A Shape Calculus for Biological Processes · Shape de nition Shape (a) is composed of four basic...

A Shape Calculus for Biological Processes

E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei

UNICAM Complex Systems (CoSy) Research Grouphttp://cosy.cs.unicam.it

29 Sep 2009, ICTCS 09 Cremona

E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes

http://cosy.cs.unicam.it

Outline

Main topics

Background

3D Shapes

Collision Detection and Response

3D Shapes behavior

3D Processes

Strong and weak splittings

Networks of 3D Processes

Conclusion and Future works


Background

Computational Systems Biology

Designing ”in-silico” drugs.

Studying molecular crowding.

Exploring biomolecularinteractions

Nanotechnology

Molecular self-assembly

Molecular recognition

Molecular motors

Virus H1N1


Three ingredients: Space, Time and Shape

Space/Time

Macromolecular crowding alters theproperties of molecules in a solution.

In not well-stirred systems, the idealway to simulate the time evolution ofthe system is to track the exactpositions and velocities of all molecules. http://www.anl.gov

Shape and communication

Contacts (collisions) and shapestransformation determinesbiomolecular interactions.


Related Works

Topological Approach

Describes the space as a set ofhierarchical and communicatingwell-stirred compartments:BioAmbients, Brane Calculi, etc...

http://lucacardelli.name/

Sphere based approach

The entities involved are modeledas spheres situated in space:Spatial CLS, SpacePI, etc..

SpacePI: Spatial extension of π calculus

Shape is not considered

The shape of a biological entity plays a very important role in his interaction.


Shapes in Shape Calculus

Shape Syntax

S ::= σ∣∣ S 〈X 〉S where σ ∈ Basic and X ⊆ R3 is a non-empty set of

points. The set X is intended to be a closed portion of the commonsurface on which the two shapes are attached.

Two examples of compound shapes in 2D

��

S

Y YS1

2

σ2

σ1σ

σ

X X X

3

4

321

(b)(a)

X

Shape definition

Shape (a) is composed of four basic shapes. A well-formed termrepresenting this shape can be ((σ1 〈X1〉σ2) 〈X2〉σ3) 〈X3〉σ4.


Trajectories of Shapes

Falling of a ball

!

Move(S,t0) = [0,"0.245,0]m /s

!

Move(S,t1) = [0,"0.49,0]m /s

!

Move(S,t2) = [0,"0.735,0]m /s

!

Move(S,t3) = [0,"0.98,0]m /s

!

Move(S,t4 ) = [0,"1.225,0]m /s

!

Move(S,ti) = [0,"1

2g(i +1)#,0]m /s

!

ti = ti"1 + #

# = 0.05s

S =

v = [vx,vy,vz ]

Time evolution and velocity update

1 The time domain T = R+0 is then divided into an infinite sequence of movement

time steps ti such that t0 = 0 and ti = ti−1 + ∆.

2 Move: Shapes× T −→ V that gives the velocity vector Move(S , t) to assign toshape S at time t


Collision Detection

First time of contact

!

S0

t0

!

S0

t1

!

S0

t2

!

S0

t3

!

S0

t4

!

S0

t5

!

S1

t0

!

S1

t1

!

S1

t2

!

S1

t3

!

S1

t4

!

S1

t5

!

S0

t4

+ t 't'< "

!"#$%&'()&*+&,*-%.,%&


Collision Response

Elastic collision (one dimensional case)

!

"(S0) = 2g

!

"(S0) = 2g

!

"(S1) =1g

!

"(S1) =1g

!

V (S0) =1cm /s

!

V (S1) = "1cm /s

!"##$%$"&'()%*"&%)'

!

V '(S0) = "1/3cm /s

!

V '(S1) = 5 /3cm /s

!

M(S0)V (S

0)2

2+M(S

1)V (S

1)2

2=M(S

0)V '(S

0)2

2+M(S

1)V '(S

1)2

2

!"#$%&'()"#*"+*,-#%).*%#%&/0*(1%&*."22-$-"#3**

4"5(2*6"6%#576*&%6(-#$*."#$5(#5*58&"7/8"75*58%*."22-$-"#3*

!

M(S0)V (S

0) + M(S

1)V (S

1) = M(S

0)V '(S

0) + M(S

1)V '(S

1)

9"2'-#/*58%$%*$-6725(#%"7$*%:7()"#$*;%*/%53*

!

V '(S0) =

V (S0)(M(S

0) "M(S

1)) " 2M(S

1)V (S

1)

M(S0) + M(S

1)

!

V '(S1) =

V (S1)(M(S

1) "M(S

0)) " 2M(S

0)V (S

0)

M(S0) + M(S

1)

Inelastic collision (one dimensional case)

!

"(S0) = 2g

!

"(S0) = 2g

!

"(S1) =1g

!

V (S0) =1cm /s

!

V (S1) = "1cm /s

!"##$%$"&'()%*"&%)'

!"#$%&'"'()#*'&+('$,)-&.")-#$)#&#/+"*0/"*#&#/(&."%%,-,")1&

!

M(S0)V (S

0) + M(S

1)V (S

1) = (M(S

0) + M(S

1))V '(S

0X S

1)

!

V '(S0X S

1) =

V (S0)M(S

0) +V (S

1)M(S

1)

M(S0) + M(S

1)

!

