MEMBRANE COMPUTING

Faculty of Informatics

Vienna University of Technology

Wien, Austria

Rudolf FREUND

MOLECULAR COMPUTING

Molecular Computing

Overview

► DNA Computing

► Membrane Computing

● Splicing, Watson-Crick complementarity● Test tube systems (universality)

● P systems - a general model

● Future Research● Related Models● P systems - selected results

• one of the most emerging fields in computer science

• brings together computer scientists and biologists

• opens new possibilities for both fields:

- computers help in decoding the sequences of DNA in genes, - allow for the simulation of biological processes etc.,

- computer scientists learn from nature,- face the challenge to bring back their new achievements to allow for a better understanding of the processes happening in nature.

Molecular Computing

Watson-Crick Complementarity

Adenine

Thymine

Cytosine

Guanine

DNA DeoxyriboNucleic Acid

double helix

3´ … A T C G … 5´

5´ … T A G C … 3´

|| || ||| |||

DNA Splicing

3´ … A T C G … 5´

5´ … T A G C … 3´ || || ||| |||

3´ … A T C C … 5´

5´ … T A G G … 3

´

|| ||| sticky ends

3´ … T T C G … 5´

5´ … A A G C … 3´ || ||| sticky ends

3´ … T T C C … 5´

5´ … A A G G … 3´ || || ||| |||

splicing 1

splicing 2

recombination 2

recombination 1

--------------------------------------------------------------------------------

DNA Computing - SplicingSPLICING RULE r = u1 # u2 $ u3 # u4

x = x1 u1 u2 x2 , y = y1 u3 u4 y2 ,

SPLICING ( x , y ) r ( z , w )

z = x1 u1 u4 y2 , w = y1 u3 u2 x2

T. Head: Formal language theory and DNA: An analysis of the generative capacity of specific recombinant behaviors. Bull. Math. Biology, 49 (1987), 737-759.

D. Pixton: Splicing in abstract families of languages. Theoretical Computer Science 234 (2000), 135-166.

E. Csuhaj-Varjú, R. Freund, L. Kari, Gh. Păun: DNA computing based on splicing: universality results. In: L. Hunter, T. Klein (Eds.): Pacific Symposium on Biocomputing '96, WSP (1996), 179-190.

SplicingSplicing: r = u1 # u2 $ u3 # u4 ,

x = x1 u1 u2 x2 , y = y1 u3 u4 y2 ,

( x , y ) r ( z , w )

z = x1 u1 u4 y2 , w = y1 u3 u2 x2

x1 u1 u2 x2

y1 u3 u4 y2

x1 u1 u4 y2

y1 u3 u2 x2

-------------------------------------------------------------------------------

x

y

z

w

Cutting and Recombination (CR)CUTTING RULE u1 # [m] $ [n] # u2

RECOMBINATION RULE ( [m] , [n] )

x = x1 u1 u2 x2 , y = x1 u1 [m] , z = [n] u2 y2

CUTTING x r ( y , z )

x = x1 u1 u2 x2 , y = x1 u1 [m] , z = [n] u2 y2

RECOMBINATION ( y , z ) r x

R. Freund, F. Wachtler: Universal systems with operations

related to splicing. Computers and Art. Intelligence 15 (4).

Splicing Systems / CR Systems

without any additional features:

- infinite number of rules

computational completeness (universality):

cannot even generate all regular languages

- multisets

- periodic sets of rules

- test tube systems

- membrane systems

- control mechnisms (control graphs,...)

Test Tube Systems - references

L. M. Adleman: Molecular computation of solutions to combinatorial problems. Science, 226 (Nov. 1994), 1021-1024.(lab solution of small travelling salesman problem)

E. Csuhaj-Varjú, L. Kari, and Gh. Păun: Test tube distributed systems based on splicing. Computers and Artificial Intelligence, Vol. 15 (2) (1996), 211-232.

R. Freund, E. Csuhaj-Varjú, and F. Wachtler: Test tube systems with cutting/recombination operations. In: R.B. Altman, A.K. Dunker, L. Hunter, T. Klein (Eds.):Pacific Symposium on Biocomputing '97 (1997), 163-174.

