Outline. Bio-inspired. Computing. Conclusions References. Living cells Membranes: structure and function Transport

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2 Outline Bio-inspired Living cells Membrnes: structure nd function Trnsport Computing Bsics Exmples Some results Some vrints Conclusions References Spiking Neurons P systems

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4 Living Cells Digrm of typicl eukryotic cell, showing subcellulr components. Orgnelles: (1) nucleolus (2) nucleus (3) ribosome (4) vesicle (5) rough endoplsmic reticulum (ER) (6) Golgi pprtus (7) Cytoskeleton (8) smooth ER (9) mitochondri (10) vcuole (11) cytoplsm (12) lysosome (13) centrioles within centrosome

5 Cell membrnes

6 Cell membrnes Cell membrnes re selective brriers tht seprte individul cells from environment nd cellulr comprtments. Some functionlities: regulte the trnsport of molecules, control informtion flow between cells, generte signls to lter cell behvior, contin molecules helping the formtion of tissues, Cell membrnes re dynmic, constntly being formed nd degrded.

7 Membrne Trnsport Three min types of trnsport: Pssive Trnsport: requires no energy from the cell. Active Trnsport: requires the cell to spend energy. Vesicle Trnsport: vesicles tht fuse with the cell membrne my be utilized to relese or trnsport chemicls out of the cell or to llow them to enter cell.

8 Pssive Trnsport Exmples include the diffusion of oxygen nd crbon dioxide, osmosis of wter, nd fcilitted diffusion. Imge from Purves et l., Life: The Science of Biology, 4th Edition, by Sinuer Assocites ( nd WH Freemn (

9 Active Trnsport Uniport Symport Antiport A A B A Uniport (fcilitted diffusion): crriers medite trnsport of single component. Symport (cotrnsport): crriers bind 2 dissimilr components (substrtes) & trnsport them together cross membrne. Antiport (exchnge diffusion): crriers exchnge one component for nother cross membrne. B

10 Vesicle Trnsport Exocytosis is the term pplied when trnsport is out of the cell. Endocytosis is the cse when molecule cuses the cell membrne to bulge inwrd, forming vesicle.

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12 A look t history Since the origins, Computer Scientist hve looked to reltionships mong mchines nd living orgnisms: 8 McCulloch nd Pitts: Neurl Networks (1943). 8 Von Neumnn: Cellulr Automt (1966). 8 Lindenmyer: L systems (1968). 8 Hollnd: Genetic Progrmming (1975). 8 Adlemn: DNA Computing (1994).

13 Membrne computing 8 Gh. Păun, Computing with Membrnes, Membrne Computing looks t the whole cell structure nd functioning s computing device. 8 Membrnes ply fundmentl role in the cell s filters nd seprtors. 8 Modeling the living cell is beyond the purpose of Membrne Computing!!

14 A membrne system (or P-system) 8 A membrne structure formed by severl membrnes embedded in unique min membrne. 8 The objects re represented s symbols of given lphbet (ech symbol denotes different object). 8 Multi-sets of objects re plced inside the regions delimited by the membrnes (one per ech region). 8 Sets of evolution rules ssocited with the regions (one set per region), which llow the system: 9 to produce new objects strting form the existing ones 9 to move objects from one region to nother

15 A membrne system (or P-system)

16 A membrne system (or P-system) Membrne structure 1: skin 2 3 membrnes

17 A membrne system (or P-system) Membrne structure 1: skin 2 regions 3 membrnes

18 A membrne system (or P-system) Membrne structure 1: skin 2 6 c 2 d regions 3 c objects d membrnes

19 A membrne system (or P-system) Membrne structure 1: skin 2 6 c 2 d 3 c c c c c d d 2 d Evolution rules dcc (NO,in 3 ) d (YES,in 3 )

20 A membrne system (or P-system) Membrne structure 1: skin 2 6 c 2 d 3 c Output membrne environment c c c c d d 2 d Evolution rules dcc (NO,in 3 ) d (YES,in 3 )

21 A membrne system (or P-system) 8 A multiset is like set, but every element cn hve more thn one copy (we cn formlly define it s n ppliction tht ssocites nturl number to every element). 8 Typiclly, n evolution rule is of the form u v nd it sys tht copy of every object in u is replced by copy of the objects ppering in v (with some extr informtion). 8 Exmple: d 2 c (b,in 3 ) The ppliction of this rule in region R, consumes 2 elements of type d nd 1 of type c, nd produces 1 element in the sme region, nd 1 element b in region 3 (supposed it is inner to R).

22 A computtion in P-system 8 We strt with n initil configurtion: n initil membrne structure nd some initil multisets of objects plced inside the regions of the system. 8 We pply the rules in non-deterministic mximl prllel mnner: in ech step, in ech region, ech object tht cn be evolved ccording to some rule must do it. 8 A computtion is sid successful if it hlts, tht is, it reches configurtion where no rules cn be pplied. 8 The result of successful computtion my be the multisets formed either by the objects contined in specific output membrne or by the objects sent out of the system during the computtion, or 8 A non-hlting computtion yields no result.

