Membrane Systems in Algebraic Biology. From a Toy to a Tool

Transcription

1 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks : From a Toy to a Tool Friedrich Schiller University Jena School of Biology and Pharmacy Department of Bioinformatics thomas.hinze@uni-jena.de January 29, 2009

2 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Membrane Systems: Inspired by Cells and Tissues = Capturing specialties of intra- and intercellular processes

3 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks (I) Nested Compartments Delimited by Permeable Membranes = Spatial regions wherein chemical reactions can occur

4 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks (II) Dynamics in Compartmental Cell Structure endocytosis exocytosis merge division dissolution separation creation gemmation = Plasticity initiated by dedicated reaction networks

5 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks (III) Complex Polymeric Biomolecules in Low Concentrations external signal endocrine (dist.) paracrine (near) autocrine (same cell) ligands hormones, factors,... receptors enzyme linked ion channel G protein linked GDP GTP cell membrane activation cascade cell response phospholipid bilayer TP DP phosphorylation activation by protein kinases gene expression signal transduction, transformation, amplification via pathways cytosol inner membrane genomic dna nucleus = Reaction pathways forming specific biomolecules

6 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Outline : From a Toy to a Tool 1. Primer: Reaction systems composed of discrete entities 2. Membrane systems: Some introductory examples 3. Features and varieties: Classification and properties 4. Π CSN : modelling framework for cell signalling networks 5. Bio-pplications: ppetizers for systems biologists 6. Membrane systems as computing devices 7. The P page: n online repository and more 8. Quo vadis: Concluding remarks and outlook # time

7 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Multiset: Molecular Configuration within a Membrane Example Definitions L = {(, 3),(B, 2),(C, 0),(D, 1)} supp(l) = {, B, D} card(l) = 6 Multiset: Let F be a set. multiset over F is a mapping F : F N { } that specifies the multiplicity of each element a F. Support: Let F : F N { } be a multiset. set S F is called supp(f) iff S = {s F F(s) > 0}. Cardinality: card(f) := F(a) a F D B B contents of a toy membrane

8 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Term Rewriting: Employ a Reaction by Set Operations D B B 2 + B C B D C Example {(, 3),(B, 2),(C, 0),(D, 1)} {(, 2),(B, 1)} {(C, 1)} = {(, 1),(B, 1),(C, 1),(D, 1)} Multiset operations Difference: F G := {(a, max(f(a) G(a), 0)) a F \ G} Sum: F G := {(a, F(a) + G(a)) a F G} Union: F G := {(a, max(f(a), G(a))) a F G} Intersection: F G := {(a, min(f(a), G(a))) a F G}

9 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Coping with Conflicts: Introducing Process Control D B B 2 + B C D B C D B B 2 + B C 2 + D E? Possible strategies to decide among satisfied reactions Nondeterminism: Maximal parallel enumeration of all potential scenarios Prioritisation of reactions: Determinisation by a predefined order for applicability of reactions Stochasticity: Randomly select a satisfied reaction

10 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Evolution over Time Time-discrete iteration scheme Starting from an initial molecular configuration L 0 Iterative term rewriting for transition(s) L t L t+1. Each iteration corresponds to a discrete period in time τ Within each iteration turn, applicable reactions are figured out and subsequentially employed once or several times (e.g. kinetic function f : L N in concert with discretised kinetic laws) Obtaining a derivation tree that lists sequences of molecular configurations as nodes Considered aspects Suitability for small amounts of reacting particles (e.g. cell signalling) Compliance with mass conservance for undersatisfied reaction scenarios

11 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks First Example: Single Membrane System Π PR = (V, T,[ 1 ] 1, L 0, R) V system alphabet T V terminal alphabet [ 1 ] compartmental structure L 0 V (N { }) multiset for initial configuration R = {r 1,...,r k } set of reaction rules Each reaction rule r i consists of two multisets (reactants E i, products P i ) such that r i = ({( 1, a 1 ),...,( h, a h )}, {(B 1, b 1 ),...,(B v, b v )}). We write in chemical denotation: r i : a a h h b 1 B b v B v = Index i specifies priority of r i : r 1 > r 2 >... > r k.

12 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks First Example: Single Membrane System Π PR = (V, T,[ 1 ] 1, L 0, R) V system alphabet T V terminal alphabet [ 1 ] compartmental structure L 0 V (N { }) multiset for initial configuration R = {r 1,...,r k } set of reaction rules Each reaction rule r i consists of two multisets (reactants E i, products P i ) such that r i = ({( 1, a 1 ),...,( h, a h )}, {(B 1, b 1 ),...,(B v, b v )}). We write in chemical denotation: r i : a a h h b 1 B b v B v = Index i specifies priority of r i : r 1 > r 2 >... > r k.