Test Tube Systems - definition

= ( B , BT , n , A , , D , f )

• B set of objects

• BT B set of terminal objects

• n number of test tubes

• A = ( A1 , ... , An ) Ai set of axioms in tube i

• = ( 1 , ... , n ) i set of operations in tube i

• D finite set of output/input relations of the form ( i , F , j ) ; F is a filter between tubes i and j

• f { 1 , ... , n } final test tube for results

TTS – begin of computation step

The computations in the system run as follows:

At the beginning of the computation the axioms are

distributed over the n test tubes according to A,

i.e., test tube Ti starts with Ai. Now let Li be the contents

of test tube Ti at the beginning of a derivation step.

Then in each test tube the rules of operate on Li, i.e.,

we obtain (elements of) i*( Li).

TTS - redistribution

The next substep is the redistribution of the

elements of i*(Li) over all test tubes according

to the corresponding output/input relations

from D, i.e., if ( i,F,j) is in D then the test tube Tj

from i*(Li) gets i*(Li) F, whereas the rest of

i*(Li) that cannot be distributed to other test

tubes remains in Ti.

TTS – final result of computation

The final result of the computations in consists of all

objects from BT that can be extracted from a final test

tube from E.The final results of the computations in consists could

also be taken all objects from B that can be extracted

from a final test tube from E thus taking B = BT .

For extended test tube systems, i. e., BT B and BT B,

the terminal results can be extracted from every test tube.

When two tubes are enough

TTS – references

Rudolf Freund, Franziska Freund:

Test Tube Systems: When two tubes are enough.

DLT '99 and in: G. Rozenberg, W. Thomas (Eds.):

Developments in Language Theory,

Foundations, Applications and Perspectives.

WSP, Singapore (2000), 338-350.

Rudolf Freund, Franziska Freund:

Test Tube Systems or

How to Bake a DNA Cake.

Acta Cybernetica, Vol. 12, Nr. 4, 445-459.

TTS – universality with CR

with only two tubes and the filters between the tubes being a finite union of sets of the form mW+n with markers m,n.

= (MW*M , [e]W+[f] , 2, (A1, Ø ) , (C1 R1 , C2), {(1, F1, 2) , (2, F2, 1)}, {2})

Proof (sketch). We simulate a grammar generating L{d}.G = (N,T,P,S) with L(G) = L{d}, where d is always generatedin the last step of the derivation of a terminal word in G.

A string w is represented by rotated versions [x]w2Bw1[y] ,w = w1w2 . Terminal strings are of the form [e]w[f] , w T+.The rules in P are labelled productions p: with1 | | 2 and 0 | | 2 .

V = N T {B} , W = V {d}.

Theorem. Any recursively enumerable language L canbe generated by a test tube system with CR rules

TTS – universality with CR (proof)M = {[e],[f],[e´],[f´], [x],[y],[x´],[y´]} { [lp], [rp], [lp´] |pLab}

{[xc],[yc],[xc´],[yc´] |cLab}

A1 = {[lp][y] |pLab, p:} {[x]c[xc´], [yc´] [y] |cV} {[x]BS[y]}

R1 = {([rp],[lp])|pLab} {([xc´], [xc] ), ([yc], [yc´]) |cV}

C1 = {u#[rp] $ [lp´]#[y] | uV, pLab, p:, ||=2}

{u#[rp] $ [lp´]#[y] | uV2 {B}, pLab, p:, ||=1}

C2 = {u#[yc] $ [y´]#c[y], [x]#[x´] $ [xc]#u | u,c V} {[x]B#[e´] $ [e]#u, u#[f] $ [f´]#d[y] | uT}

D = {(1, F1, 2), (2, F2, 1)}

F1 = [x]W+[y]

F2 = cV [xc]W+[yc] �

TTS – universality with splicingTheorem. Any recursively enumerable language L canbe generated by a test tube system with splicing rules

with only two tubes and the filters between the tubes being a finite union of sets of the form {A}W+{B} with A,B W.

