INVARIANT Priese/Wimmel 2008 ANALYSIS Lautenbachs miracle - - - PowerPoint PPT Presentation

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INVARIANT ANALYSIS

• EXAMPLES

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INDEX

❑

Starke 90, p.121 exponential number of invariants

❑

Starke 90, p. 111 CPI & CTI, but not live

❑

Priese/Wimmel 2008

❑

Lautenbach‘s miracle

❑

pathway analysis T-invariants - elementary modes - extreme pathways

❑

carbon oxidation P/T-invariants and their interpretation

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[STARKE 121]

❑

k = 4, 24 = 16 minimal P-invariants: (p1a, p2a, p3a, p4a), (p1b, p2a, p3a, p4a), (p1a, p2a, p3a, p4b), (p1b, p2a, p3a, p4b), (p1a, p2a, p3b, p4a), (p1b, p2a, p3b, p4a), (p1a, p2a, p3b, p4b), (p1b, p2a, p3b, p4b), (p1a, p2b, p3a, p4a), (p1b, p2b, p3a, p4a), (p1a, p2b, p3a, p4b), (p1b, p2b, p3a, p4b), (p1a, p2b, p3b, p4a), (p1b, p2b, p3b, p4a), (p1a, p2b, p3b, p4b), (p1b, p2b, p3b, p4b)

• >

generally 2k P-invariants

❑

analogously for T-invariants

pka pkb tk p3a p3b t3 p2a p2b t2 t1 p1b p1a

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[STARKE, P. 111]

(2) (2) (2) t1 t3 t3 t2 t1 t4 t3 t2 t1 p1 p1 p2 p3 p3 p1 p2 p2 p2 p1 p3 p2 p1

INA: ORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 pF0 MG SM FC EFC ES N N N N Y N Y Y N N N N N N N N N DTP CPI CTI B SB REV DSt BSt DTr DCF L LV L&S ? Y Y Y Y N N ? N N N N N

t-invariant= {t1, t2, t3, t3}

?

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[STARKE, P. 111],

INVARIANTS ❑

incidence matrix

• >

side conditions, here p2 for t4, are not reflected in C

❑

CPI

• >

the only P-invariant (p1, p2, p3) covers the net

❑

CTI

• >

T-inv1: (1, 1, 2, 0) -> (t1, t2, 2 t3) -> {t1, t2, t3, t3}

• >

T-inv2: (t4)

❑

but not live

• >

t4 - the only live transition

❑

state equation, counter example

• >

m0 = (1, 0, 0), m1 = (0, 0, 1) m1 = m0 + Cx Cx = m1 - m0 Cx = (-1, 0, 1)

• >

x = (t1, t2, t3)

• >

BUT, no permutation of t1, t2, t3 can be fired (t2 needs two tokens)

• 1
• 1

1 1

• 1

2

• 1

C =

dependability engineering & Petri nets November 2013 monika.heiner@b-tu.de 7 - 6 / 28

[PRIESE 2003, P. 80]

❑

the system

❑

its basic steps

2 2 t4 t3 t2 t1 p5 p4 p3 p2 p1 ORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 pF0 MG SM FC EFC ES N Y N Y N Y Y N N Y N N N N Y Y Y DTP CPI CTI B SB REV DSt BSt DTr DCF L LV L&S ? N Y N N ? ? ? N Y ? ? N t4 p3 p3 p5 p4 p1 p4 p2 t2 p1 p2 p4 p5 t1 p5 t3 p3

dependability engineering & Petri nets November 2013 monika.heiner@b-tu.de 7 - 7 / 28

[PRIESE 2003, P. 80],

T-INVARIANTS ❑

The net is covered by one minimal T-invariants. t-inv = (1, 1, 2, 2) = {t1, t2, t3, t3, t4, t4}

❑

two possible runs

t3 t4 p3 t3 p3 p5 p1 p5 p1 p4 p2 t2 p1 p2 p4 p5 t1 t4 t4 t1 p5 p4 p2 p1 t2 p2 p4 p1 p5 p1 p5 t3 p3 t4 p3 t3

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[PRIESE 2003, P. 80],

T-INVARIANTS ❑

possible runs, short notation

❑

required terminus ?!

