Membrane Systems with Marked Membranes
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- Rodger Clinton Allison
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1 Membrne Systems with Mrked Membrnes Mtteo Cvliere Microsoft Reserch - University of Trento Centre for Computtionl nd Systems Biology Itly Dep. of Computer Science nd Artificil Intelligence University of Seville Spin
2 Nturl Computing Nturl\Biologicl processes cn be used: For implementing computtions (Adlemn s experiment, 1994).
3 Nturl Computing Nturl\Biologicl processes cn be used: For implementing computtions (Adlemn s experiment, 1994). Constructing new computing devices (Benenson et l., 2001).
4 Nturl Computing Nturl\Biologicl processes cn be used: For implementing computtions (Adlemn s experiment, 1994). Constructing new computing devices (Benenson et l., 2001). Get inspirtions for new computtionl prdigms.
5 Nturl Computing Nturl\Biologicl processes cn be used: For implementing computtions (Adlemn s experiment, 1994). Constructing new computing devices (Benenson et l., 2001). Get inspirtions for new computtionl prdigms. Membrne Computing Computtionl prdigm inspired by the structure nd functioning of living cells.
6 Forml lnguge theory Alphbet finite set of symbols. V * set of ll strings of symbols from n lphbet V. V + = V * - {λ}. L V * lngugeover V.
7 Forml lnguge theory Alphbet finite set of symbols. V * set of ll strings of symbols from n lphbet V. V + = V * - {λ}. L V * lngugeover V. V = {, b, c} is n lphbet. x = bbbc = 3 b 3 c 2 is string over V. L = { n b n c n > 0} = {bc, bbc, bbbc,... } is lnguge over V.
8 Forml lnguge theory Alphbet finite set of symbols. V * set of ll strings of symbols from n lphbet V. V + = V * - {λ}. L V * lngugeover V. V = {, b, c} is n lphbet. x = bbbc = 3 b 3 c 2 is string over V. L = { n b n c n > 0} = {bc, bbc, bbbc,... } is lnguge over V. How cn we generte lnguges?
9 Forml (Chomsky) grmmrs A grmmr is finite device generting in specified wy the strings of lnguge.
10 Forml (Chomsky) grmmrs A grmmr is finite device generting in specified wy the strings of lnguge. A grmmr is G = (N, T, S, P) N set of nonterminls T set of terminls S xiom P productions u v u (N T) * N (N T) *, v (N T) *
11 Forml (Chomsky) grmmrs A grmmr is finite device generting in specified wy the strings of lnguge. A grmmr is G = (N, T, S, P) N set of nonterminls T set of terminls S xiom P productions u v u (N T) * N (N T) *, v (N T) * x y iff x = x 1 u x 2 x 1, x 2 (N T) * u v P
12 Forml (Chomsky) grmmrs A grmmr is finite device generting in specified wy the strings of lnguge. A grmmr is G = (N, T, S, P) N set of nonterminls T set of terminls S xiom P productions u v u (N T) * N (N T) *, v (N T) * x y iff x = x 1 u x 2 y = x 1 v x 2 x 1, x 2 (N T) * u v P
13 Forml (Chomsky) grmmrs A grmmr is finite device generting in specified wy the strings of lnguge. A grmmr is G = (N, T, S, P) N set of nonterminls T set of terminls S xiom P productions u v u (N T) * N (N T) *, v (N T) * bcd bd bc b x y iff x = x 1 u x 2 y = x 1 v x 2 x 1, x 2 (N T) * u v P
14 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }.
15 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }. G = ({S}, {,b}, S, {S Sb, S b}).
16 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }. G = ({S}, {,b}, S, {S Sb, S b}). S Sb
17 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }. G = ({S}, {,b}, S, {S Sb, S b}). S Sb Sbb
18 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }. G = ({S}, {,b}, S, {S Sb, S b}). S Sb Sbb bbb
19 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }. G = ({S}, {,b}, S, {S Sb, S b}). S Sb Sbb bbb bb
20 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }. G = ({S}, {,b}, S, {S Sb, S b}). S Sb Sbb bbb bb Sbbb...
