Class 38 Fixed Points and Biological Computing CS

Class 38: Fixed Points and Biological Computing CS 200: Computer Science David Evans University of Virginia 24 April 2002 CS 200 Spring 2002 http: //www. cs. virginia. edu/~evans Computer Science

Menu • Making Recursive Definitions without define • Computing with DNA • How Biology Programs 24 April 2002 CS 200 Spring 2002 2

Lambda Calculus term : : = variable |term | (term)| variable. term -reduction (renaming) y. M v. (M [y v]) where v does not occur in M. -reduction (substitution) ( x. M)N M [ x N ] 24 April 2002 CS 200 Spring 2002 3

Lambda Calculus is a Universal Computer z z z z z ), X, L ), #, R (, #, L 2: look for ( 1 Start (, X, R #, 1, - HAL T #, 0, - Finite State Machine We have this, but we cheated using to make recursive definitions! 24 April 2002 • Read/Write Infinite Tape ü Mutable Lists • Finite State Machine ü Numbers to keep track of state • Processing ü Way of making decisions (if) [ Way to keep going CS 200 Spring 2002 4

Fixed Point Theorem • The fixed point of a function f, is a value x such that f(x) = x • If we can find the fixed point of our Turing Machine simulator, then we have something that keeps going until it halts! fixed-point TM input result of running TM on input 24 April 2002 CS 200 Spring 2002 5

All Lambda Calculus Terms have Fixed Points! • For any Lambda Calculus term F, there exists a Lambda Calculus Term X such that FX = X • Proof: Let W = x. F(xx) and X = WW. We can X = WW = ( x. F(xx))W make F a parameter! F (WW) = FX 24 April 2002 CS 200 Spring 2002 6

Why of Y? • Y is f. WW: Y f. ( x. f (xx)) • Y calculates a fixed point of any lambda term! • Hence: we don’t need define to do recursion! • Works in Scheme too - check the “lecture” from the Adventure Game 24 April 2002 CS 200 Spring 2002 7

Lambda Calculus is Turing Universal! • All you need is beta-reduction and you can compute anything • This is just one way of representing numbers, if, etc. – many others are possible • Integers, booleans, if, while, +, *, =, <, classes, define, inheritance, etc. are for wimps! Real programmers only use . 24 April 2002 CS 200 Spring 2002 8

Models of Computation • Mechanical: Turing Machine • Symbolic: Lambda Calculus • Next: Biological 24 April 2002 CS 200 Spring 2002 9

Computing with DNA Leonard Adleman (Mathematical Consultant for Sneakers), 1995 24 April 2002 CS 200 Spring 2002 10

DNA • Sequence of nucleotides: adenine (A), guanine (G), cytosine (C), and thymine (T) • Two strands, A must attach to T and G must attach to C 24 April 2002 CS 200 Spring 2002 G C T A 11

Hamiltonian Path Problem • Input: a graph, start vertex and end vertex • Output: either a path from start to end that touches each vertex in the graph exactly once, or false indicating no such path exists RIC start: CHO end: BWI CHO IAD 24 April 2002 CS 200 Spring 2002 Hamiltonian Path is NP-Complete 12

Encoding The Graph • Make up a two random 4 -nucleotide sequences for each city: CHO: RIC: IAD: BWI: CHO 1 = ACTT RIC 1 = TCGG IAD 1 = GGCT BWI 1 = GATC CHO 2 = gcag RIC 2 = actg IAD 2 = atgt BWI 2 = tcca • If there is a link between two cities (A B), create a nucleotide sequence: A 2 B 1 CHO RIC CHO 24 April 2002 gcag. TCGG actg. ACTT CS 200 Spring 2002 Based on Fred Hapgood’s notes on Adelman’s talk http: //www. mitre. org/research/nanotech/hapgoo d_on_dna. html 13

