deepens
• Simulated annealing (asexual reproduction) works fine
• Genetic algorithms (sexual reproduction) don’t work
• In Nature, the opposite happens: Sex is successful and ubiquitous
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?
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A Radical Thought
• What if sex is a mediocre optimizer of fitness (= expectation of offspring)?
• What if sex optimizes something else? • And what if this something else is its
raison d’ être?
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Mixability!
• In a recent paper [LPDF, PNAS 2008] we establish through simulations that:
• Natural selection under asex optimizes fitness
• But under sex it optimizes mixability: • The ability of alleles (gene variants) to
perform well with a broad spectrum of other alleles
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Explaining Mixability
• Fitness landscape of a 2-gene organism
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Rows: alleles of gene A
Columns: alleles of gene B
Entries: fitness of the combination
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Explaining Mixability (cont)
• Asex will select the largest numbers
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Explaining Mixability (cont)
• But sex will select the rows and columns with the largest average
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In Pictures
alleles (variants) of gene A
alleles of gene B
peaks
troughs
“plateau”
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Sex favors plateaus over peaks
Theorem [Livnat, P., Feldman 11] In landscapes of this form
• Unless peak > 2 × plateau, in sexual reproduction the plateau will dominate and the peaks will become extinct
• In asexual reproduction, the peaks will always dominate and the plateau will become extinct
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And plateaus accelerate evolution
• They act as springboards allowing alternatives to be explored in parallel…
• …and this acceleration promotes speciation (the creation of new species)…
• …which results in an altered landscape… • …in which sex selects more plateaus… • …and life goes on…
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Pointer Dogs
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Pointer Dogs
C. H. Waddington
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Waddington’s Experiment (1952)
Generation 1 Temp: 20o C
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Waddington’s Experiment (1952)
Generation 2-4 Temp: 40o C ~15% changed Select and breed those
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Waddington’s Experiment (1952)
Generation 5 Temp: 40o C ~60% changed Select and breed those
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Waddington’s Experiment (1952)
Generation 6 Temp: 40o C ~63% changed Select and breed those
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Waddington’s Experiment (1952)
(…) Generation 20 Temp: 40o C ~99% changed
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Surprise!
Generation 20 Temp: 20o C ~25% stay changed!!
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Genetic Assimilation
• Adaptations to the environment become genetic!
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A Genetic Explanation
• Suppose that “red head” is this Boolean function of 10 genes and “high temperature”
“red head” = “x1 + x2 + … + x10 + 3t ≥ 10” • Suppose also that the genes are independent
random variables, with pi initially half, say.
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A Genetic Explanation (cont.)
• In the beginning, no fly is red (the probability of being red is 2-n)
• With the help of t = 1, a few become red • If you select them and breed them, ~60%
will be red!
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Why 60%?
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A Genetic Explanation (cont.)
• Eventually, the population will be very biased towards xi = 1 (the pi’s are close to 1)
• And so, a few flies will have all xi = 1 for all i, and they will stay red.
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Generalize!
• Let B is any Boolean function • n variables x1 x2 … xn (no t)
• Independent, with probabilities p = (p1 p2 … pn)
• Now, generate a population of bit vectors, and select the ones that make B(x) = 1
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(cont.)
• In expectation, p à p’, where pi’ = probp (xi = 1 | B(x) = 1)
(Looking under the rug: Linkage?) Conjecture: This solves SAT Theorem (with Greg Valiant, 2011): If instead p à ε p’ + (1 – ε) p, then it converges to a satisfying assignment with prob. 1 (for large enough population and small enough ε).
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Interpretation
• If there is any Boolean combination of a modestly large number of alleles that creates an unanticipated trait conferring even a small advantage, then this combination will be discovered and eventually fixed in the population.
• “With sex, all moderate-sized Boolean functions are evolvable.”
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Sooooo…
• The theory of life is deep and fascinating • The point of view of a computer scientist
makes it even more tantalizing • Mixability helps understand the role of sex • A natural stochastic process on Boolean
functions may help illuminate genetic assimilation and the emergence of novel traits
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Thank You!