• >

“maximally unordered” t1 t4 t3 t3 t2 t4 t1 t4 t3 t3 t2 t4

dependability engineering & Petri nets November 2013 monika.heiner@b-tu.de 7 - 9 / 28

[PRIESE 2003, P. 80],

P-INVARIANTS ❑

The net is NOT covered by semipositive P-invariants. Covered places: p1, p2, p3, p5,

• >

semipositive place invariants = 1 | p2 : 2, | p3 : 1, | p5 : 1 2 | p1 : 2, | p3 : 1, | p5 : 1

❑

Karp-Milller graph

• >

18 nodes

• >

capacities needed: p1 p2 p3 p4 p5 2 1 3 oo 3

(2) (2) t4 t3 t2 t1 p5 p4 p3 p2 p1 2 (2) (2) t4 t3 t2 t1 p5 p4 p3 p2 p1 2

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LAUTENBACH’S MIRACLE ❑

the system

❑

its basic steps

2 2 2 3 2 tb ta t6 t5 t4 t3 t2 t1 p4 p3 p2 p1

ORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 pF0 MG SM FC EFC ES N N N Y N N Y N Y Y N N N N Y Y Y DTP CPI CTI B SB REV DSt BSt DTr DCF L LV L&S ? N Y N N ? N ? N ? Y Y N

tb p4 p4 p1 p1 ta t5 t2 t6 t3 t4 t1 p4 p3 p2 p2 p1 p1 p2 p2 p2 p4 p3 p1 p3 p2 p2 p3 p1 p2

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LAUTENBACH’S MIRACLE ❑

two minimal T-invariants

• >

not realizable under the empty marking

• >

not reproducing the empty marking

❑

unique runs

ta t1 t1 t3 t3 t5 t5 tb p1 p1 p2 p1 p2 p2 p2 p3 p2 p3 p4 p2 p4 p3 p2 ta t2 t2 t4 t4 t6 t6 tb p1 p1 p2 p2 p1 p1 p3 p1 p3 p4 p4

t-inv1 = {ta, t2, t2, t4, t4, t6, t6, tb} t-inv2 = {ta, t1, t1, t3, t3, t5, t5, tb} trapped token trapped token ta t1 t1 tb t3 t5 t3 t5 ta t4 t2 tb t6 t6 t2 t4

dependability engineering & Petri nets November 2013 monika.heiner@b-tu.de 7 - 12 / 28

LAUTENBACH’S MIRACLE ❑

a non-minimal T-invariant

• >

covering the net

• >

reproducing the empty marking

❑

T-inv3 = {ta, t1, t2, t3, t4, t5, t6, tb}

• >

T-inv3 = (T-inv1 + T-inv2) / 2

• >

non-negative linear combination of minimal ones

❑

two possible runs

tb t6 t3 t4 t1 ta t5 t2 p4 p3 p3 p1 p2 p2 p1 p1 p2 p3 p4 p2 p1 p1 ta p2 p4 p2 p2 p2 p1 p3 p3 p4 t2 t5 t1 t4 t3 t6 tb

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LAUTENBACH’S MIRACLE ❑

short notation of the two possible runs

❑

comparison ?

