Mecbic2009 pres2

Modelling Cell Cycle at DifferentLevels of Representation

Thomas Anung Basuki, Antonio Cerone and Rafael V. Carvalho

Bologna, September 5, 2009

Modelling Cell Cycle at Different Levels of Representation – p. 1/32

Motivation

Many formalisms and tools have been produced to helpbiologists for in silico experiments (P Systems, Biocham,Virtual Cell)

are based on deterministic approach, while biologicalsystems 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


Architecture of Our Approach

Levels of Representation

Visual�

Molecular�

6

� : horizontal rules

6 : vertical rules

State of the system is represented as Spatial CLS terms

Horizontal rules:

• control behaviour at one level

• Spatial CLS rewrite rules

with rate constant

PLk7→ PR

Vertical rules:

• link behaviour between levels

• Instantaneous rewrite rules

PL∞7→ PR


Using Spatial CLS for Visualisation

Two-level modelling, using positional terms at visuallevel 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 visualstage;information on whether that stage has beenvisualised.

Two kinds of rewrite rule:horizontal rules to define behaviour in one levelvertical rules to link the behaviour in the differentlevels


Visual State for Cell Cycle

Spatial information: mp, 3r4

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 ofphase S)cell with two nuclei (phase M before cytokinesis)

described by symbol stagei, with i = 1, ..., 4

Visual rules introduce symbol visualisedi, whichactivates vertical rules to change from stagei tostage(i+1)mod4












small cell before growth (beginning of phase G1)

big cell after growth (end of phase G1)replicated chromosomes inside the nucleus (end ofphase S)cell with two nuclei (phase M before cytokinesis)







small cell before growth (beginning of phase G1)big cell after growth (end of phase G1)

replicated chromosomes inside the nucleus (end ofphase S)cell with two nuclei (phase M before cytokinesis)







small cell before growth (beginning of phase G1)big cell after growth (end of phase G1)replicated chromosomes inside the nucleus (end ofphase S)

cell with two nuclei (phase M before cytokinesis)

























Visual Level/Cellular Level

Horizontal rules:

R1 :(

m)L

p, 3r4c (X | stage1)

0.0257−→

(

m)L

p,r c (X | stage1 | visualised1)

R2 :(

m)L

p,r c ((n)Lu c (cr.x | cr.y) | stage2)

0.0337−→

(m)Lp,r c ((n)L

u c (2cr.x | 2cr.y) | stage2 | visualised2)

R3 :(

n)L(0,0,0), 2r

5c (2cr.x | 2cr.y) | stage3

0.047−→

(n)L(− r

2 ,0,0), 2r5c(cr.x | cr.y) |

(

n)L( r

2 ,0,0), 2r5c (cr.x | cr.y) | stage3 | visualised3

R4 :(

m)L

p,r c ((

n)L

u c X |(

n)L

v c Y | stage4)0.27−→

(

m)L

p, 3r4c (

(

n)L

u c X | stage4 | visualised4) |(

m)L

getpos, 3r4c (

(

n)L

u c Y | stage4 | visualised4)

Initial state:

(b)L.,R c (m)L

(0,0,0), 3r4c ((n)L c (cr.gN2.gB5 | cr.gB2.gC20)|stage1 |molecules)


Molecular Level

Reaction rates at molecular level are classified into 4categories:

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 reactionsat cellular level.We define a speeding factor s, and multiply it by thereaction rates to control reaction speed at molecular level.


