A Shape Calculus for Biological Processes

Transcription

1 A Shape Calculus for Biological Processes E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei UNICAM Complex Systems (CoSy) Research Group 29 Sep 2009, ICTCS 09 Cremona

2 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

3 Background Computational Systems Biology Designing in-silico drugs. Studying molecular crowding. Exploring biomolecular interactions Virus H1N1 Nanotechnology Molecular self-assembly Molecular recognition Molecular motors

4 Three ingredients: Space, Time and Shape Space/Time Macromolecular crowding alters the properties of molecules in a solution. In not well-stirred systems, the ideal way to simulate the time evolution of the system is to track the exact positions and velocities of all molecules. Shape and communication Contacts (collisions) and shapes transformation determines biomolecular interactions.

5 Related Works Topological Approach Describes the space as a set of hierarchical and communicating well-stirred compartments: BioAmbients, Brane Calculi, etc... Sphere based approach The entities involved are modeled as 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.

6 Shapes in Shape Calculus Shape Syntax S ::= σ S X S where σ Basic and X R 3 is a non-empty set of points. The set X is intended to be a closed portion of the common surface on which the two shapes are attached. Two examples of compound shapes in 2D σ 1 σ3 01 X X X σ4 S2 X Y Y S 1 σ 2 (a) (b) Shape definition Shape (a) is composed of four basic shapes. A well-formed term representing this shape can be ((σ 1 X 1 σ 2 ) X 2 σ 3 ) X 3 σ 4.

7 Trajectories of Shapes Falling of a ball Move(S,t 0 ) = [0,"0.245,0]m /s Move(S,t 1 ) = [0,"0.49,0]m /s Move(S,t 2 ) = [0,"0.735,0]m /s Move(S,t i ) = [0," 1 g(i +1)#,0]m /s 2 t i = t i"1 + # # = 0.05s Move(S,t 3 ) = [0,"0.98,0]m /s S = Move(S,t 4 ) = [0,"1.225,0]m /s v = [v x,v y,v z ] Time evolution and velocity update 1 The time domain T = R + 0 is then divided into an infinite sequence of movement time steps t i such that t 0 = 0 and t i = t i Move: Shapes T V that gives the velocity vector Move(S, t) to assign to shape S at time t

8 Collision Detection First time of contact S 0 t 1 S 0 t 2 S 0 t 3 S 0 t 4 S 0 t 5 S 0 t 0 S 0 t 4 +t' t'< "!"#$%&'()&*+&,*-%.,%& t S 5 1 t S 4 1 t S 3 1 t S 2 1 t S 1 1 S 1 t 0

9 Collision Response Elastic collision (one dimensional case) V (S 0 ) =1cm /s V (S 1 ) = "1cm /s "(S 0 ) = 2g!"##$%$"&'()%*"&%)' V '(S 0 ) = "1/3cm /s V '(S 1 ) = 5 /3cm /s "(S 0 ) = 2g "(S 1 ) =1g "(S 1 ) =1g!"#$%&'()"#*"+*,-#%).*%#%&/0*(1%&*."22-$-"#3** M(S 0 )V (S 0 ) 2 + M(S 1)V (S 1 ) 2 = M(S 0)V '(S 0 ) 2 + M(S 1)V '(S 1 ) "5(2*6"6%#576*&%6(-#$*."#$5(#5*58&"7/8"75*58%*."22-$-"#3* M(S 0 )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 '(S 0 ) = V (S 0 )(M(S 0 ) " M(S 1 )) " 2M(S 1 )V (S 1 ) M(S 0 ) + M(S 1 ) V '(S 1 ) = V (S 1 )(M(S 1 ) " M(S 0 )) " 2M(S 0 )V (S 0 ) M(S 0 ) + M(S 1 ) Inelastic collision (one dimensional case) V (S 0 ) =1cm /s V (S 1 ) = "1cm /s X "(S 0 ) = 2g X "(S 1 ) =1g!"#$%&'"'()#*'&+('$,)-&.")-#$)#&#/+"*0/"*#&#/(&."%%,-,")1& M(S 0 )V (S 0 ) + M(S 1 )V (S 1 ) = (M(S 0 ) + M(S 1 ))V '(S 0 X S 1 )!"##$%$"&'()%*"&%)' V '(S 0 X S 1 ) =1/3cm /s V '(S 0 X S 1 ) = V (S 0 )M(S 0 ) + V (S 1 )M(S 1 ) M(S 0 ) + M(S 1 ) "(S 0 ) = 2g X "(S 1 ) =1g

10 3D Processes for HEX, GLC and ATP Representation of enzymatic reaction in Shape Calculus ADP ATP Binding Site Glucose Glucose-6-phosphate Binding Site Y hg X ha Hexokinase Approximation Y gh Hexokinase Real Shape X ah

11 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 ɛ(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 R 3 closed, bounded, connected and with volume zero. Modeling Hexokinase in Shape Calculus S h [HEX] where HEX = atp, X ha.ha + glc, Y hg.hg. Temporal behavior of B s terms Delayt Nilt t nil v nil Preft t t ɛ(t ).B v t ɛ(t t).b X = X + (t v) α, X.B v t α, X.B t B 1 v B 1 Choicet B 1 + B 2 Splitt t v B 2 B 2 t v B 1 + B 2 X = X + (t v) ω(α, X ).B v t ω(α, X ).B B v t B Deft K v t B if K def = B

12 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 ɛ(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, X ha ).HEX + ( glc, X hg.ρ({ atp, X ha, glc, Y hg }).HEX) Functional behavior of B-terms Reac a2 Delaya Prefa µ C ω(c) {τ} µ.b µ B { α, X } L ρ(l).b ρ(α,x ) ρ(l\{ α, X }).B B µ B ɛ(0).b µ B Choicea B 1 µ B B 1 + B 2 µ B Reac a1 L = { α, X } ρ(l).b ρ(α,x ) B B ρ(α,x ) B Reac a3 ρ(l).b ρ(α,x ) ρ(l).b Defa B µ B K a B if K def = B

13 Modeling ATP and Glucose processes ATP process S a [ atp, X ah.ɛ(t atp ).(ρ({ atp, X ah }).ADP + ω(atp, X ah ).ATP)]. GLC process S g [ glc, X gh.ɛ(t glc ).(ρ({ glc, X gh }).G6P + ω(glc, X gh ).GLC)]. Functional behavior of B-terms Reac a2 Delaya Prefa µ C ω(c) {τ} µ.b µ B { α, X } L ρ(l).b ρ(α,x ) ρ(l\{ α, X }).B B µ B ɛ(0).b µ B Choicea B 1 µ B B 1 + B 2 µ B Reac a1 L = { α, X } ρ(l).b ρ(α,x ) B B ρ(α,x ) B Reac a3 ρ(l).b ρ(α,x ) ρ(l).b Defa B µ B K a B if K def = B

14 Strong and Weak Splitting Weak Spitting Strong Spitting

15 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 Emptyt t Nil Nil Part N t N M t M N M t N M Par a1 N µ N N N µ N M

16 Conclusion and future work We have defined a calculus that takes into account: space and time collision and communication Biosignalling by shape transformation!"#"$%&'(( %)'&*+,"(-+,.*"( compound aggregation and splitting Working in progress Manage the Shape transfomation A simulator for our Shape Calculus Equivalences and abstractions E. Bartocci, F. Corradini, M.R. Di Berardini, E. Merelli and L. Tesei /0( /,( A Shape Calculus for Biological Processes