Mecbic2009 pres2

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Mecbic2009 pres2

  1. 1. Modelling Cell Cycle at Different Levels of Representation Thomas Anung Basuki, Antonio Cerone and Rafael V. Carvalho Bologna, September 5, 2009 Modelling Cell Cycle at Different Levels of Representation – p. 1/3
  2. 2. Motivation Many formalisms and tools have been produced to help biologists for in silico experiments (P Systems, Biocham, Virtual Cell) are based on deterministic approach, while biological systems are stochastic use text/plot to express result is often inadequate =⇒ needs visualisation/animation Our approach is stochastic supports simulation and visualisation of the model supports model-checking Modelling Cell Cycle at Different Levels of Representation – p. 2/3
  3. 3. Architecture of Our Approach Levels of Representation Visual : horizontal rules 6 6 : vertical rules Molecular State of the system is represented as Spatial CLS terms Horizontal rules: Vertical rules: • control behaviour at one level • link behaviour between levels • Spatial CLS rewrite rules • Instantaneous rewrite rules ∞ with rate constant PL → PR k PL → PR Modelling Cell Cycle at Different Levels of Representation – p. 3/3
  4. 4. Using Spatial CLS for Visualisation Two-level modelling, using positional terms at visual level and non positional terms at molecular level A visual state describes three kinds of information: spatial information; a stage of the system evolution, which we call visual stage; information on whether that stage has been visualised. Two kinds of rewrite rule: horizontal rules to define behaviour in one level vertical rules to link the behaviour in the different levels Modelling Cell Cycle at Different Levels of Representation – p. 4/3
  5. 5. Visual State for Cell Cycle Spatial information: m p, 3r 4 Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  6. 6. Visual State for Cell Cycle Spatial information: m p, 3r 4 4 phases (G1 - S - G2 - M) =⇒ 4 visual stages: Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  7. 7. Visual State for Cell Cycle Spatial information: m p, 3r 4 4 phases (G1 - S - G2 - M) =⇒ 4 visual stages: small cell before growth (beginning of phase G1 ) Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  8. 8. Visual State for Cell Cycle Spatial information: m p, 3r 4 4 phases (G1 - S - G2 - M) =⇒ 4 visual stages: small cell before growth (beginning of phase G1 ) big cell after growth (end of phase G1 ) Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  9. 9. Visual State for Cell Cycle Spatial information: m p, 3r 4 4 phases (G1 - S - G2 - M) =⇒ 4 visual stages: small cell before growth (beginning of phase G1 ) big cell after growth (end of phase G1 ) replicated chromosomes inside the nucleus (end of phase S) Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  10. 10. Visual State for Cell Cycle Spatial information: m p, 3r 4 4 phases (G1 - S - G2 - M) =⇒ 4 visual stages: small cell before growth (beginning of phase G1 ) big cell after growth (end of phase G1 ) replicated chromosomes inside the nucleus (end of phase S) cell with two nuclei (phase M before cytokinesis) Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  11. 11. Visual State for Cell Cycle Spatial information: m p, 3r 4 4 phases (G1 - S - G2 - M) =⇒ 4 visual stages: small cell before growth (beginning of phase G1 ) big cell after growth (end of phase G1 ) replicated chromosomes inside the nucleus (end of phase S) cell with two nuclei (phase M before cytokinesis) described by symbol stagei , with i = 1, ..., 4 Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  12. 12. Visual State for Cell Cycle Spatial information: m p, 3r 4 4 phases (G1 - S - G2 - M) =⇒ 4 visual stages: small cell before growth (beginning of phase G1 ) big cell after growth (end of phase G1 ) replicated chromosomes inside the nucleus (end of phase S) cell with two nuclei (phase M before cytokinesis) described by symbol stagei , with i = 1, ..., 4 Visual rules introduce symbol visualised i, which activates vertical rules to change from stagei to stage(i+1)mod4 Modelling Cell Cycle at Different Levels of Representation – p. 5/3