21 The lnguge generted by grmmr G = (N, T, S, P) is L(G) = {x T * S * x }. G = ({S}, {,b}, S, {S Sb, S b}). S Sb Sbb bbb bb Sbbb... L(G) = { n b n n 1} = {b, bb, bbb,... }.
22 Chomsky hierrchy RE = Turing mchines CS = LBA CF = PDA REG =FA
23 Chomsky hierrchy RE = Turing mchines CS = LBA CF = PDA REG =FA { x x L} length set of L (set of numbers).
24 Chomsky hierrchy RE = Turing mchines CS = LBA CF = PDA REG =FA L= () * = {λ,,,,...} Length-set (L) = {x x is n even number} { x x L} length set of L (set of numbers).
25 Chomsky hierrchy RE = Turing mchines CS = LBA CF = PDA REG =FA L= () * = {λ,,,,...} Length-set (L) = {x x is n even number} { x x L} length set of L (set of numbers). NREG, NCF,..., NRE
26 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)}
27 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)} A multiset M cn be represented by string w = M( 1 1 ) M( 2 2)... M( n n )
28 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)} A multiset M cn be represented by string w = M( 1 1 ) M( 2 2)... M( n n ) Any permuttion of w is vlid representtion of M.
29 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)} A multiset M cn be represented by string w = M( 1 1 ) M( 2 2)... M( n n ) Any permuttion of w is vlid representtion of M. The multiset {(,3), (b,2), (c,4)} cn be represented by:
30 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)} A multiset M cn be represented by string w = M( 1 1 ) M( 2 2)... M( n n ) Any permuttion of w is vlid representtion of M. The multiset {(,3), (b,2), (c,4)} cn be represented by: bbcccc= 3 b 2 c 4
31 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)} A multiset M cn be represented by string w = M( 1 1 ) M( 2 2)... M( n n ) Any permuttion of w is vlid representtion of M. The multiset {(,3), (b,2), (c,4)} cn be represented by: bbcccc= 3 b 2 c 4 bbcccc
32 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)} A multiset M cn be represented by string w = M( 1 1 ) M( 2 2)... M( n n ) Any permuttion of w is vlid representtion of M. The multiset {(,3), (b,2), (c,4)} cn be represented by: bbcccc= 3 b 2 c 4 bbcccc...
33 Multisets A multiset is set where ech element hs multiplicity. {,,, b, b, c, c, c, c} = {(,3), (b,2), (c,4)} A multiset M cn be represented by string w = M( 1 1 ) M( 2 2)... M( n n ) Any permuttion of w is vlid representtion of M. The multiset {(,3), (b,2), (c,4)} cn be represented by: bbcccc= 3 b 2 c 4 bbcccc... The order in the string does not count
34 Membrne systems
35 Membrne systems Membrne systems re models of computtion inspired by some bsic fetures of living cells.
36 Membrne systems Membrne systems re models of computtion inspired by some bsic fetures of living cells. 1 3 b 4 b out b c c 2 c here c out b bout bc b 3 2
37 Membrne systems Membrne systems re models of computtion inspired by some bsic fetures of living cells. membrnes 1 3 b 4 b out b c c 2 c here c out b bout bc b 3 2
38 Membrne systems Membrne systems re models of computtion inspired by some bsic fetures of living cells. membrnes b b c c 2 c here c out multisets of objects (biomolecules) inserted in regions b out b bout bc b 3 2
39 Membrne systems Membrne systems re models of computtion inspired by some bsic fetures of living cells. membrnes b b c c 2 c here c out multisets of objects (biomolecules) inserted in regions b out b bout bc b 3 2 evolution rules with trgets (biochemicl rections)
40 Membrne systems Membrne systems re models of computtion inspired by some bsic fetures of living cells. membrnes b b c c 2 c here c out multisets of objects (biomolecules) inserted in regions b out environment b bout bc b 3 2 evolution rules with trgets (biochemicl rections)
41 Membrne systems Membrne systems re models of computtion inspired by some bsic fetures of living cells. membrnes b b c c 2 c here c out multisets of objects (biomolecules) inserted in regions b out environment b bout bc b 3 2 evolution rules with trgets (biochemicl rections) evolution rules re locl to the regions where they re present
42 Membrne systems A membrne system is construct Π = (V, µ, w 1, w 2,..., w m, R 1, R 2,..., R m, i 0 )
43 Membrne systems A membrne system is construct Π = (V, µ, w 1, w 2,..., w m, R 1, R 2,..., R m, i 0 ) V lphbet - its elements re clled objects.
