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
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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
Background
Computational Systems Biology
Designing ”in-silico” drugs.
Studying molecular crowding.
Exploring biomolecularinteractions
Nanotechnology
Molecular self-assembly
Molecular recognition
Molecular motors
Virus H1N1
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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.
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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.
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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.
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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'< "
!"#$%&'()&*+&,*-%.,%&
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
Collision Response
Elastic collision (one dimensional case)
!
"(S0) = 2g
!
"(S0) = 2g
!
"(S1) =1g
!
"(S1) =1g
!
V (S0) =1cm /s
!
V (S1) = "1cm /s
!"##$%$"&'()%*"&%)'
!
V '(S0) = "1/3cm /s
!
V '(S1) = 5 /3cm /s
!
M(S0)V (S
0)2
2+M(S
1)V (S
1)2
2=M(S
0)V '(S
0)2
2+M(S
1)V '(S
1)2
2
!"#$%&'()"#*"+*,-#%).*%#%&/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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
Strong and Weak Splitting
Weak Spitting
Strong Spitting
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes
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( /,(
E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei A Shape Calculus for Biological Processes