Rule Application

The state of the system:

(b)L.,R c ((m|GFR)L

(0,0,0), 3r4c ((n)L c (cr.gN2.gB5 | cr.gB2.gC20)|stage1 |

iSBF|iMBF|Sic1|Net1|Cdc14)|GF)

Molecular rewrite rules:

S1 : (Y | GFR)LcX | GF 20·s7−→ (Y | iGFR)Lc(Cln3 | X)

S2 : Cln3 | iSBF | iMBF 1·s7−→ Cln3 | SBF | MBF

Vertical rewrite rules:

T1 : visualised1|Cln2mc(Cln2,2)|stage1∞

7−→ Cln2mc(Cln2,2)|stage2

Visual rewrite rules:

R1 :(

m)L


0.0257−→

(

m)L



Rule Application


(b)L.,R c ((m|iGFR)L


Cln3|iSBF|iMBF|Sic1|Net1|Cdc14))


S1 : (Y | GFR)LcX | GF 20·s7−→ (Y | iGFR)Lc(Cln3 | X)






R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|iSBF|iMBF|Sic1|Net1|Cdc14))


S1 : (Y | GFR)LcX | GF 20·s7−→ (Y | iGFR)Lc(X | Cln3)






R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|SBF|MBF|Sic1|Net1|Cdc14))


S1 : (Y | GFR)LcX | GF 20·s7−→ (Y | iGFR)Lc(X | Cln3)

S2 : Cln3 | iSBF | iMBF 1·s7−→Cln3 | SBF | MBF





R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|SBF|MBF|Sic1|Net1|Cdc14))


S3 : (n)Lc(y.gN2.x |Y) | SBF0.25·s7−→ (n)Lc(y.gN2.x |Y) | Cln2 | SBF

S4 : (n)Lc(y.gB5.x |Y) | MBF 0.25·s7−→ (n)Lc(y.gB5.x |Y) | Clb5 | MBF





R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|Cln2|SBF|MBF|Sic1|Net1|Cdc14))


S3 : (n)Lc(y.gN2.x |Y) | SBF 0.25·s7−→ (n)Lc(y.gN2.x |Y) | Cln2 | SBF






R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|Cln2|SBF|MBF|Sic1|Net1|Cdc14))



S4 : (n)Lc(y.gB5.x |Y) | MBF0.25·s7−→ (n)Lc(y.gB5.x |Y) | Clb5 | MBF





R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|Cln2|SBF|Clb5|MBF|Sic1|Net1|Cdc14))








R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|Cln2|SBF|Clb5|MBF|Sic1|Net1|Cdc14))


S5 : Clb5| Sic1 5·s7−→ Sic1 − Clb5

S9 : Cln2 | Sic1 − Clb5 5·s7−→ Cln2 | pSic1 | Clb5





R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|Cln2|SBF|MBF|Sic1−Clb5|Net1|Cdc14))


S5 : Clb5| Sic1 5·s7−→Sic1 − Clb5






R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|Cln2|SBF|MBF|Sic1−Clb5|Net1|Cdc14))


S5 : Sic1 | Clb5 5·s7−→ Sic1 − Clb5






R1 :(

m)L


0.0257−→

(

m)L



Rule Application




Cln3|Cln2|SBF|MBF|pSic1|Clb5|Net1|Cdc14))


S5 : Sic1 | Clb5 5·s7−→ Sic1 − Clb5

S9 : Cln2 | Sic1 − Clb5 5·s7−→Cln2 | pSic1 | Clb5





R1 :(

m)L


0.0257−→

(

m)L



Rule Application






S10 : pSic1 | SCF 1·s7−→ SCF

S11 : Cln2 | SCF 1·s7−→ SCF





R1 :(

m)L


0.0257−→

(

m)L

p,rc (X | stage1 | visualised1)


Rule Application




visualised1|Cln3|Cln2|SBF|MBF|pSic1|Clb5|Net1|Cdc14))








R1 :(

m)L


0.0257−→

(

m)L

p,rc (X | stage1 | visualised1)


Rule Application



(0,0,0),r c ((n)L c (cr.gN2.gB5 | cr.gB2.gC20)|stage1 |

visualised1|Cln3|Cln2|SBF|MBF|pSic1|Clb5|Net1|Cdc14))





