Special course in Computer Science: Molecular Computing

Transcription

1 Special course in Computer Science: Molecular Computing Lecture 7: DNA Computing by Splicing and by Insertion-Deletion, basics of formal languages and computability Vladimir Rogojin Department of IT, Åbo Akademi Fall 2013

2 Storing data with DNA DNA/RNA molecules: Data carrier media Alphabet implemented via: Adenine (Uracil, in case of RNA) Cytosine Guanine Thymine WK complementarity: Antimorphism θ(uv)=θ(v)θ(u): θ(a)=t, θ(t)=a θ(c)=g, θ(g)=c θ(w)=w - A/U C G T

3 Encoding of information with DNA Designing a set of good DNA strands: Positive design problem: Design a set of input DNA molecules such that there is a sequence of reactions that produces the correct result. Negative design problem: Design a set of input DNA molecules that do not interact in undesirable ways i.e., do not produce incorrect outputs, and do not consume molecules necessary for other programmed interactions

4 DNA complementarity and formal language theory Within framework of formal language theory the positive and negative design problems are addressed when creating libraries of oligonucleotides Avoiding intramolecular undesired hybridization Avoiding intermolecular undesired hybridization

5 Tube languages Tube languages: Formalizes set of molecules in a test tube Tube language L is equal to, or a subset of K+, where K is a finite language whose elements are called codewords In majority cases K consists of words of some fixed length k, i.e., K a code of length k

6 Why compute with DNA? Advantages: Disadvantages: Massive parallelism Building and controlling nano-machines Expensive and complex experimental setup Slow biochemical processes Difficult to program Difficult to interface automatic data input/output Building nanostructures, self-assembly For instance, smart drugs For instance, selfassembled microcircuits Reprogramming cells Synthetic biology Molecular computers cannot be used to play Tetris or Doom

7 What can we compute with DNA? In theory any algorithm can be encoded and executed with DNA In practice, there is a large number of issues: Molecular computers are slow, require more involved programming approach and difficult to interact with However, molecular computers complement the electronic ones: It is way more easy to use molecular computers at microscopicscale environments Molecular computers are capable for self-assembly and selfreplication One could get easily millions of copies Could be naturally integrated into living organisms and organic systems

8 What can we compute with DNA? We will concentrate here on the theoretical aspect: What can we compute with DNA at least hypothetically? The answer could be given by biologically motivated formal computational paradigms in the field of formal language theory and computability

9 Computability Intuitively, studies question of what can be computed by various theoretical and real computing devices Not every computational device can compute any algorithm: For instance: conventional pocket calculator cannot be used for anything else except of performing arithmetics Arithmometer fully mechanical arithmetic calculator. No programming at all. Every operation is directly determined by a button push and by wheel rotation direction

10 Turing universality Theoretically, any computer can execute any algorithm Actually, in practice any computer is a very-very large finite automata with an astronomical number of states Turing machines class of formal theoretical computational devices that can compute any algorithm Turing universality (or Turing completeness) the ability of a computational device to compute any algorithm One can simulate a Turing machine to show the Turing universality for his/her computational model

11 Computing with DNA Can we build a Turing complete computing device basing solely on DNA operations? The answer: YES! There exist many formal DNA-based computational devices/paradigms and even some of their particular implementations that can execute any algorithm (at least in theory) We will focus on a number of computing paradigms inspired by DNA manipulation and investigate their computational power (i.e., what they can compute)

12 Formal language theory We will work in the framework of formal language theory A (formal) language: A word: Set of words (could either finite or infinite) Sequence of letters from some fixed alphabet A family (class) of languages: A type/category of languages sharing some properties

13 Formal language Can be enumerated (in practice this can be done for finite languages) Can be generated By a formal grammar By a finite/pushdown automata or Turing machine Etc. Can be recognized: By a finite/pushdown automata or Turing machine Etc. Can represent concepts from other computational theories/paradigms For instance, sets, multsets, algorithms, etc.

14 Formal grammar A finite set of production rules (rewriting rules): Left-hand-side right-hand-side Each side consists of a sequence of symbols (words) A finite set of nonterminal symbols A finite set of terminal symbols Indicates that some production rule yet can be applied. Denoted by capitals Indicates that no production rule can be applied. Denoted by low-case letters A start symbol: A distinguished nonterminal symbol

15 Generating language with formal grammar A rule is applied to a word as follows: A substring from the word that matches the lefthand-side of the rule is replaced by the right-handside of the rule Generating (defining) a language: Given start symbol S and the set of rules: One applies a sequence of rewriting rules (derivation) from the given set of rules to S and eventually produces a word consisting solely of terminal symbols The set of all words of terminal symbols that can be obtained from S through a sequence rewritings dictated by the set of rules is the language accepted by the grammar

16 Formal grammar, example Nonterminal symbols: Start symbols: S Terminal symbols: S a, b Production rules: S asb S λ

17 Formal grammar, example Derivation: Production rules: S asb S λ S asb or S λ; asb aasbb or asb ab; aasbb aaasbbb or aasbb aabb; ansbn an+1sbn+1 or ansbn anbn Generated language: anbn, n 0

18 (Chomsky) hierarchy of languages. Finite and regular languages Finite language (FIN) finite set of words Regular language (REG) Type-3 grammar: A a and A ab (or A Ba) One nonterminal on the left and one terminal on the right, and at most one nonterminal on the right Also defined by regular expressions Recognized by finite automata Used in defining search patterns and lexical structure of programming languages