44 Membrne systems A membrne system is construct Π = (V, µ, w 1, w 2,..., w m, R 1, R 2,..., R m, i 0 ) V lphbet - its elements re clled objects. µ is membrne structure (hierrchicl rrnged) with m membrnes. Ech membrne hs n unique lbel.
45 Membrne systems A membrne system is construct Π = (V, µ, w 1, w 2,..., w m, R 1, R 2,..., R m, i 0 ) V lphbet - its elements re clled objects. µ is membrne structure (hierrchicl rrnged) with m membrnes. Ech membrne hs n unique lbel. w 1, w 2,..., w m strings representing the initil contents of regions 1, 2,..., m of µ.
46 Membrne systems A membrne system is construct Π = (V, µ, w 1, w 2,..., w m, R 1, R 2,..., R m, i 0 ) V lphbet - its elements re clled objects. µ is membrne structure (hierrchicl rrnged) with m membrnes. Ech membrne hs n unique lbel. w 1, w 2,..., w m 1, 2,..., m of µ. strings representing the initil contents of regions R 1, R 2,..., R m finite sets of evolution rules ssocited with regions 1, 2,..., m of µ.
47 Membrne systems A membrne system is construct Π = (V, µ, w 1, w 2,..., w m, R 1, R 2,..., R m, i 0 ) V lphbet - its elements re clled objects. µ is membrne structure (hierrchicl rrnged) with m membrnes. Ech membrne hs n unique lbel. w 1, w 2,..., w m 1, 2,..., m of µ. strings representing the initil contents of regions R 1, R 2,..., R m finite sets of evolution rules ssocited with regions 1, 2,..., m of µ. i 0, output region.
48 Computtion of membrne system There exists globl clock mrking the time of ech step for the whole system.
49 Computtion of membrne system There exists globl clock mrking the time of ech step for the whole system. Configurtion µ (membrne structure) + contents of the regions.
50 Computtion of membrne system There exists globl clock mrking the time of ech step for the whole system. Configurtion µ (membrne structure) + contents of the regions. Step of computtion The evolution rules re used in nondeterministic mximlly prllel wy, in ech region. Next configurtion
51 Computtion of membrne system There exists globl clock mrking the time of ech step for the whole system. Configurtion µ (membrne structure) + contents of the regions. Step of computtion The evolution rules re used in nondeterministic mximlly prllel wy, in ech region. Next configurtion The computtion hlts (successful) when n hlting configurtion is reched.
52 Computtion of membrne system There exists globl clock mrking the time of ech step for the whole system. Configurtion µ (membrne structure) + contents of the regions. Step of computtion The evolution rules re used in nondeterministic mximlly prllel wy, in ech region. Next configurtion The computtion hlts (successful) when n hlting configurtion is reched. The result of successful computtion is the number of objects in the output region.
53 Nondeterministic mximlly prllel Objects nd rules re chosen in nondeterministic wy. The choice is exhustive: fter choice is done no rule cn be pplied to the non ssigned objects.