  13. 13. Visual Level/Cellular Level Horizontal rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 L L 0.033 R2 : m p,r ((n)u ˜ ˜ (cr. x | cr.y) | stage2 ) −→ (m) L p,r L ((n)u ˜ ˜ (2cr. x | 2cr.y) | stage2 | visualised2 ) L 0.04 R3 : n (0,0,0), 2r ˜ ˜ (2cr. x | 2cr.y) | stage3 −→ 5 L L ˜ ˜ (n)(− r ,0,0), 2r (cr. x | cr.y) | n ( 2 ,0,0), 2r r ˜ ˜ (cr. x | cr.y) | stage3 | visualised3 2 5 5 L L L 0.2 R4 : m p,r ( n u X| n v Y | stage4 ) −→ L L m p, 3r ( n u X | stage4 | visualised4 ) | 4 L L m getpos, 3r ( n u Y | stage4 | visualised4 ) 4 Initial state: L L (b).,R (m)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 |molecules) 4 Modelling Cell Cycle at Different Levels of Representation – p. 6/3
  14. 14. Molecular Level Reaction rates at molecular level are classified into 4 categories: very fast, with rate constant 20 fast, with rate constant 5 slow, with rate constant 1 very slow, with rate constant 0.25 Reactions at molecular level are much faster than reactions at cellular level. We define a speeding factor s, and multiply it by the reaction rates to control reaction speed at molecular level. Modelling Cell Cycle at Different Levels of Representation – p. 7/3
  15. 15. Rule Application The state of the system: L (b).,R L ((m| GFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 iSBF |iMBF |Sic1| Net1|Cdc14)| GF ) Molecular rewrite rules: 20· s S1 : (Y | GFR) L X | GF −→ (Y | iGFR) L (Cln3 | X ) 1· s S2 : Cln3 | iSBF | iMBF −→ Cln3 | SBF | MBF Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 8/3
  16. 16. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|iSBF |iMBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 20· s S1 : (Y | GFR) L X | GF −→ (Y | iGFR) L (Cln3 | X ) 1· s S2 : Cln3 | iSBF | iMBF −→ Cln3 | SBF | MBF Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 9/3
  17. 17. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|iSBF |iMBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 20· s S1 : (Y | GFR) L X | GF −→ (Y | iGFR) L ( X | Cln3) 1· s S2 : Cln3 | iSBF | iMBF −→ Cln3 | SBF | MBF Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 10/3
  18. 18. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|SBF | MBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 20· s S1 : (Y | GFR) L X | GF −→ (Y | iGFR) L ( X | Cln3) 1· s S2 : Cln3 | iSBF | iMBF −→Cln3 | SBF | MBF Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 11/3
  19. 19. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|SBF | MBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 0.25· s S3 : (n) L (y.gN2.x | Y ) | SBF −→ (n) L (y.gN2. x | Y ) | Cln2 | SBF ˜ ˜ ˜ ˜ 0.25· s S4 : (n) L (y.gB5. x | Y ) | MBF −→ (n) L (y.gB5. x | Y ) | Clb5 | MBF ˜ ˜ ˜ ˜ Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 12/3
  20. 20. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|Cln2|SBF | MBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 0.25· s S3 : (n) L (y.gN2. x | Y ) | SBF −→ (n) L (y.gN2.x | Y ) | Cln2 | SBF ˜ ˜ ˜ ˜ 0.25· s S4 : (n) L (y.gB5. x | Y ) | MBF −→ (n) L (y.gB5. x | Y ) | Clb5 | MBF ˜ ˜ ˜ ˜ Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 13/3
  21. 21. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|Cln2|SBF | MBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 0.25· s S3 : (n) L (y.gN2. x | Y ) | SBF −→ (n) L (y.gN2. x | Y ) | Cln2 | SBF ˜ ˜ ˜ ˜ 0.25· s S4 : (n) L (y.gB5.x | Y ) | MBF −→ (n) L (y.gB5. x | Y ) | Clb5 | MBF ˜ ˜ ˜ ˜ Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 14/3