23 One more exmple: Non-determinism

24 More thn evolution nd communiction 8 Membrne dissolution: specil opertor which cn be used for dissolving membrne. r: u vδ if the rule r is used inside membrne, such membrne is dissolved fter the ppliction of the rule. 8 Membrne thickness: two opertors δ, τ for vrying the permebility of the membrnes. 8 Priority: prtil order mong the rules, which define priority reltionship: In ech step, if rule with high priority is pplied then no rule with lower priority cn be pplied in the sme step.

25 An exmple: divisibility predicte c 2 d 3 Output membrne c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

26 An exmple: divisibility predicte c 2 d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

27 An exmple: divisibility predicte c 2 d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

28 An exmple: divisibility predicte c 2 d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

29 An exmple: divisibility predicte c 2 d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

30 An exmple: divisibility predicte 1 3 c 2 d dcc (NO,in 3 ) d (YES,in 3 ) >

31 An exmple: divisibility predicte 1 3 YES dcc (NO,in 3 ) d (YES,in 3 ) >

32 An exmple: divisibility predicte c 2 d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

33 An exmple: divisibility predicte c 2 d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

34 An exmple: divisibility predicte c 2 d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

35 An exmple: divisibility predicte c c d c c c c > d dδ dcc (NO,in 3 ) > d (YES,in 3 )

36 An exmple: divisibility predicte 1 3 c c d dcc (NO,in 3 ) > d (YES,in 3 )

37 An exmple: divisibility predicte 1 3 NO dcc (NO,in 3 ) > d (YES,in 3 )

38 Using ctlysts 8 Coopertion restricted to some specil objects clled ctlysts c cv Ctlysts cnnot be modified by ny rule nd cnnot be moved from one region to nother, but they trigger the ppliction of the rule. 8 Bi-stble ctlysts: ctlysts with two sttes c c v c c v

39 Some universlity results P-systems with ctlysts re computtionlly universl

40 Some universlity results Using ctlysts nd bi-stble ctlysts

41 Summry 1 8 P-systems re bio-inspired distributed nd prllel computing devices. 8 The objects re locted inside specific regions delimited by membrnes. 8 The system opertes on multisets of objects. 8 The objects evolve ccording to locl rules ssocited with the regions. 8 The rules cn modify the objects or move them through the membrnes.

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43 V1: P-systems with string objects 8 Now, the objects re strings over given lphbet. 8 The regions hve ssocited lnguges insted of multisets of objects. The rules encode string-opertions: 9 rewriting: X (y, tr), with tr ε {here, in, out} 9 replicted rewriting: X (y 1, tr 1 ) (y n, tr n ), with tr 1,, tr n ε {here, in, out} 9 splicing More ingredients: membrne dissolution, membrne thickness, priority, etc

44 V2: Communictive P-systems 8 Communiction of objects through membrnes is one of the most importnt ingredients of every P-system. 8 Wht cn we do if we llow only communiction? Purely communictive systems: the objects re not chnged during computtion, but they just chnge their plce inside the system. 8 In order to hve enough elements for computtion, we suppose tht the system is embedded in n infinite environment, which contins n rbitrry number of copies of ech object. The environment provides the objects the system needs to perform its internl computtions.

45 V2: Communictive P-systems

46 Bio-justifiction :Membrne trnsport of smll molecules Uniport Symport Antiport A A B A B

47 P-systems with symport/ntiport 8 Rules encode symport/ntiport mechnism: 8 Initilly they use only symbol objects. 8 Generliztion: (x,in), (x,out), (x,in; y,out), for x, y multisets of rbitrry size

48 The power of communiction P-systems with symport/ntiport re computtionlly universl (Păun, A., Frisco, P., Păun, Gh., 2003) (1,2) + 1 Mem = T.M. (2,0) + 4 Mem = T.M. (3,0) + 2 Mem = T.M. (3,0) + 1 Mem = T.M. (n,m) denotes the size of the rules: for ech (x,in), (x,out), x n, nd for ech (x,in; y,out), mx{ x, y } m

49 Evolution-Communiction P-systems P-systems with boundry rules (rules ssocited to membrnes, more thn regions): 8 Communiction rules: xx [ i y y xy [ i x y (x,in), (y,out), (x,in; y,out) 8 Evolution rules: [ i y [ i y

50 The Power of EC P-systems EC P systems re computtionlly universl

51 V3: P-systems with ctive membrnes 8 Rules re ble to perform opertion for modifying the membrne structure: 9 membrne cretion: [ i ] i [ j b ] j 9 membrne division: [ i ] i [ k b ] k [ j c ] j 9 membrne dupliction: [ i ] i [ k b [ j c ] j ] k 9 membrne dissolution: [ i ] i where, b re objects nd i, j, k re lbels of possible membrnes. 8 Communiction nd Evolution rules ssume the form: [ i v ] i, [ i ] i [ i ] i b, [ i ] i [ i b ] i where, b re objects nd i, j, k re lbels of possible membrnes.