13 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Dynamical Behaviour of Π PR Iteration scheme for configuration update incrementing discrete time points t N L t+1 = { (a,α a,k ) a V α a,0 = card (L t {(a, )}) β a,i = card (E i {(a, )}) γ a,i = card (P i {(a, )}) { αa,i 1 + f α a,i = i γ a,i f i β a,i iff a V : α a,i 1 f i β a,i α a,i 1 else i {1,...,k} } System output (distinction empty/nonempty configuration): supp ( t=0 (L t {(w, ) w T }) ) T

14 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation Study of a Concrete Toy System Π PR Π PR = ({Z 0, Z 1, Z 2, Z 3, Z 4, N, Y, 0, 1, Φ, C}, {Φ}, [ 1 ] 1, L 0, {r 1,..., r 12 }) L 0 = {(Z 0, 2),(0, 1), (1, 1),(C, 1000)} r 1 : Z C Y + Φ + 1 r 5 : Z C Z Z 0 r 9 : Z C Z r 2 : Z C Y + Φ + 0 r 6 : Z C Z Z 0 r 10 : Z C N + 0 r 3 : Y C Y + Φ + 1 r 7 : Z C N + 1 r 11 : Z C N + 0 r 4 : Y C Y + Φ + 0 r 8 : Z C Z r 12 : Z C N + 1 number Objektanzahl of particles C Φ time Zeitskala course N Y Dynamical simulation was carried out using MatLab ( τ = 10 4 ). Example comes from transformation of a finite automaton.

15 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks From a Single Membrane to a Membrane Structure Hierarchically nested membranes denoted as a tree Representation as character string (or graph) µ = [ 1 [ 2 ] 2 [ 3 ] 3 [ 4 [ 5 ] 5 [ 6 [ 8 ] 8 [ 9 ] 9 ] 6 [ 7 ] 7 ] 4 ] 1 G. Păun. Membrane Computing: n Introduction. Springer Verlag, 2002

16 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Extensions of Rules Beyond Reactions: Transportation Transportation of molecules through membranes 5 4 D B B 5 4 D B B [ 4 ] 4 : {(, 2)} [ 5 ] 5 lgebraic settings available for different scenarios like Diffusion (unspecific transport through neighboured membranes) Receptors (acting as molecular filters) Symport/antiport (interactive molecular exchange)

17 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks ctive Membranes Rules for dynamical changes in membrane structure G. Păun. Introduction to Membrane Computing. Romanian cademy, 2001 Specification [ h1 a] h1 [ h2 ] h2 [ h2 [ h1 b] h1 ] h2 Change in membrane structure µ effects further system components djustment of configuration L and rules R attached to concerned membranes Membrane properties like electric charge (e.g. [] +,[] )

18 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks T T C G T G G DN-Strang 5 C C T 5 T Structuring the Objects 3 3 linearisierte Darstellung 5 T C T G C G T 3 G T C T G C Erkennen der zugrunde liegenden DN-Einzelstränge H-TCTGCGT-H H-CGTCTG-H 3 5 Notation als Wörter einer formalen Sprache T. Hinze, M. Sturm. Rechnen mit DN. 316pp., Oldenbourg Wissenschaftsverlag, 2004 lgebraic representation of molecular entities or reactants as character strings in terms of formal languages Placeholders ( ) for handling regular expressions in conjunction with matching strategies

19 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Classification of Membrane Systems cell like hierarchically nested membranes tissue like graph based compartmental topology neural spiking pulses instead of reactions population / conformons interacting autonomous agents G. Păun. Membrane Computing: n Introduction. Springer Verlag, 2002 Main classes with respect to compartmental topology and principle of operation

20 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Membrane System for Cell Signalling Processes Capture significant aspects of cellular signalling: Components, topology, modularity Protein activation states Dynamical behaviour (kinetics) Signal coding and transduction Coping with incomplete protein information Based on tissue-like membrane systems Keep formalism tractable Balance detailedness with computational needs Facilitate system modification, recombination, and construction ab initio # time T. Hinze, T. Lenser, P. Dittrich. Protein Substructure Based P System for Description and nalysis of Cell Signalling Networks. In H.J. Hoogeboom, G. Paun, G. Rozenberg,. Salomaa (Eds.), Membrane Computing. Series Lecture Notes in Computer Science 4361: , Springer Verlag, 2006

21 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks System Definition Π CSN System Π CSN = (V, V, E, M, n) V : alphabet of protein identifiers V : alphabet of protein substructure/property identifiers M: modules functional reaction units E: graph transduction channels between modules n: number of modules (degree of the P system) Modules M i = (R i1,...,r iri, f i1,... f iri, i ) M R ij : reaction rule multisets of educts and products may contain meta-symbols matching required f ij : function corresponding to kinetics of R ij, number of educt objects taken from module within one reaction step i : multiset of axioms initial contents of M i Channels e ij = (i, j, I ij, d ij ) E Weighted directed channel from module i to module j I ij : filter interface (receptor pattern and conc. gradient) d ij : time delay (number of system steps) for passage