19 (Chomsky) hierarchy of languages. Linear and context-free languages Linear languages (LIN): Defined by linear grammar: A ubv and A w One nonterminal on the left and at most one nonterminal on the right. Any number of terminals on the right Recognized by pushdown automata Example: anbn All regular languages are linear, but not all linear languages are regular Context-free languages (CF): Type-2 grammar (context-free): A γ One nonterminal symbol on the left and sequence of any number of terminals and nonterminals on the right

20 (Chomsky) hierarchy of languages. Context-free languages Context-free languages (CF): Type-2 grammar (context-free): A γ One nonterminal symbol on the left and sequence of any number of terminals and nonterminals on the right Recognized by push-down automata Serve as theoretical basis for the phrase structure of most programming languages All linear languages are also context-free languages, but not viceversa Example: Dyck language the language consisting of balanced words of parenthesis This language is important in parsing of expressions that need correctly nested parenthesis

21 (Chomsky) hierarchy of languages. Context-sensitive languages Context-sensitive languages (CS): Type-1 grammars: αaβ αγβ α, β, γ contain any number of terminals and nonterminals. α, β may be empty, but γ must be nonempty α, β the context Recognized by a linear bounded automation a special type of Turing machine Every context-free language is also a contextsensitive language, but not viceversa

22 (Chomsky) hierarchy of languages. Recursively enumerable languages Recursively enumerable languages: Type-0 grammars (unrestricted grammars): α γ Sequence of any number of terminals and nonterminals at both sides Include all formal grammars Generate exactly all languages that can be recognized by a Turing machine

23 Finite automation Set of states and conditional transitions between them A transition switches the automata to another state A transition is chosen basing on the automata state and the input symbol S1(a) S2 S2(b) S3 S3(c) S1 Initial state: S1 Terminal state: S1 Recognizes: (abc)* S1(abcabc) S2(bcabc) S3(cabc) S1(abc) S2(bc) S3(c) S1 ACCEPT! S1(abb) S2(bb) S3(b) REJECT! a S1 S2 b c S3

24 Pushdown automata Pushdown automata recognize context-free languages: Difference from a FA: equipped with stack: Stack: a variable length vector where only the top-most element is accessible Operations on the top element: Push Pop Read Transition is chosen basing on: Incoming signal Current state Top-most symbol in the stack

25 Pushdown automata Transition is chosen basing on: Incoming signal Current state Top-most symbol in the stack During transition: a diagram of the pushdown automaton -Wikipedia New state is chosen Optionally: a modification on top of the stack is performed (either push a symbol or pop) Nondeterministic PDA recognize all CF languages

26 Turing machines A hypothetical device representing a computing machine Manipulates symbols on a strip of tape according to a table of rules Basically, a Turing machine can simulate logic of any computer algorithm Turing machines help scientists to understand the limits of mechanical computation Digital physics assumption: Any process in the Universe can be simulated by a Turing machine

27 Turing machine Introduced in 1936 by Alan Turing He called it an a-machine (automatic machine) A Turing machine that is able to simulate any other Turing machine is called a Universal Turing Machine Church-Turing thesis states that Turing machines capture the informal notion of effective method in logic and mathematics and provide a precise definition of an algorithm (or mechanical procedure )

28 Turing machine Consists of: A tape divided into cells. Each cell contains a symbol from some finite alphabet. The tape is arbitrarily extensible to the left and right A head can read/write a symbol from/to the current cell on the tape, one per time. A head can move along the tape left/right one cell per time A state register stores one of the finite number of the machine's states An instruction table a finite table of instructions

29 Turing machine An instruction: Chosen in condition of the machine's state and the input symbol Dictates to which state to switch the machine Dictates what to print to the tape (overwrite the symbol in current cell on the tape with a new symbol, or just erase the symbol from the current cell, or do not modify the content of the current cell) Dictates where to move the head (left/right/stay)

30 Turing machine Execution: Input: the tape content Starts from a tape (could be either empty or contain some word), some initial head position and some initial state Follows instructions from the instruction table When reached the halting state (or when no instruction is applicable) halt Output: the tape content after halt On some inputs a T machine can work forever It is undecidable whether a machine can halt on a given input

31 Turing machine, some implementations

32 Chomsky hierarchy Finite languages, included in Regular languages, included in Linear languages, included in Context-free languages, included in Context-sensitive languages, included in Recursive enumerable languages: Include any other language family Can express any algorithm Can be computed by Universal Turing Machines

33 Splicing in DNA Computing Splicing: sequence of cut by restriction enzymes, recombination and paste by ligase operations Restriction enzimes recognize sites and perform cut Recombination Ligation α1 α1 A T C G T A α2 α2 β1 β1 A T C G T A β2 β2

34 Splicing in DNA Computing Formally: String rewriting operation Splicing rule: r = u1#u2$u3#u4 Splicing transformation: (x,y) r (z,w), iff x=x1u1u2x2, y=y1u3u4y2 z=x1u1u4y2, w=y1u3u2x2 For some x1, x2, y1, y2 words over V

35 Insertion/Deletion in DNA Computing More complex DNA biochemistry Basic concept: Insertion transformation: x ins y iff x=x1uvx2, y=x1uwvx2, for some insertion rule (u,w,v), where x1, x2 are strings over V Deletion transformation: x del y iff x=x1uwvx2, y=x1uvx2, for some deletion rule (u,w,v), where x 1, x2 are strings over V

36 Computing with DNA: H-systems and Ins/Del-systems We will overview results in the field of computer science and DNA computing related to the computational power and universality of formalized splicing operations and insertion/deletion