54 Nondeterministic mximlly prllel Objects nd rules re chosen in nondeterministic wy. The choice is exhustive: fter choice is done no rule cn be pplied to the non ssigned objects. r 1 : bcc cc r 2 : c d r 3 : bc bb b c c
55 Nondeterministic mximlly prllel Objects nd rules re chosen in nondeterministic wy. The choice is exhustive: fter choice is done no rule cn be pplied to the non ssigned objects. Choice 1: (r 1, 1) r 1 : bcc cc r 2 : c d r 3 : bc bb b c c
56 Nondeterministic mximlly prllel Objects nd rules re chosen in nondeterministic wy. The choice is exhustive: fter choice is done no rule cn be pplied to the non ssigned objects. Choice 1: (r 1, 1) Choice 2: (r 2, 2) r 1 : bcc cc r 2 : c d r 3 : bc bb b c c
57 Nondeterministic mximlly prllel Objects nd rules re chosen in nondeterministic wy. The choice is exhustive: fter choice is done no rule cn be pplied to the non ssigned objects. Choice 1: (r 1, 1) Choice 2: (r 2, 2) Choice 3: (r 2, 1), (r 3, 1) r 1 : bcc cc r 2 : c d r 3 : bc bb b c c
58 Nondeterministic mximlly prllel Objects nd rules re chosen in nondeterministic wy. The choice is exhustive: fter choice is done no rule cn be pplied to the non ssigned objects. Choice 1: (r 1, 1) Choice 2: (r 2, 2) Choice 3: (r 2, 1), (r 3, 1) r 1 : bcc cc r 2 : c d r 3 : bc bb b c c Choice 4: (r 2, 1) is not mximlly prllel
59 Nondeterministic mximlly prllel Objects nd rules re chosen in nondeterministic wy. The choice is exhustive: fter choice is done no rule cn be pplied to the non ssigned objects. Choice 1: (r 1, 1) One of the three choices is selected in nondeterministic wy. Choice 2: (r 2, 2) Choice 3: (r 2, 1), (r 3, 1) r 1 : bcc cc r 2 : c d r 3 : bc bb b c c
60 Computtion in membrne system N(Π ) = {} 1 c h h h 2 2
61 Computtion in membrne system N(Π ) = {} 1 c h h h c h h h 2 2
62 Computtion in membrne system N(Π ) = {1} 1 c h h h c h h h hlting c h c h h 2 2
63 Computtion in membrne system N(Π ) = {1} 1 c h h h c h h h c h c h h 2 2
64 Computtion in membrne system N(Π ) = {1} 1 c h h h c h h h c h c h h c h h h 2 2
65 Computtion in membrne system N(Π ) = {1} 1 c h h h c h h h c h c h h c h h h c h h h 2 2
66 Computtion in membrne system N(Π ) = {1, 2} 1 c h h h c h h h c h c h h c h h h c h h h c h c c h h 2 2 hlting
67 Computtion in membrne system N(Π ) = {1, 2} 1 c h h h c h h h c h c h h c h h h c h h h c h c c h h 2 2
68 Computtion in membrne system N(Π ) = {1, 2,...} =N 1 c h h h c h h h c h c h h c h h h c h h h c h c c h h
69 Computtionl power of membrne systems coopertive u v u V +, v V + tr ex: bc 2 1 b here N 1 (coo) = NRE.
70 Computtionl power of membrne systems coopertive u v u V +, v V + tr ex: bc 2 1 b here N 1 (coo) = NRE. noncoopertive v V, v V + tr ex: 2 1 b here N m (ncoo) = NCF, m 1.
71 Computtionl power of membrne systems coopertive u v u V +, v V + tr ex: bc 2 1 b here N 1 (coo) = NRE. noncoopertive v V, v V + tr ex: 2 1 b here N m (ncoo) = NCF, m 1. ctlytic c cv c C, V, v V + tr ex: c c 2 1 b here N 2 (ct) = NRE.
72 Membrne systems with mrked membrnes Membrnes re crucil in living cells: they seprte (protect) the internls of the cell from the environment. Membrnes re: continers coordintors nd loctions of ctivities.
73 The presence of proteins embedded in membrnes is crucil for the ctivities of the membrnes.
74 Membrne system with mrked membrnes is model of membrne systems where membrnes cn evolve depending on the ttched objects (proteins).
75 Membrne system with mrked membrnes is model of membrne systems where membrnes cn evolve depending on the ttched objects (proteins). objects cn move through the regions of the system.