T1 : stage1|visualised1|Cln2mc(Cln2,2) ∞7−→ stage2|Cln2mc(Cln2,2)


R2 :(

m)L


0.0337−→

(m)Lp,r c ((n)L



Rule Application



(0,0,0),r c ((n)L c (cr.gN2.gB5 | cr.gB2.gC20)|stage2 |







7−→stage2|Cln2mc(Cln2,2)


R2 :(

m)L


0.0337−→

(m)Lp,r c ((n)L



Propensity

Propensity aµ

measures the probability of reaction Rµ to be chosen asnext reaction

aµ = kµ × hµ

where hµ = number of possible combinations ofreactants

Based on assumption that molecules arehomogeneously distributed in the system

Ex: X1 molecules of A and X2 molecules of B

R1 : A + B → 2Aa1 = k1(

X11 )(X2

1 ) = k1X1X2


Compartments and Propensity

compartments make molecules not homogeneouslydistributed in the system

molecules are contained in compartments and arehomogeneously distributed within each compartment

each reaction can only involves reactants from onecompartment

aσµ measures the probability of reaction Rµ to be chosen

as next reaction and occurs at compartment σ

aσµ = kµ × hσ

µ

where hσµ = number of possible combinations of

reactants at compartment σ


Modified Gillespie’s Direct Method

Given reactions {R1, . . . , RM} and molecular populationX1, . . . , XN and C compartments, where Xi = ∑

Cv=1 Xv

i

Step 0 Initialise time variable t to 0. Calculate a1, . . . aM.Calculate ∑

Mv=1 ∑

Cw=1 aw

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 byτ.

Step 3 Generate r2 and calculate (µ, σ).

Step 4 Execute Rµ. Update X1, . . . , XN and a1, . . . , aN.

Step 5 Calculate ∑Mv=1 av. Return to Step 2.


Computing τ, µ and σ

If awv is the propensity of reaction Ri in compartment w and

a0 = ∑Mv=1 ∑

Cw=1 aw

v then

τ =1a0

ln(1r1

) (1)

(µ, σ) = the integers for whichµ

∑v=1

σ−1

∑w=1

awv < r2a0 ≤

µ

∑v=1

σ

∑w=1

awv (2)

where r1, r2 ∈ [0, 1] are two real values generated by a random

number generator.


Conclusion

defined an approach to model biological systems atdifferent levels of representation

molecular level and one or more visual levelscase study budding yeast cell cycle

defined a modified Gillespie’s algorithm to deal withcompartmentalisation and spatial information

implemented a tool for visualisation


Spatial CLS Terms

We assume an alphabet E . Terms T, Branes B and

Sequences S are given by the following grammar:

T ::= λ∣

∣ (S)d∣

∣

(

Bd)L

c T∣

∣ T | T

B ::= λ∣

∣ (S)d∣

∣ B | B

S ::= ε∣

∣ a∣

∣ S · S

where a is an element of E , ε 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


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 ::= ε∣

∣ a∣

∣ SP.SP∣

∣ x∣

∣ x

where u ∈ PV, x ∈ X , x ∈ SV and g ∈ T


Left and Right Patterns

Left Patterns PL and Right Patterns PR

are given by the following grammar:

PL ::= (SP)u∣

∣

(

BPLX)L

u c PLX∣

∣ PL | PL

BPLX ::= BPL∣

∣ BPL | X∣

∣ X

PLX ::= PL∣

∣ PL | X

PR ::= ε∣

∣ (SP)g∣

∣

(

BPRX)L

g c PR∣

∣ PR | PR∣

∣ X∣

∣ X

BPRX ::= BPR∣

∣ BPR | X∣

∣ X

where u ∈ PV, x ∈ X , x ∈ SV and g ∈ T.


Rewrite Rules

A rewrite rule is a 4-tuple ( fc, PL, PR, k), usually written as

[ fc]PLk7→ PR

where fc : 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).


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Mecbic2009 pres2