52 Trding time for spce 8 The Hmiltonin Pth Problem (HPP) cn be solved in qudrtic time nd the SAT problem cn be solved in liner time by P-systems with ctive membrnes, by using membrne division nd dissolution. 8 HPP cn be solved in liner time by using membrne cretion. Ide: generte in n efficient mnner ll pths from specified initil node, then checking whether or not t lest one of these pths is Hmiltonin.

53 Wht else Whtever you wnt 8 Energy-Controlled P-systems. 8 P-systems with promoters/inhibitors. 8 P-systems with crriers. 8 P-systems with gemmtion of mobile membrnes. 8 Tissue P-systems. 8 Probbilistic P-systems. 8 P-systems with elementry grph productions. 8 Prllel Rewriting P-systems

54 V5: Spiking Neurl P systems Bsed on the specific ides of spiking neurons. A set of neurons (only one membrne) plced on the nodes of (directed) grph nd sending signls using the edges of the grph. Rules of two types: Spiking rules (firing): c ;d (c 1, d 0). Forgetting rules: s.

55 How the rules work Spiking (firing) rules: c ;d 3 c inputs c ;0 3 Step: 0

56 How the rules work Spiking (firing) rules: c ;d c inputs 2 c c ;0 2 Step: 1

57 How the rules work Spiking (firing) rules: c ;d c inputs c c ;0 Step: 2

58 How the rules work Spiking (firing) rules: c ;d c inputs c c ;0 Step: 3

59 How the rules work Spiking (firing) rules: c ;d 3 c inputs c ;2 3 Step: 0

60 How the rules work Spiking (firing) rules: c ;d c inputs 2 c c ;2 2 Step: 1

61 How the rules work Spiking (firing) rules: c ;d c inputs c c ;2 Step: 2

62 How the rules work Spiking (firing) rules: c ;d c inputs c c ;2 Step: 3

63 How the rules work Spiking (firing) rules: c ;d c inputs c ;2 Step: 4

64 How the rules work Forgetting rules: c λ c inputs c λ Step: 0

65 How the rules work Forgetting rules: c λ c inputs c c λ Step: 1

66 How the rules work Forgetting rules: c λ c inputs c λ Step: 2

67 Some results Theorem. Any Boolen function, f:{0,1} k {0,1}, cn be computed by n SNP with k input neurons (lso using further 2k+4 neurons, one being the output one). Theorem. The universlity of SNP cn be obtined for systems tht: 1. Do not use delys in the rules, 2. Do not use forgetting rules, 3. (1 ó 2) + the restriction of hving grph with output degree less or equl to (1 ó 2) + the restrcition of hving grph with input degree less or equl to 2.

68 One exmple input 1 input 2 input 3 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

69 One exmple: Computing f(1,1,1) input 1 input 2 input 3 Time=0 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

70 One exmple: Computing f(1,1,1) input 1 input 2 input 3 Time=1 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

71 One exmple: Computing f(1,1,1) input 1 input 2 input 3 Time=2 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

72 One exmple: Computing f(1,1,1) input 1 input 2 input 3 Time=3 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

73 One exmple: Computing f(1,1,1) input 1 input 2 input 3 Time=4 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

74 One exmple: Computing f(1,1,0) input 1 input 2 input 3 Time=0 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

75 One exmple: Computing f(1,1,0) input 1 input 2 input 3 Time=1 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

76 One exmple: Computing f(1,1,0) input 1 input 2 input 3 Time=2 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

77 One exmple: Computing f(1,1,0) input 1 input 2 input 3 Time=3 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

78 One exmple: Computing f(1,1,0) input 1 input 2 input 3 Time=4 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

79 One exmple: Computing f(1,1,0) input 1 input 2 input 3 Time=5 Auxilir neurons output ;0 2 ;0 3 ;0 4 5 ; ;0 SNP computing: f:{0,1} 3 {0,1} Defined by: f(b 1,b 2,b 3 )=1 iff b 1 +b 2 +b 3 2

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81 Conclusions 8 Membrne Computing provides computtionl models tht bstrct from the living cells structure nd functioning. 8 Such models hve been proved to be computtionlly powerful (equiv. to T.M.) nd efficient (solving NP-Complete problems in polynomil time). 8 Membrne Computing defines n bstrct frmework for resoning bout: 9 distribute rchitectures 9 communiction 9 prllel informtion processing 8 Such fetures re relevnt both for Computer Science (Distributed Computing Models, Multi-Agent Systems) nd Biology (Modeling nd Simultion of Biologicl Systems).

82 Conclusions 8 Hrd to mke rel implementtions: some ttempts were mde, nd some re in development now. 8 Non-Determinism nd Mximl Prllelism re not lwys desirble fetures nd some vrints try to control them.

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84 Min References: books

85 Min References Web Pge: 8 The P systems Web Pge: Where you cn find lot of ppers, reports, softwre, etc. Conferences/Workshops (every yer): 8 (BWMC) Brinstorming Week on Membrne Computing. 8 (WMC) Workshop on Membrne Computing. 8 (BIC-TA) Interntionl Conference on Bio-Inspired Computing: Theory nd Applictions.

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