"(S1) =1g

!

X

!

V '(S0X S

1) =1/3cm /s!

X

!

X


3D Processes for HEX, GLC and ATP

Representation of enzymatic reaction in Shape Calculus

GlucoseGlucose-6-phosphateBinding SiteATPADP

Hexokinase

Real Shape

Hexokinase

Approximation

Binding Site Yhg Xha

Ygh

Xah


Modeling Hexokinase behavior

The set B of shapes’ behaviors is generated by the grammar

B ::= nil˛̨τ.B

˛̨〈α, X〉.B

˛̨ω(α, X ).B

˛̨ρ(L).B

˛̨ε(t).B

˛̨B + B

˛̨K

where 〈α, X〉 ∈ C, L is a non-empty subset of C whose channels are pairwise incompatible, t ∈ T and K is a process name in K.

The set 3DP of 3D processes is generated by the following grammar:

P ::= S[B]˛̨

P 〈a, X〉 P, where S ∈ Shapes, B ∈ B, a ∈ Λ and X ⊆ R3 closed, bounded, connected and with volume zero.

Modeling Hexokinase in Shape Calculus

Sh[HEX] where HEX = 〈atp,Xha〉.HA + 〈glc,Yhg 〉.HG.

Temporal behavior of B’s terms

Niltnil

t−→v nilPreft

X ′ = X + (t · v)

〈α, X〉.B t−→v 〈α, X ′〉.BSplitt

X ′ = X + (t · v)

ω(α, X ).Bt−→v ω(α, X ′).B

Delaytt′ ≥ t

ε(t′).B t−→v ε(t′ − t).BChoicet

B1t−→v B′1 B2

t−→v B′2B1 + B2

t−→v B′1 + B′2Deft

Bt−→v B′

Kt−→v B′

if Kdef= B


Modeling Hexokinase behavior

The set B of shapes’ behaviors is generated by the grammar

B ::= nil˛̨τ.B

˛̨〈α, X〉.B

˛̨ω(α, X ).B

˛̨ρ(L).B

˛̨ε(t).B

˛̨B + B

˛̨K

where 〈α, X〉 ∈ C, L is a non-empty subset of C whose channels are pairwise incompatible, t ∈ T and K is a process name in K.

Modeling Hexokinase in Shape Calculus

HA = ω(〈atp,Xha).HEX + (〈glc,Xhg 〉.ρ({〈atp,Xha〉, 〈glc,Yhg 〉}).HEX)

Functional behavior of B-terms

Prefaµ ∈ C ∪ ω(C) ∪ {τ}

µ.Bµ−→ B

Reaca1

L = {〈α, X〉}

ρ(L).Bρ(α,X )−−−−−→ B

Reaca2

{〈α, X〉} ⊂ L

ρ(L).Bρ(α,X )−−−−−→ ρ(L\{〈α, X〉}).B

Reaca3Bρ(α,X )−−−−−→ B′

ρ(L).Bρ(α,X )−−−−−→ ρ(L).B′

DelayaBµ−→ B′

ε(0).Bµ−→ B′

ChoiceaB1

µ−→ B′

B1 + B2µ−→ B′

DefaBµ−→ B′

Ka−→ B′

if Kdef= B


Modeling ATP and Glucose processes

ATP process

Sa[〈atp,Xah〉.ε(tatp).(ρ({〈atp,Xah〉}).ADP + ω(atp,Xah).ATP)].

GLC process

Sg [〈glc,Xgh〉.ε(tglc).(ρ({〈glc,Xgh〉}).G6P + ω(glc,Xgh).GLC )].

Functional behavior of B-terms

Prefaµ ∈ C ∪ ω(C) ∪ {τ}

µ.Bµ−→ B

Reaca1

L = {〈α, X〉}

ρ(L).Bρ(α,X )−−−−−→ B

Reaca2

{〈α, X〉} ⊂ L

ρ(L).Bρ(α,X )−−−−−→ ρ(L\{〈α, X〉}).B

Reaca3Bρ(α,X )−−−−−→ B′

ρ(L).Bρ(α,X )−−−−−→ ρ(L).B′

DelayaBµ−→ B′

ε(0).Bµ−→ B′

ChoiceaB1

µ−→ B′

B1 + B2µ−→ B′

DefaBµ−→ B′

Ka−→ B′

if Kdef= B


Strong and Weak Splitting

Weak Spitting

Strong Spitting


Network of 3D processes

Network of 3DP

The set of networks of 3D processes is generated by the grammar:

N ::= Nil∣∣ P

∣∣ N ‖N

Functional and Temporal semantics of the networks

EmptytNil

t−→ NilPart

Nt−→ N′ M

t−→ M′

N ‖Mt−→ N′ ‖M′

Para1Nµ−→ N′

N ‖ Nµ−→ N′ ‖M


Conclusion and future work

We have defined a calculus thattakes into account:

1 space and time

2 collision and communication

3 compound aggregation andsplitting

Working in progress

Manage the Shapetransfomation

A simulator for our ShapeCalculus

Equivalences andabstractions

Biosignalling by shapetransformation

!"#"$%&'((

%)'&*+,"(-+,.*"(

/0( /,(


A Shape Calculus for Biological Processes · Shape de nition Shape (a) is composed of four basic...

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