76 Membrne system with mrked membrnes is model of membrne systems where membrnes cn evolve depending on the ttched objects (proteins). pino rules pinocytosis drip rules cellulr dripping objects cn move through the regions of the system.
77 Membrne system with mrked membrnes is model of membrne systems where membrnes cn evolve depending on the ttched objects (proteins). pino rules pinocytosis drip rules cellulr dripping objects cn move through the regions of the system. protein movement rules ttchment/de-ttchment movement of proteins
78 Membrne system with mrked membrnes Ech membrne hs embedded proteins. [ ] u the multiset u (of proteins) mrks the membrne.
79 Membrne system with mrked membrnes Ech membrne hs embedded proteins. [ ] u the multiset u (of proteins) mrks the membrne. [ [ [ ] b bb ] b [ ] bb ] b b b b b b b b
80 Membrne opertions Protein-membrne rules [ Q ] uv [ [ ] ux Q ] v pino [ Q ] uv [ ] ux [ Q] v drip V u, v, x V * Non-involved proteins re rndomly distributed.
81 Pino ccc bb b [Q ] bb [ [ ] b Q ] b
82 Pino ccc bb b [Q ] bb [ [ ] b Q ] b
83 Pino ccc bb b c b b cc b [Q ] bb [ [ ] b Q ] b
84 Drip ccc bb b [Q ] bb [ ] b [Q ] b
85 Drip ccc bb b [Q ] bb [ ] b [Q ] b
86 Drip cc b ccc bb b b c b [Q ] bb [ ] b [Q ] b
87 Protein movement rules [ ] u [ ] u ttch i [ ] u [ ] u ttch o
88 Protein movement rules [ ] u [ ] u ttch i [ ] u [ ] u ttch o [ ] u [ ] u de-ttch i [ ] u [ ] u de-ttch o
89 Protein movement rules [ ] u [ ] u ttch i [ ] u [ ] u ttch o [ ] u [ ] u de-ttch i [ ] u [ ] u de-ttch o [ ] u [ ] u move out [ ] u [ ] u move in
90 ttch i c ccbb b [ b ] c [ ] cb
91 ttch i c ccbb b [ b ] c [ ] cb
92 ttch i c ccbb c ccbb b b [ b ] c [ ] cb
93 move in c ccbb c b [ ] c b [b] c
94 move in c ccbb c b [ ] c b [b] c
95 move in c ccbb c b c ccbb c b [ ] c b [b] c
96 Computtion The computtion of membrne system with mrked membrnes is obtined by pplying t ech step, to ech membrne:
97 Computtion The computtion of membrne system with mrked membrnes is obtined by pplying t ech step, to ech membrne: either the protein movement rules in the non-deterministic mximlly prllel wy (ttch, de-ttch, move)
98 Computtion The computtion of membrne system with mrked membrnes is obtined by pplying t ech step, to ech membrne: either the protein movement rules in the non-deterministic mximlly prllel wy (ttch, de-ttch, move) or one of the protein-membrne rule (pino, drip)
99 Output of n hlting (successful) computtion The numbers of proteins mrking the output membrnes in the hlting configurtion.