  22. 22. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|Cln2|SBF |Clb5| MBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 0.25· s S3 : (n) L (y.gN2. x | Y ) | SBF −→ (n) L (y.gN2. x | Y ) | Cln2 | SBF ˜ ˜ ˜ ˜ 0.25· s S4 : (n) L (y.gB5. x | Y ) | MBF −→ (n) L (y.gB5.x | Y ) | Clb5 | MBF ˜ ˜ ˜ ˜ Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 15/3
  23. 23. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|Cln2|SBF |Clb5| MBF |Sic1| Net1|Cdc14)) Molecular rewrite rules: 5· s S5 : Clb5| Sic1 −→ Sic1 − Clb5 5· s S9 : Cln2 | Sic1 − Clb5 −→ Cln2 | pSic1 | Clb5 Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 16/3
  24. 24. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|Cln2|SBF | MBF | Sic1 − Clb5| Net1|Cdc14)) Molecular rewrite rules: 5· s S5 : Clb5| Sic1 −→Sic1 − Clb5 5· s S9 : Cln2 | Sic1 − Clb5 −→ Cln2 | pSic1 | Clb5 Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 17/3
  25. 25. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|Cln2|SBF | MBF |Sic1 − Clb5| Net1|Cdc14)) Molecular rewrite rules: 5· s S5 : Sic1 | Clb5 −→ Sic1 − Clb5 5· s S9 : Cln2 | Sic1 − Clb5 −→ Cln2 | pSic1 | Clb5 Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 18/3
  26. 26. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)|stage1 | 4 Cln3|Cln2|SBF | MBF | pSic1|Clb5| Net1|Cdc14)) Molecular rewrite rules: 5· s S5 : Sic1 | Clb5 −→ Sic1 − Clb5 5· s S9 : Cln2 | Sic1 − Clb5 −→Cln2 | pSic1 | Clb5 Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 19/3
  27. 27. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)| stage1 | 4 Cln3|Cln2|SBF | MBF | pSic1|Clb5| Net1|Cdc14)) Molecular rewrite rules: 1· s S10 : pSic1 | SCF −→ SCF 1· s S11 : Cln2 | SCF −→ SCF Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 20/3
  28. 28. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0), 3r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)| stage1 | 4 visualised1 |Cln3|Cln2|SBF | MBF | pSic1|Clb5| Net1|Cdc14)) Molecular rewrite rules: 1· s S10 : pSic1 | SCF −→ SCF 1· s S11 : Cln2 | SCF −→ SCF Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ Cln2mc(Cln2,2) |stage2 Visual rewrite rules: L 0.025 L R1 : m p, 3r ( X | stage1 ) −→ m p,r ( X | stage1 | visualised1 ) 4 Modelling Cell Cycle at Different Levels of Representation – p. 21/3
  29. 29. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0),r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)| stage1 | visualised1 |Cln3|Cln2|SBF | MBF | pSic1|Clb5| Net1|Cdc14)) Molecular rewrite rules: 1· s S10 : pSic1 | SCF −→ SCF 1· s S11 : Cln2 | SCF −→ SCF Vertical rewrite rules: ∞ T1 : stage1 |visualised1 |Cln2mc(Cln2,2) −→ stage2 |Cln2mc(Cln2,2) Visual rewrite rules: L L 0.033 R2 : m p,r ((n)u ˜ ˜ (cr. x | cr.y) | stage2 ) −→ (m) L p,r L ((n)u ˜ ˜ (2cr. x | 2cr.y ) | stage2 | visualised2 ) Modelling Cell Cycle at Different Levels of Representation – p. 22/3
  30. 30. Rule Application The state of the system: L (b).,R L ((m|iGFR)(0,0,0),r ((n) L (cr.gN2.gB5 | cr.gB2.gC20)| stage2 | Cln3|Cln2|SBF | MBF | pSic1|Clb5| Net1|Cdc14)) Molecular rewrite rules: 1· s S10 : pSic1 | SCF −→ SCF 1· s S11 : Cln2 | SCF −→ SCF Vertical rewrite rules: ∞ T1 : visualised1 |Cln2mc(Cln2,2) |stage1 −→ stage2 |Cln2mc(Cln2,2) Visual rewrite rules: L L 0.033 R2 : m p,r ((n)u ˜ ˜ (cr. x | cr.y) | stage2 ) −→ (m) L p,r L ((n)u ˜ ˜ (2cr. x | 2cr.y ) | stage2 | visualised2 ) Modelling Cell Cycle at Different Levels of Representation – p. 23/3