= (W*,{E}W+{F}, 2, (A1,A2), (R1 R2), {(1,F1,2),(2,F2,1)}, {2})

Proof (sketch). We simulate a grammar generating L{d}like in the proof before the CRTTS; as there we only cut at the ends of a string, we can construct as follows:For a cutting rule like [x] # [r] $ [l] # u we take the axiom LZ and the splicing rule L # Z $ X # u . For a cutting rule like c # [r] $ [l] # u[y] we take the axiom ZR and the splicing rule c # uY $ Z # R . For a recombination rule like ( [r] $ [l] ) involving the axiom [l]w[y] we take the axiom ZwY and the splicing rule u # R $ Z # wY, etc.

introduced by Gheorghe PǍUN (1998)

- brought new impact into the area of theoretical computer science, especially into the area of formal languages;

- incorporate several features of living cells;

- allow for constructing various formal models of universal computers,- restricted models allow for the characterization of well-known families of languages.

Membrane Systems

(invented by Gheorghe PǍUN , 1998)

membrane structuremultisets of objectsevolution / communication rules applied in • the maximally parallel mode• the sequential mode

many variants computationally completedissolving / generating membranes

P Systems (Membrane Systems)

Gheorghe Păun: Membrane Computing - An Introduction. Springer-Verlag, Berlin, 2002.

The P Systems Web Page: http://psystems.disco.unimib.it

1

2 3

4 5

skin membrane

elementarymembrane

region

Membrane structure [1 [2 [4 ]4 [5 ]5 ]2 [3 ]3 ]1

Grammar schemes / grammarsA grammar scheme G is a construct

(O,OT,P, ) where

∙ O is the set of objects;

∙ OT O is the set of terminal objects;

∙ P is a finite set of productions p O×O or Om×On;

∙ is the derivation relation of G induced by the productions in P;

A grammar G is a construct (O,OT,P, ,w) where

∙ w is the axiom or the initial multiset over O.

L(G) = { v OT | w * v } or

L(G) = { v M(OT) | w * v }

String grammarsA string grammar G is a construct (N,T,P,S) where ∙ N is the alphabet of non-terminal symbols; ∙ T is the alphabet of terminal symbols, N ∩ T = { }; ∙ P is a finite set of productions of the form

u → v with u V* and v V+, where V := N T; ∙ S N is the start symbol.

A string grammar G now is a construct

(V*,T*,P, G ,S) where the derivation relation for

u → v P is defined as usual by xuy u→v xvy

for all x,y V*, thus yielding the well-known derivation

relation G for the string grammar G.

L(G) = { v T* | w G* v } .

Types of string grammarsAs special types of string grammars we consider string grammars with arbitrary productions, context-free productions of the form A → v with A N and v V*, and λ-free context-free productions of the form A → v with A N and v V*, the corresponding types of

grammars denoted by ENUM, CF, and CF-λ, thus yielding

the families of languages L(ENUM), i.e, the family of recursively enumerable languages, as well as L(CF) and

L(CF-λ), i.e., the families of context-free and λ-free

context-free languages, respectively.

Matrix grammars

is a grammar of type X, M is a finite set of finite sequences of productions (an element of M is called a

matrix), and F P. For a matrix m(i) = [mi,1,…,mi,n(i)] in M

and v,u O we define v m(i) u if and only if there are

w0,w1,…,wn(i) O such that w0 = v, wn(i) = u,

and for each j, 1 ≤ j ≤ n(i),

∙ either wj-1 m(i,j) wj according to G,

∙ or m(i,j) is not applicable to wj-1 according to G,

wj = wj-1, m(i,j) F.

A matrix grammar G with appearance checking (with ac

for short) GM of type X is a construct

(O,OT,P, G,w,M,F) where (O,OT,P, G ,w)

Language generated by a matrix grammar

The language generated by GM is

L(GM) = {v OT : w m(i,1) w1… m(i,k) wk,

wk = v, wj O, m(i,j) M for 1 ≤ j ≤ k ,k ≥ 1}.

The matrix grammar GM is said to be of type MATac; it is

said to be of type MAT - to be without appearance checking (without ac) - if F = { }. The corresponding families of languages are denoted by L(X-MATac) and L(X-

MAT), respectively.