22 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Matching and Matching Strategies String-based representation of proteins String-object s: representation of a protein, its properties, substructure, binding domains, activation state, ligands Structure: s V + ({#} ((V ) + { } (V ) + {*})) Meta-symbols: placeholder (wild-card) * appropriate or unknown substructure/property; exclusion Test whether two string-objects identify same molecule Matching strategies loose: two stringobjects match iff there is at least one common wildcard free representation Cα#GDP#p Cα#GDP# p Cα#GTP#p Cα#GTP# p Cα#*#p Cα#GDP#* Cα#*#* Dβ:E#*#* loose matching V = {Cα, Dβ:E} V = {GDP, GTP, p} Cα#GDP#p Cα#GDP# p Cα#GTP#p Cα#GTP# p Cα#*#p Cα#GDP#* Cα#*#* Dβ:E#*#* Cα#GDP#p Cα#GDP# p concretions Cα#GTP#p Cα#GTP# p Cα#*#p Cα#GDP#* Cα#*#* Dβ:E#*#* strict matching patterns strict: participating string-objects interpreted as a pattern and a candidate (concretion of the pattern)

23 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Matching and Matching Strategies concretions Cα#GDP#p Cα#GDP# p Cα#GTP#p Cα#GTP# p Cα#*#p Cα#GDP#* Cα#*#* Dβ:E#*#* V = {Cα, Dβ:E} V = {GDP, GTP, p} Cα#GDP#p Cα#GDP# p Cα#GTP#p Cα#GTP# p Cα#*#p Cα#GDP#* Cα#*#* Dβ:E#*#* loose matching Cα#GDP#p Cα#GDP# p Cα#GTP#p Cα#GTP# p Cα#*#p Cα#GDP#* Cα#*#* Dβ:E#*#* strict matching patterns

24 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks System Behaviour and Properties Definition of dynamical system behaviour Contents of module M i at global time t N: multiset L i (t) System step by module M i : L i (0) = i L i(t) = L i (t) Educts i (t) Products i (t) L i (t + 1) = L i(t) Outgoing i (t) Incoming i (t) 1. Determine multiset of educts using L i (t), R i1,..., R iri, f i1,... f iri ; involves matching 2. Remove educt objects from module contents 3. Determine and add multiset of reaction products, obtain L i(t) 4. Determine and separate objects leaving host module evaluate L i(t) and I for each outgoing channel, matching 5. dd objects received from incoming channels, consider d System properties Modularity static system topology ability to identify objects/substructures flexibility in level of abstraction Determinism computational tractability universality

25 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Example: Yeast Pheromone Pathway Signal transduction in Saccharomyces cerevisiae α M 1 Ste2 α Ste2 Gα Gα Gβγ GTP GDP Gα Gβγ GDP Ste11 Ste5 Ste7 Fus3 Fus3 p Ste11 Ste5 p Ste7 Ste20 Gβγ Fus3 Ste11 Ste5 Ste7 Fus3 p p p p Ste11 Ste5 Ste7 Fus3 Ste20 Gβγ p p Ste11 Ste5 Ste7 Fus3 p Ste11 Ste5 Ste7 Fus3 Ste20 Gβγ M 2 M 4 p p Gβγ Ste11 Ste5 Ste7 Fus3 Ste20 p Ste11 Ste5 p Ste7 Fus3 Ste20 Gβγ p Ste11 Ste5 p Ste7 Fus3 Ste20 Gβγ p p Ste11 Ste5 Ste7 Fus3 Gβγ Ste11 Ste5 Ste7 Fus3 Ste20 Gβγ p Ste11 Ste5 Ste7 Fus3 Ste20 Gβγ p Ste11 Ste5 p Ste7 Fus3 Ste20 Gβγ p M 3 Π=(V, V, E, M, 4) V = {Ste2, α, Gβγ, Gα,...} V = {a, GDP, GTP, p} M = {M 1, M 2, M 3, M 4 } M 1 =(R 11, R 12, R 13, R 14, R 15, f 11, f 12, f 13, f 14, f 15, 1 ) R 11 = Ste2# a + α Ste2#a R 12 = Ste2#a Ste2# a f 11 = k 11 [Ste2# a][α]/v T. Hinze, T. Lenser, P. Dittrich. Protein Substructure Based P System for Description and nalysis of Cell Signalling Networks. In H.J. Hoogeboom, G. Paun, G. Rozenberg,. Salomaa (Eds.), Membrane Computing. Series Lecture Notes in Computer Science 4361: , Springer Verlag, 2006

26 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Bio-pplications: Further ppetizers books.google.de G. Ciobanu, G. Păun, M.J. Perez-Jimenez (Eds.). pplications of Membrane Computing. Springer Verlag, 2005