  31. 31. Propensity Propensity aµ measures the probability of reaction R µ to be chosen as next reaction aµ = k µ × hµ where hµ = number of possible combinations of reactants Based on assumption that molecules are homogeneously distributed in the system Ex: X1 molecules of A and X2 molecules of B R1 : A + B → 2A a1 = k1 (X1 )(X2 ) = k1 X1 X2 1 1 Modelling Cell Cycle at Different Levels of Representation – p. 24/3
  32. 32. Compartments and Propensity compartments make molecules not homogeneously distributed in the system molecules are contained in compartments and are homogeneously distributed within each compartment each reaction can only involves reactants from one compartment aσ measures the probability of reaction R µ to be chosen µ as next reaction and occurs at compartment σ aσ = k µ × hσ µ µ where hσ = number of possible combinations of µ reactants at compartment σ Modelling Cell Cycle at Different Levels of Representation – p. 25/3
  33. 33. Modified Gillespie’s Direct Method Given reactions { R1 , . . . , R M } and molecular population X1 , . . . , X N and C compartments, where Xi = ∑C=1 Xiv v Step 0Initialise time variable t to 0. Calculate a1 , . . . a M . Calculate ∑v=1 ∑C =1 aw . M w v Step 1 Execute any applicable vertical rules. Step 2 If the space is fully occupied then stop simulation. Otherwise generate r1 and calculate τ. Increment t by τ. Step 3 Generate r2 and calculate (µ, σ). Step 4 Execute Rµ . Update X1 , . . . , X N and a1 , . . . , a N . M Step 5 Calculate ∑v=1 av . Return to Step 2. Modelling Cell Cycle at Different Levels of Representation – p. 26/3
  34. 34. Computing τ, µ and σ If aw is the propensity of reaction Ri in compartment w and v a0 = ∑v=1 ∑C =1 aw then M w v 1 1 τ = ln( ) (1) a0 r1 µ σ −1 µ σ (µ, σ) = the integers for which ∑∑ a w r2 a0 ≤ v ∑∑ aw (2) v v =1 w =1 v =1 w =1 where r1 , r2 ∈ [0, 1] are two real values generated by a random number generator. Modelling Cell Cycle at Different Levels of Representation – p. 27/3
  35. 35. Conclusion defined an approach to model biological systems at different levels of representation molecular level and one or more visual levels case study budding yeast cell cycle defined a modified Gillespie’s algorithm to deal with compartmentalisation and spatial information implemented a tool for visualisation Modelling Cell Cycle at Different Levels of Representation – p. 28/3
  36. 36. Spatial CLS Terms We assume an alphabet E . Terms T, Branes B and Sequences S are given by the following grammar: L T ::= λ (S)d Bd T T|T B ::= λ (S)d B|B S ::= a S·S where a is an element of E , is the empty sequence, and d ∈ D = ((Rn ) ∪ {.}) × R+ . Two kinds of term: positional terms, has position and size, and non positional terms, only has size Modelling Cell Cycle at Different Levels of Representation – p. 29/3
  37. 37. Brane and Sequence Patterns Left Brane Patterns BPL , Sequence Patterns SP and Right Brane Patterns BPR are given by the following grammar: BPL ::= (SP)u BPL | BPL BPR ::= (SP) g BPR | BPR SP ::= a SP.SP ˜ x x ˜ where u ∈ PV, x ∈ X , x ∈ SV and g ∈ T Modelling Cell Cycle at Different Levels of Representation – p. 30/3
  38. 38. Left and Right Patterns Left Patterns PL and Right Patterns PR are given by the following grammar: L PL ::= (SP)u BPLX u PLX PL | PL BPLX ::= BPL ¯ BPL | X ¯ X PLX ::= PL PL | X L PR ::= (SP) g BPRX PR PR | PR X ¯ X g BPRX ::= BPR ¯ BPR | X ¯ X ˜ where u ∈ PV, x ∈ X , x ∈ SV and g ∈ T. Modelling Cell Cycle at Different Levels of Representation – p. 31/3
  39. 39. Rewrite Rules A rewrite rule is a 4-tuple ( f c , PL , PR , k), usually written as k [ f c ] PL → PR where f c : T → {tt, f f }, k ∈ R + , Var(PR ) ⊆ Var(PL ), and each function g appearing in PR refers only to position variables in Var(PL ). Modelling Cell Cycle at Different Levels of Representation – p. 32/3
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