Graph-controlled grammars

A graph-controlled grammar G of type X is a construct

(O,OT,P, G,w,R,L(in),L(fin)) where

(O,OT,P, G ,w) is a grammar of type X,

R is a finite set of rules r of the form ( l(r): p(l(r)), σ(l(r)), φ(l(r)) ), where l(r) Lab(G), Lab(G) being a set of labels associated (in a one-to-one manner) with the rules r in R, p(l(r)) P, σ(l(r)) Lab(G) is the success field of the rule r, and φ(l(r)) Lab(G) is the failure field of the rule r; L(in) Lab(G) is the set of initial labels, and L(fin) Lab(G) is the set of final labels.

Graph-controlled grammar - languages

For r = ( l(r): p(l(r)), (l(r)), (l(r)) ) and v,u O we define

(v,l(r)) G (u,k) if and only if

∙ either p(l(r)) is applicable to v, v G u, and k σ(l(r)),

∙ or p(l(r)) is not applicable to v, u=v, and k φ(l(r)).

The language generated by G is

L(G) ={v OT : (w0,l0) G (w1,l1)… G (wk,lk), k ≥ 1,

wj O and lj Lab(G) for 0 ≤ j ≤ k,

w0 = w, wk = v, l0 L(in), lk L(fin).

The graph-controlled grammar G is said to be of type

GCac; it is said to be of type GC - to be without

appearance checking (without ac) - if φ(l)= { } for all l Lab(G). The corresponding families of languages are

denoted by L(X-GCac) and L(X-GC), respectively.

Graph-controlled and matrix grammars

Theorem 1. For any arbitrary type X,

L(X-MAT) L(X-GC) and L(X-MATac) L(X-GCac).

Proof. Let G = (O,OT,P, G,w,M,F) be a matrix grammar of

type X; then we construct the graph-controlled grammar

G’ = (O,OT,P, G,w,R,L(in),L(fin)) with

Lab(G’) = { (i,j) : 1 ≤ j ≤ n(i), 1 ≤ i ≤ n },L(in) = L(fin) = { (i,1) : 1 ≤ i ≤ n },

R = { ( (i,j): mi,j, σ(i,j), φ(i,j) ) : 1 ≤ j ≤ n(i), 1 ≤ i ≤ n },

σ(i,j) = { (i,j+1) }, 1 ≤ j < n(i), σ(i,n(i)) = { (i,1) : 1 ≤ i ≤ n } ( = L(fin) = L(in) ), 1 ≤ i ≤ n;

φ(i,j) = σ(i,j) if mi,j F and φ(i,j) = { } otherwise,

1 ≤ j ≤ n(i), 1 ≤ i ≤ n . □

P system - definitionA P system (with priorities) of type X is a construct

= ( O,OT,P, , μ, wμ, Rμ, ρμ ) where

- ( O,OT,P, ) is a grammar scheme G() of type X;

- μ is the membrane structure of ; usually, we shall label the membranes with 1,...,n in a bijective way; the outermost membrane is labelled by 1 (skin membrane);

- wμ ( = (w0,w1,...,wn) ) is a function assigning a multiset

over O to the environment (w0) and to each region

inside membrane i, 1 ≤ i ≤ n;

- Rμ ( = (R1,...,Rn) ) is a function assigning a finite set

of rules to each membrane i, 1 ≤ i ≤ n, of μ;

- ρμ ( = (ρ1,...,ρn) ) is a function assigning a priority

relation for the rules in Ri to each membrane i of μ.

is one of the most common features of many models introduced so far in the area of P systems. A universal clock is assumed to control the parallel application of rules which may be quite unrealistic, but is relevant for many interesting theoretical results, especially when proving computational completeness and solving (NP-)hard problems.

Applying the rules assigned to the membranes in the maximally parallel derivation mode means choosing a multiset of rules in such a way that after assigning objects from the environment and the regions to these rules not enough objects are left to add another rule which could be applied together with the chosen rules.

maximally parallel derivation mode

sequential derivation mode

Biological processes in living organisms happen in parallel, but they may not be synchronized by a universal clock, instead many processes involve several objects in parallel, but the processes themselves are carried out in an unsynchronized or asynchronous way.