27 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Membrane Systems as Universal Computing Devices Models and concepts for biologically inspired computing (Bio)molecular computation Genetic circuits Cell-based computing Neural networks Gene assembly Evolutionary computing morphous computing = Covered by variants of P systems = Computational completeness shown by simulation of Turing machines or equivalents lan Turing

28 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Random ccess Machine (RM) Established Turing-Complete Model for Computation Syntactical denotation of components RM = (R, L, P, l 1 ) jump label of first instruction program (finite set of instructions) finite set of jump labels L = {l 1,..., l c} finite set of registers R = {r 1,..., r m}, vailable instructions l i : INCR(r k ), l j increment register r k, jump to l j l i : DECR(r k ), l j decrement register r k, jump to l j l i : r k > 0, l j, l p if r k > 0 jump to l j else jump to l p l i : HLT terminate program and output Useful assumptions Consecutive indexing of jump labels and registers Determinism Initialisation of registers at start Output of all m registers when HLT r k Z

29 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN Consider for simulation and transformation RM = (R, L, P, l 1 ) R = {r 1, r 2 } L = {l 1, l 2, l 3, l 4 } P = {l 1 : r 1 > 0, l 2, l 4, l 2 : DECR(r 1 ), l 3, l 3 : INCR(r 2 ), l 1, l 4 : HLT} Initialisation: r 1 = 1 and r 2 = 0 (arbitrarily chosen) Program moves contents of r 1 cumulatively to r 2 RM consists of m = 2 registers and c = 4 instructions Each register and each instruction forms separate module of membrane system Π CSN = (V, V, E, M, c + m)

30 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#1 M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 B + D E 1 M 5 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + M 6 B + D E 1 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

31 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#X M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 M 3 l 3 :INCR(r 2 ),l 1 PC#3 PC#1 +, D 1 B + D E 1 M 5 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 6 B + D E 1 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4

32 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#W, E 1 M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 M 3 l 3 :INCR(r 2 ),l 1 PC#3 PC#1 +, D 1 B + D E 1 M 5 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 6 B + D E 1 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4

33 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#W, 2E 1 M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 B + D E 1 M 5 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + M 6 B + D E 1 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

34 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 B + D E 1 M 5 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + M 6 B + D E 1 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

35 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 PC#2 E 1, H 1, B B + D E 1 M 5 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B M 2 l 2 :DECR(r 1 ),l 3 PC#2 PC#3 + B PC#3 PC#3 PC#3 PC#1 + M 6 B + D E 1 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

36 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#1 M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 M 5 B + D E 1 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + B + D E 1 M 6 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

37 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#X M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 D 1 E 1, H 1 PC#2 M 5 B + D E 1 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + B + D E 1 M 6 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

38 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#W M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 2D 1 B + D E 1 M 5 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + B + D E 1 M 6 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

39 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN PC#W, H 1 M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 M 5 B + D E 1 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + B + D E 1 M 6 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

40 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 M 5 B + D E 1 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + B + D E 1 M 6 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

41 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Simulation of Register Machines by Π CSN M 1 l 1 :r 1 > 0,l 2,l 4 PC#4 PC#4 M 4 l 4 :HLT all functions f = 1, g = 1, d = 0 PC#1 PC#1 PC#X + D 1 PC#X PC#W + D 1 PC#W + 2E 1 PC#2 PC#W + H 1 PC#4 D 1 E 1, H 1 PC#2 M 5 B + D E 1 register r 1 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 B PC#2 PC#3 + B M 2 l 2 :DECR(r 1 ),l 3 PC#3 PC#3 PC#1 + B + D E 1 M 6 register r 2 D E 2 2D 2 H 2 D E 3 2D 3 H 3 D E 4 2D 4 H 4 M 3 l 3 :INCR(r 2 ),l 1

42 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks The P Page: Repository and More...

43 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Membrane Systems: International Dimension Pioneered in 1998 by Gheorghe Paun 10 years scientific evolution 100 active researchers in community 1, 500 publications up to now Gheorghe Păun paun

44 Motivation Prerequisites Membrane Systems Bio-pplications Computing Devices Concluding Remarks Quo Vadis Membrane Systems? Trends and Visions State-of-the-art Establish modelling technique in Systems Biology Capturing aspects of inter/ intracellular information proc. pproximation towards further bio-applications Upcoming contributions Framework for reverse engineering (e.g. artificial evolution) Backtracking P systems Molecular tracing Envisioned: Membrane Theory" Kai Kai successive dephosphorylation P P P KaiB P P B B P P P P P Kai P KaiB B B? P P P KaiB Kai P Kai B P P KaiB P B P P successive phosphorylation gamma configuration Figures are parts of recently submitted material. B molecular amount alpha beta time