Paris/Sources

{{Paris_menu_modeling}}<br>

=Cellular Automaton for DAP Diffusion=
<code><pre>
!include "gbviewOutput.mgs" ;;

N := 30;;

gbf grille = <nord, est ; 30 * nord, 30 * est>
and record Bact = { DAP:float, DAPe:float }
and record BactG = Bact + { bactG }
and record BactS = Bact + { bactS} ;;

fun iota_v(v,n) = map((\x.v), iota(n,seq:())) ;;
fun new_grid(v) = iota_v(iota_v(v,N), N) following |nord>, |est> ;;

fun majBactG[DAPexport = 0.2, DAPimport = 0.2, DAPconso= 1, SupDiff = 2.4, InfDiff = 2.02 ](x) = 
 	if ( ( x.DAP > InfDiff ) & ( x.DAP < SupDiff ) & (random(1) < 0.8) ) then
		{DAP = x.DAP , DAPe = x.DAPe, bactS} else
		let dap = if x.DAP - DAPconso  > 0 then  x.DAP - DAPconso  else  0 fi
		 and dape= x.DAPe
		 in if 	dap > dape then 
                    x + {DAP = dap - DAPexport * dap, DAPe = dape + DAPexport * dap }
		   else x + {DAP = dap + DAPimport * dape, DAPe = dape - DAPimport * dape }
                  fi
        fi 
;;

fun majBactS[DAPexport = 0.2, DAPimport = 0.2, DAPprod=10](x) =  
		let dap =  x.DAP + DAPprod
		 and dape= x.DAPe
		 in if 	dap > dape then 
                    x + {DAP = dap - DAPexport * dap, DAPe = dape + DAPexport * dap }
		   else x + {DAP = dap + DAPimport * dape, DAPe = dape - DAPimport * dape }
                  fi

;;


trans evol[Delta_t=0.1,DAPeDiff=1, DAPeDegrad = 0.2] = {



  (* production of DAP *)
  x:BactS => ( 
	let d = neighborsfold(
      (\y.\acc.( DAPeDiff*Delta_t*(y.DAPe-x.DAPe) + acc)),
      x.DAPe,
      x)
    in majBactS( x + {DAPe = if ( d - DAPeDegrad * x.DAPe) > 0 then d - DAPeDegrad * x.DAPe else 0 fi } )
      );

  (* Diffirentiation and Consumption *)
 x:BactG => ( 
	let d = neighborsfold(
      (\y.\acc.( DAPeDiff*Delta_t*(y.DAPe-x.DAPe) + acc)),
      x.DAPe,
      x)
    in majBactG(x + {DAPe = if ( d - DAPeDegrad * x.DAPe) > 0 then  d - DAPeDegrad * x.DAPe else 0 fi } )
    

  )

} ;;





g := new_grid({DAP=6.0, DAPe=3.0, bactG}) ; 0 ;;
g := set_gbfpos(g, (random(30)*|nord> + random(30)*|est>), {DAP = 6.0, DAPe=3.0, bactS}) ;;
g := set_gbfpos(g, (random(30)*|nord> + random(30)*|est>), {DAP = 6.0, DAPe=3.0, bactS}) ;;
g := set_gbfpos(g, (random(30)*|nord> + random(30)*|est>), {DAP = 6.0, DAPe=3.0, bactS}) ;;
g := set_gbfpos(g, (random(30)*|nord> + random(30)*|est>), {DAP = 6.0, DAPe=3.0, bactS}) ;; 
evol[iter=1000,interlude=GBVexport((\v.("DAPe = "+(12 * v.DAPe )+if BactS(v) then ", bactS= 255" else  ", bactG = 255"fi )))](g) ;;
</pre></code>

= Cells Spatial Organization =

==Constant rate of differentiation==

<code><pre>
record MecaBact = {x, y, vx, vy, fx, fy, radius}
and record CellBact  = {dap:float, soma:bool}
and record Bact = MecaBact + CellBact;;


delaunay(2) D2 = (\e.(e.x, e.y)) ;;

fun noise(x) = x + 0.0005 - random(0.001) ;;
// --- Mechanistic-------------------------------------------------------

DT    := 0.05;;
K     := 1.0;;
MU    := 1.8;;
R0_Gm := 0.50;; //germinal cells minmal rayon 
R0_G  :=0.75;;
R0_S  := 1.00;;

// interaction : the effect of the cells between each others
fun interaction(ref, src) = (
 let X      = ref.x - src.x
 and Y      = ref.y - src.y in
 let dist   = sqrt(X*X+Y*Y) in
 let spring = 0.0-K*(dist-(ref.radius+src.radius))/dist in
   {fx=X*spring - ref.vx*MU, fy = Y*spring - ref.vy*MU}
) ;;

fun add_vect(u, v) = { fx = u.fx + v.fx, fy = u.fy + v.fy } ;;
fun sum(x, u, acc) = add_vect(acc, interaction(x,u)) ;;


trans Meca = {
 e => (
   let f = neighborsfold(sum(e), {fx=0,fy=0}, e) in
     e+{ x  = noise(e.x  + DT*e.vx),
         y  = noise(e.y  + DT*e.vy),
         vx = e.vx + DT*f.fx,
         vy = e.vy + DT*f.fy,
         fx = f.fx,
         fy = f.fy
       }
 )
} ;;