Encoding The Problem • Each city nucleotide sequence binds with its complement (A T, G C) : CHO 1 = ACTT CHO 2 = gcag CHO’: TGAA cgtc RIC: TCGGactg RIC’: AGCCtgac IAD: GGCTatgt IAD’ = CCGAtaca BWI: GATCtcca BWI’ = CTAGaggt • Mix up all the link and complement DNA strands – they will bind to show a path! 24 April 2002 CS 200 Spring 2002 14

Path Binding BWI’ RIC’ IAD’ CHO’ TGAAcgtc. CCGAtaca. AGCCtgac. CTAGaggt gcag. GGCTatgt. TCGG actg. GATC CHO IAD RIC BWI TCGGactg RIC BWI GATCtcca CHO ACTTgcag IAD GGCTatgt 24 April 2002 CS 200 Spring 2002 15

Getting the Solution • Extract DNA strands starting with CHO and ending with BWI – Easy way is to remove all strands that do not start with CHO, and then remove all strands that do not end with BWI • Measure remaining strands to find ones with the right weight (7 * 8 nucleotides) • Read the sequence from one of these strands 24 April 2002 CS 200 Spring 2002 16

Why don’t we solve NPComplete problems this way? • Speed: shaking up the DNA strands does 1014 operations per second ($400 M supercomputer does 1010) • Memory: we can store information in DNA at 1 bit per cubic nanometer • How much DNA would you need? – Volume of DNA needed grows exponentially with input size – To solve ~45 vertices, you need ~20 M gallons 24 April 2002 CS 200 Spring 2002 17

DNA-Enhanced PC 24 April 2002 CS 200 Spring 2002 18

How does Nature program? 24 April 2002 CS 200 Spring 2002 19

How Big is the Make-a-Human Program? • 3 Billion Base Pairs – Each nucleotide is 2 bits (4 possibilities) – 3 B pairs * 1 byte/4 pairs = 750 MB 1 CD ~ 650 MB 24 April 2002 CS 200 Spring 2002 20

Encoding is Redundant • DNA encodes proteins • Every sequence of 3 base pairs one of 20 amino acids (or stop codon) – 21 possible codons, but 43 = 64 possible values – So, really only 750 MB * (21/64) ~ 250 MB 24 April 2002 CS 200 Spring 2002 21

People are almost all the Same • Genetic code for 2 humans differs in only 2. 1 million bases – 4 million bits = 0. 5 MB 24 April 2002 CS 200 Spring 2002 22

How big is. 5 MB? • 1/3 of a floppy disk • <1% of Windows 2000 • ~22 times the size of the PS 6 adventure game code 24 April 2002 CS 200 Spring 2002 23

Is DNA Really a Programming Language? 24 April 2002 CS 200 Spring 2002 24

Nerdy Linguist’s Definition A description of pairs (S, M), where S stands for sound, or any kind of surface forms, and M stands for meaning. A theory of language must specify the properties of S and M, and how they are related. 24 April 2002 CS 200 Spring 2002 25

Programming Language (Definition from Lecture 1) A description of pairs (S, M), where S stands for sound, or any kind of surface forms, and M stands for meaning intended to be read and written by humans and processed by machines. 24 April 2002 CS 200 Spring 2002 26

Stuff Programming Languages are Made Of • Primitives codons (sequence of 3 nucleotides that encodes a protein) • Means of Combination ? ? Morphogenesis? Not well understood (by anyone). This is where most of the expressiveness comes from! • Means of Abstraction DNA itself – separate proteins from their encoding Genes – group DNA by function (sort of) Chromosomes – package Genes together Organisms – packages for reproducing Genes 24 April 2002 CS 200 Spring 2002 27

Biology is (becoming) a subfield of Computer Science • Biological mechanisms are mostly understood (proteomics still has a way to go) • What is not understood is how those are combined to create meaning 24 April 2002 CS 200 Spring 2002 28

Charge • Noon (now): President Casteen’s State of the University in Old Cabal Hall – Extra credit question: “Given that Computer Science is the most liberal art, how come UVa College students are not able to major in Computer Science? ” • Friday: review – Chance to ask questions about anything you want 24 April 2002 CS 200 Spring 2002 29