100 b ccc bb c d [Q ] bb [ [ ] b Q ] b [b] [ ] b [ ] d b [ ] bd [Q] bd [ [ ] b Q]
101 b ccc bb c d c b cc b c d b [Q ] bb [ [ ] b Q ] b [b] [ ] b [ ] d b [ ] bd [Q] bd [ [ ] b Q]
102 b ccc bb c d c b cc b c d b [Q ] bb [ [ ] b Q ] b [b] [ ] b [ ] d b [ ] bd [Q] bd [ [ ] b Q]
103 b ccc bb c d c b cc c b cc b c d b b c d b [Q ] bb [ [ ] b Q ] b [b] [ ] b [ ] d b [ ] bd [Q] bd [ [ ] b Q]
104 ccc bb c b b c d b c cc b c b cc c b cc b c d b b c d b [Q ] bb [ [ ] b Q ] b [b] [ ] b [ ] d b [ ] bd [Q] bd [ [ ] b Q]
105 ccc bb c b b c d b c cc b c b cc c b cc Hlting flg = b output = {2,3,5} b c d b b c d b [Q ] bb [ [ ] b Q ] b [b] [ ] b [ ] d b [ ] bd [Q] bd [ [ ] b Q]
106 Computtionl power Membrne systems using protein movement rules = NFIN (move, ttch, de-ttch)
107 Computtionl power Membrne systems using protein movement rules = NFIN (move, ttch, de-ttch) protein membrne rules NCF (pino, drip)
108 Computtionl power Membrne systems using protein movement rules = NFIN (move, ttch, de-ttch) protein membrne rules NCF (pino, drip) protein movement rules + = NRE protein membrne rules
109 Rechbility b b
110 Rechbility b b cd b
111 Rechbility b b cd b * b w b b
112 Rechbility b b cd b * b w b Cn we predict whether or not w is obtined? b
113 Rechbility Problem M membrne system, w multiset, rbitrry. Does M rech configurtion contining membrne mrked with w?
114 Rechbility Problem M membrne system, w multiset, rbitrry. Does M rech configurtion contining membrne mrked with w? pino (drip) + move/ttch/de-ttch undecidble
115 Rechbility Problem M membrne system, w multiset, rbitrry. Does M rech configurtion contining membrne mrked with w? pino (drip) + move/ttch/de-ttch undecidble move/ttch/de-ttch decidble
116 Rechbility Problem M membrne system, w multiset, rbitrry. Does M rech configurtion contining membrne mrked with w? pino (drip) + move/ttch/de-ttch undecidble move/ttch/de-ttch pino + drip decidble decidble
117 Boundness A membrne system is bounded if there exists n integer k, such tht, ny rechble configurtion of the system contins less thn k membrnes.
118 Boundness A membrne system is bounded if there exists n integer k, such tht, ny rechble configurtion of the system contins less thn k membrnes. Problem M rbitrry membrne system. Is M bounded?
119 Boundness A membrne system is bounded if there exists n integer k, such tht, ny rechble configurtion of the system contins less thn k membrnes. Problem M rbitrry membrne system. Is M bounded? pino + drip decidble
120 Rechbility of configurtion Problem M membrne system, C configurtion, rbitrry. Does M rech the configurtion C?
121 Rechbility of configurtion Problem M membrne system, C configurtion, rbitrry. Does M rech the configurtion C? pino + drip + move/ttch/de-ttch decidble
122 Perspectives
123 Perspectives Model closer to relity: pino + drip movement of objects evolution rules (inside the regions) trnsport rules (cross the membrnes) symport/ntiport rules
124 Perspectives Model closer to relity: pino + drip movement of objects evolution rules (inside the regions) trnsport rules (cross the membrnes) symport/ntiport rules (x,in) (x,out) (x,in; y,out) x x y x
125 Perspectives Model closer to relity: pino + drip movement of objects evolution rules (inside the regions) trnsport rules (cross the membrnes) symport/ntiport rules Adding time to the model: time of execution, witing time, time-free systems..
126 Perspectives Model closer to relity: pino + drip movement of objects evolution rules (inside the regions) trnsport rules (cross the membrnes) symport/ntiport rules Adding time to the model: time of execution, witing time, time-free systems.. A non-stndrd wy to look t the output: Observing the evolution of (bio) system.
127 References R. Brijder, M. Cvliere, G. Rozenberg, A. Riscos-Núñez, D. Sburln, Membrne Systems with Mrked Membrnes. Submitted. L. Crdelli, Brne Clculi. Interctions of Biologicl Membrnes. Proc. Computtionl Methods in Systems Biology 2004, LNCS 3082, L. Crdelli, Gh. Păun, An Universlity Result for (Mem)Brne Clculus Bsed on Mte/Drip Opertions. Proc. ESF Workshop on Cellulr Computing, Seville, Gh. Păun, Membrne Computing - An Introduction, Springer-Verlg, 2002.
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