❑

guess

• >

no chance for uniqueness of non-minimal T-invariants’runs ta tb t3 t6 t2 t4 t5 t1 ta tb t3 t6 t1 t4 t5 t2

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PATHWAY ANALYSIS ❑

substances involved

• >

input substance A

• >
• utput substance C
• >

auxiliary substance B

❑

steady state substance flows

• >

T-invariants

❑

all flow behaviour under the steady state assumption

• >

non-negative linear combination

• f minimal T-invariants

gB rB gA rC ac ab bc A C B

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T-INVARIANTS AND EXTREME PATHWAYS

gB rB rB gA ab gB rC bc ab gA rB gB rB ab ab rb gA ac rC gA bc rC gB B A B C B B A B B C A C A B C

inv4 = inv2 + inv5 - inv3 inv5 inv4 inv3 inv2

inv1

+

• dependability engineering & Petri nets

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CARBON OXIDATION, BASIC REACTIONS r1) 2 C + O2 --> 2 CO r2) C + O2 --> CO2 r3) C + CO2 <--> 2 CO

2 r3b CO2 CO C 2 r3a CO2 CO C r2 CO2 O2 C 2 2 r1 CO O2 C carbon monoxide carbon dioxide

O2 oxygen

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CARBON OXIDATION, COMPOSITION

2 2 2 2 C O2 CO CO2 r1 r3a r2 r3b

1) 2 C + O2

• > 2 CO

2) C + O2

• >

CO2 3) C + CO2 <-> 2 CO 3 C + 2 O2

• > 2 CO +

CO2

BASIC MODEL MODEL OF THE SYSTEM’S TOTAL EQUATION SYSTEM’S TOTAL EQUATION 2 CO2 CO 3 2 sum O2 C

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CARBON/BND, INCIDENCE MATRIX

CO2 O2 C CO init r1 r2 r3a r3b P T

• 1
• 1
• 2
• 1

+2 +1

• 1
• 1

+1 2

• 2

+1 2 2 2 2 3 2 2 r3b start repeat r2 r3a r1 init CO2 CO O2 C start repeat

• 1

+2 +3

• 2
• 1

+1

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CARBON/BND, P-INVARIANTS

2 2 2 2 3 2 2 r3b start repeat r2 r3a r1 init CO2 CO O2 C 2 2 2 2 3 2 2 r3b start repeat r2 r3a r1 init CO2 CO O2 C

4x 2x 2x P-inv1 = (3 init, C, CO, CO2) -> carbon preservation P-inv2 = (4 init, 2 O2, CO, 2 CO2) -> oxygen preservation 3X

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CARBON/BND, T-INVARIANTS 1, 2

r3a repeat r3b r3a r2 r1 r2 r3b r2 r1 start s6 s5 s4 s3 s2 s1 r3a repeat r3b r3a r2 r1 r2 r3b r2 r1 start s6 s5 s4 s3 s2 s1

T-inv1 = (r3a, r3b) -> inner cycle T-inv2 = (start, 2 r1, r3b, repeat) -> input/output cycle

2 2 2 2 3 2 2 r3b start repeat r2 r3a r1 init CO2 CO O2 C 2 2 2 2 3 2 2 r3b start repeat r2 r3a r1 init CO2 CO O2 C 2x

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CARBON/BND, T-INVARIANTS 3

r3a repeat r3b r3a r2 r1 r2 r3b r2 r1 start s6 s5 s4 s3 s2 s1

T-inv3 = (start, 2 r2, r3a, repeat)

r3a repeat r3b r3a r2 r1 r2 r3b r2 r1 start s6 s5 s4 s3 s2 s1

start r2 r2 r3a repeat

2 2 2 2 3 2 2 r3b start repeat r2 r3a r1 init CO2 CO O2 C

2X

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CARBON/BND, T-INVARIANTS 4

r3a repeat r3b r3a r2 r1 r2 r3b r2 r1 start s6 s5 s4 s3 s2 s1

T-inv4 = (start, r1, r2, repeat)

r3a repeat r3b r3a r2 r1 r2 r3b r2 r1 start s6 s5 s4 s3 s2 s1

start r1 r2 repeat

2 2 2 2 3 2 2 r3b start repeat r2 r3a r1 init CO2 CO O2 C

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CARBON/UNBOUNDED, T-INVARIANTS 1 - 3