// --- Growth ------------------------------------------------------

DIFF  := 1.0  ;;
CONS  := 10.0  ;;
DiffP := 0.00023  ;; 
DeathSP := 0.00001 ;;
DivG := 0.00600 ;;
DEPOT := 15.0 ;; 
CroitG :=0.004  ;; 
CroitS :=0.007;;  


//cell division function
fun divide(b) = (
 b+{dap= b.dap/2, radius=R0_Gm },
 b + {dap= b.dap/2,x=noise(b.x),y=noise(b.y), radius=R0_Gm}
) ;;



trans Evol = {

 x / x.soma => if (random(1.0) <= DeathSP) then <undef> else if ( ( x.radius < R0_S ) & ( random(1.0) < CroitS) ) then x + 
{radius = x.radius + (R0_S - R0_G)/4} else  x fi fi ;  

 x => (
   let dap_diff = neighborsfold((\y.\acc.( {dap=DT*DIFF*(y.dap-x.dap) + acc.dap, n=acc.n+1} )), {dap=0.0,n=0}, x) in
   let dap' = if (x.dap + dap_diff.dap/dap_diff.n) - DT*CONS >= 0 then (x.dap + dap_diff.dap/dap_diff.n) - DT*CONS else 0 fi in

       if (random(1.0) <= DiffP)
       then x + {dap=DEPOT,soma=true,radius=x.radius} //la nouvelle cellule S a la taille de la cellule G dont elle provient
       else  if x.radius  >= R0_G then if  ( (random(1.0) <= DivG) & dap'> 1 )then divide(x + {dap=dap'}) else (x + {dap=dap'}) fi
 else  if random(1.0) < CroitG then x + {radius = x.radius + (R0_G - R0_Gm)/5 , dap=dap' } else x+{dap=dap'} fi fi fi
 ) 

} ;;




// --- Visualisation ----------------------------------------------------

outname := "/tmp/sheet.imo" ;;
outfile := (if (is_opened(outname)) then close(outname) else <undef> fi ; open(outname,1)) ;;




fun show_cell(e) = (
 "\tTranslated { Translation <" + e.x + ", " + e.y + ", " + 0.0 + "> Geometry Sphere { Radius " + e.radius + " Slices 16 
Stacks 16 " + "Color<" + if(e.soma) then 0 + ", " + 1.0 + ", " + 0 else if ((e.dap+0.4)*(0.5+R0_G/e.radius))< 1.0 then 
((e.dap+0.4)*(0.5+R0_G/e.radius)) else  1.0  fi+ ", " + 0 + ", " + 0 fi + "> } }"
) ;;



fun show[cpt=0](f, freq, c) = (
 cpt := 1 + cpt ;
 if (0 == cpt % freq)
 then (
   stdout << cpt << "\n" ;
   print_coll(f,
              c,
              show_cell,
              "Scene a"+cpt+" {\n",
              "\n",
              "}\nReplace { Show a"+cpt+"}\n\n")
 ) else <undef> fi ;
 c
) ;;

// --- Etat initial ----------------------------------------------------

v0 := {vx=0, vy=0, vz=0, fx=0, fy=0, fz=0, dap=14.0,soma=false,radius=R0_Gm} ;;
v1 :={vx=0, vy=0, vz=0, fx=0, fy=0, fz=0, dap=DEPOT,soma=true,radius=R0_S} ;;
pre_init :=
       v1+{x = 1.01,  y = 1.023, z = 0.101},
       v1+{x = 0.07,  y = 1.0,   z = 0.095},
       v1+{x = 1.0,   y = 0.01,  z = 0.098},
       v1+{x = 0.52,  y = 0.53,  z = 0.1},
       v1+{x = 0.01,  y = 0.02,  z = 0.099},
       v0+{x = -0.4,  y = 0.02,  z = 0.099},
       v0+{x = 1.4,  y = 0.02,  z = 0.099}
;;

init := Meca[iter=1000](delaunayfy(D2:(), pre_init)) ;;

//show(outfile, 1, init);;
//system("imoview "+outname);;

// --- Evolution -------------------------------------------------------

fun step(sys) = (
 Evol(Meca(sys))
) ;;

step[iter=100000,interlude=show(outfile,10)](init) ;;

system("./imoview_black "+outname);;
!quit ;;
</pre></code>

==Differentiation DAP dependent==

<code><pre>
record MecaBact = {x, y, vx, vy, fx, fy, radius}
and record CellBact  = {dap:float, soma:bool}
and record Bact = MecaBact + CellBact;;


delaunay(2) D2 = (\e.(e.x, e.y)) ;;

fun noise(x) = x + 0.0005 - random(0.001) ;;
// --- Mechanistic-------------------------------------------------------

