Modelling Cell Cycle Visualisation

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

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


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

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


Visual State for Cell Cycle
Spatial information: m p, 3r
4


4

4 phases (G1 - S - G2 - M) =⇒ 4 visual stages:


4

small cell before growth (beginning of phase G1 )


4

big cell after growth (end of phase G1 )


4

replicated chromosomes inside the nucleus (end of
phase S)


4

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


4

phase S)
described by symbol stagei , with i = 1, ..., 4


4

phase S)
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


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.


Rule Application
L
(b).,R L
4

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

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


Rule Application
L
(b).,R L
4

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

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


Rule Application
L
(b).,R L
4

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

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


Rule Application
L
(b).,R L
4

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

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


Rule Application
L
(b).,R L
4

Cln3|Cln2|SBF |Clb5| MBF |Sic1| Net1|Cdc14))
5· s
S5 : Clb5| Sic1 −→ Sic1 − Clb5
5· s
S9 : Cln2 | Sic1 − Clb5 −→ Cln2 | pSic1 | Clb5

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


Rule Application
L
(b).,R L
4

Cln3|Cln2|SBF | MBF | Sic1 − Clb5| Net1|Cdc14))
5· s
S5 : Clb5| Sic1 −→Sic1 − Clb5
5· s

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


Rule Application
L
(b).,R L
4

Cln3|Cln2|SBF | MBF |Sic1 − Clb5| Net1|Cdc14))
5· s
S5 : Sic1 | Clb5 −→ Sic1 − Clb5
5· s

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


Rule Application
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))
1· s
1· s

∞
L 0.025 L
R1 : m p, 3r
( X | stage1 ) −→ m p,r
4


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


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


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.


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.


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


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


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


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.


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 Visualisation

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