2 2 2 2 r1 r3a r2 r3b

• ut_CO2
• ut_CO

in_C in_O2 C O2 CO CO2

T-inv1 = (r3a, r3b)

2 2 2 2 r1 r3a r2 r3b

• ut_CO2
• ut_CO

in_C in_O2 C O2 CO CO2 2x 2x

T-inv2 = (in_O2 , 2 in_C, r1, 2 out_CO)

2 2 2 2 r1 r3a r2 r3b

• ut_CO2
• ut_CO

in_C in_O2 C O2 CO CO2

T-inv3 = (in_O2, in_C, r2, out_CO2)

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CARBON/UNBOUNDED, T-INVARIANTS 4, 5

2 2 2 2 r1 r3a r2 r3b

• ut_CO2
• ut_CO

in_C in_O2 C O2 CO CO2 2 2 2 2 r1 r3a r2 r3b

• ut_CO2
• ut_CO

in_C in_O2 C O2 CO CO2 2x 2x

T-inv4 = (in_O2, 2 in_C, r2, r3a, 2 out_CO) T-inv5 = (in_O2, in_C, r1, r3b, out_CO2)

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CARBON/UNBOUNDED, T-INVARIANTS, INTERPRETATION ❑ steady state = constant token distribution ❑ preservation of a given system state under continuous firing requires

• >

relative transition firing rates = T-invariant’s entries

• >

ex T-inv2: a given state is preserved, if in_C and out_CO fire twice as often as in_O2 and r1; ❑ the in- / out-components of the T-invariant

• >

sum equation of the T-invariants remaining transitions T-inv1: --

• > inner cycle

T-inv2: O2 + 2 C -> 2 CO

• > stoichiometric equation of r1

T-inv3: C + O2 -> CO2

• > stoichiometric equation of r2

T-inv4: 2 C + O2 -> 2 CO

• > sum of the stoichiometric equations of r2, r3a

T-inv5: C + O2 -> CO2

• > sum of the stoichiometric equations of r1, r3b

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T-INVARIANTS, SUMMARY TWO INTERPRETATIONS ❑ state-reproducing transition sequence (partial order)

• f transitions occuring one after the other

❑ relative transition firing rates

• f transitions occuring permanently & concurrently

BASIC TYPES IN BIO NETWORKS ❑ trivial minimal T-invariants

• >

boundary transitions of auxiliary compounds

• >

reversible reactions ❑ non-trivial minimal T-invariants

• >

i/o-T-invariants covering boundary transitions of input / output compounds

• >

inner cycles

dependability engineering & Petri nets November 2013 monika.heiner@b-tu.de 7 - 27 / 28

PROBLEM ❑

given

• >

(minimal) T-invariant

❑

wanted

• >

partial order run, according to net structure

❑

purpose

• >

visualization

• >

decision on realizability

• >

looking for a minimal marking, making the invariant realizable

❑

questions

• >

unique solution ?

• >

preconditions,

• e. g. minimal T-invariants ?
• >

required termini ?

❑

preliminary remark

• >
• bviously, there is no unique solution for

an interleaving sequence of concurrent transitions

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REFERENCES

[Desel 1998] Desel, J.: Petrinetze, lineare Algebra und lineare Programmierung;

• B. G. Teubner 1998.
• > additional material, not discussed here

[Heiner 2008] M Heiner, D Gilbert and R Donaldson: Petri Nets for Systems and Synthetic Biology; SFM 2008, Bertinoro, Springer, LNCS 5016, pages 215–264. [Heiner 2009] M Heiner: Understanding Network Behaviour by Structured Representations of Transition Invariants – A Petri Net Perspective on Systems and Synthetic Biology; Algorithmic Bioprocesses, Springer, pages 367–389, 2009. [Starke 1990] Starke, P. H.: Analyse von Petri-Netz-Modellen;

• B. G. Teubner 1990.