Formally, this feature an be captured by letting an arbitrary number of processes happen in parallel in an unsynchronized / asynchronous way(asynchronous derivation mode).

In the sequential derivation mode, only one rule is applied in one derivation step.

P system - derivation

A derivation in the P system works as follows:

We start with the initial multisets wi in the environment

and the regions inside the membranes, respectively.

At any stage of the derivation, the rules assigned to the membranes are used according to the derivation mode

(observe that a rule from Ri can only be applied if no other

rule of higher priority according to the priority relation ρi

could be applied, too, to any combination of objects inside and outside membrane i).

P system - languageAll terminal objects from OT ever appearing at any step

in any membrane contribute to the language L() generated by . The family of languages generated by X-P(pri) systems (with membrane structure μ) in the derivation mode m (sequential: m = s, maximally parallel: m = p) is denoted by L(X-mP(pri)) ( L(X-mP(μ,pri)), respectively). If all priority relations in an X-P(pri) system are empty, we call it an X-P system and denote the corresponding families of languages generated by X-P systems (with membrane structure μ) in the derivation mode d by L(X-mP) ( L(X-mP(μ) ), respectively).

Rules in a P system of type X

A rule in a P system of type X is of the form

<(x1,or1), …, (xm,orm)> → <(y1,tar1), …, (yn,tarn)>

where orj (origin) and tark (target) are from {in, out} and

<x1, …, xm> → <y1, …, yn> is a rule over the

underlying grammar scheme G() of type X.

The objects x1, …, xm to be affected as in the rule

<x1, …, xm> → <y1, …, yn> are chosen according to the

origins or1, …, orm and the resulting objects y1, …, yn

are released according to the targets tar1, …, tarn.

P systems with splicing rulesTheorem. Any recursively enumerable language L canbe generated by a P system with splicing rules with only one membrane and even without priorities on the rulesin the sequential derivation mode, i.e.,

L(splicing-sP( [1 ]1 ) = L(ENUM).

| |skin membrane | |

axioms(unbounded)

axioms(unbounded)

in out

| | splicing rules | |

P systems with string rewriting rulesTheorem. Any recursively enumerable language L canbe generated by a P system with priorities and context-free rules in a membrane structure of two membranes in the sequential mode, i.e.,

L(CF-MATac) =

L(CF-GCac) =

L(CF-sP( [1 [2 ]2 ]1 , pri ) ) =

L(ENUM).

P systems with string rewriting rulesbut without priorities

Theorem. Without priorities, P systems withcontext-free rules (in a linear membrane structure of three membranes) in the sequential modecan only generate languages generated bymatrix grammars without appearance checking, i.e.,

L(CF-MAT) = L(CF-sP( [1 [2 [3 ]3 ]2 ]1 ) ).

Theorem. Without priorities, P systems withcontext-free rules (in a linear membrane structure of three membranes) in the maximally parallel modecan generate any recursively enumerable language, i.e.,

L(CF-MATac) = L(CF-GCac) =

L(CF-pP( [1 [2 [3 ]3 ]2 ]1 ) ) = L(ENUM).

Variants of P systems

used for implementing parallel algorithms (usually linear in time, but trading time for space)for solving (NP-)hard problems

► generate/dissolve membranes

► tissue(-like) P systemsarbitrary graph structure for connections betweencells (not necessarily a tree as in P systems);e.g., useful for describing neural networks

► ...

Related models

Luca Cardelli:

Brane Calculi: process calculi with computation “on” themembranes, not inside them.

BioAmbients: a stochastic calculus with compartments.

http://www.luca.demon.co.uk

Future research► investigate the complexity of various models of P systems, especially searching for the borderline between systems that are / are not universal (or computationally complete, respectively);

► find new (parallel) algorithms for solving (NP-)hard problems based on P systems;

► investigate the potential of various models of P systems to describe biological processes;

► ...

► implement various models of P systems “in silicio” and/or “in vitro” (in the lab);