DT    := 0.05;;
K     := 1.0;;
MU    := 1.8;;
R0_Gm := 0.50;; //germinal cells minmal rayon 
R0_G  :=0.75;;
R0_S  := 1.00;;

// interaction : the effect of the cells between each others
fun interaction(ref, src) = (
 let X      = ref.x - src.x
 and Y      = ref.y - src.y in
 let dist   = sqrt(X*X+Y*Y) in
 let spring = 0.0-K*(dist-(ref.radius+src.radius))/dist in
   {fx=X*spring - ref.vx*MU, fy = Y*spring - ref.vy*MU}
) ;;

fun add_vect(u, v) = { fx = u.fx + v.fx, fy = u.fy + v.fy } ;;
fun sum(x, u, acc) = add_vect(acc, interaction(x,u)) ;;


trans Meca = {
 e => (
   let f = neighborsfold(sum(e), {fx=0,fy=0}, e) in
     e+{ x  = noise(e.x  + DT*e.vx),
         y  = noise(e.y  + DT*e.vy),
         vx = e.vx + DT*f.fx,
         vy = e.vy + DT*f.fy,
         fx = f.fx,
         fy = f.fy
       }
 )
} ;;


// --- Growth -----------------------------------------------------


DIFF  := 1.0  ;;
CONS  := 10.0  ;;
DiffP := 0.005  ;;
DeathSP := 0.00001 ;;
DivG := 0.00300 ;;
DEPOT := 16 ;;
CroitG :=0.002  ;; re
CroitS :=0.007;;  

fun divide(b) = (
 b+{dap= b.dap/2, radius=R0_Gm },
 b + {dap= b.dap/2,x=noise(b.x),y=noise(b.y), radius=R0_Gm}
) ;;



trans Evol = {

 x / x.soma => if (random(1.0) <= DeathSP) then <undef> else if ( ( x.radius < R0_S ) & ( random(1.0) < CroitS) ) then x + {radius
 = x.radius + (R0_S - R0_G)/4} else  x fi fi ; r)

  x => (
   let dap_diff = neighborsfold((\y.\acc.( {dap=DT*DIFF*(y.dap-x.dap) + acc.dap, n=acc.n+1} )), {dap=0.0,n=0}, x) in
   let dap' = if (x.dap + dap_diff.dap/dap_diff.n) - DT*CONS >= 0 then (x.dap + dap_diff.dap/dap_diff.n) - DT*CONS else 0 fi in
     if (dap'<=0.0)
     then (
       if (random(1.0) <= DiffP)
       then x + {dap=DEPOT,soma=true,radius=x.radius} //la nouvelle cellule S a la taille de la cellule G dont elle provient
       else x + {dap=dap'} fi
     ) else if x.radius  >= R0_G then if  ( (random(1.0) <= DivG) & dap'> 2 )then divide(x + {dap=dap'}) else (x + {dap=dap'}) fi 
else  if random(1.0) < CroitG then x + {radius = x.radius + (R0_G - R0_Gm)/5 , dap=dap' } else x+{dap=dap'} fi fi fi 
 ) 
} ;;




// --- Visualisation ----------------------------------------------------

outname := "/tmp/sheet.imo" ;;
outfile := (if (is_opened(outname)) then close(outname) else <undef> fi ; open(outname,1)) ;;




fun show_cell(e) = (
 "\tTranslated { Translation <" + e.x + ", " + e.y + ", " + 0.0 + "> Geometry Sphere { Radius " + e.radius + " Slices 16 
Stacks 16 " + "Color<" + if(e.soma) then 0 + ", " + 1.0 + ", " + 0 else if ((e.dap+0.4)*(0.5+R0_G/e.radius))< 1.0 then
 
((e.dap+0.4)*(0.5+R0_G/e.radius)) else  1.0  fi+ ", " + 0 + ", " + 0 fi + "> } }"
) ;;



fun show[cpt=0](f, freq, c) = (
 cpt := 1 + cpt ;
 if (0 == cpt % freq)
 then (
   stdout << cpt << "\n" ;
   print_coll(f,
              c,
              show_cell,
              "Scene a"+cpt+" {\n",
              "\n",
              "}\nReplace { Show a"+cpt+"}\n\n")
 ) else <undef> fi ;
 c
) ;;

// --- Etat initial ----------------------------------------------------

v0 := {vx=0, vy=0, vz=0, fx=0, fy=0, fz=0, dap=14.0,soma=false,radius=R0_Gm} ;;
v1 :={vx=0, vy=0, vz=0, fx=0, fy=0, fz=0, dap=DEPOT,soma=true,radius=R0_S} ;;
pre_init :=
       v1+{x = 1.01,  y = 1.023, z = 0.101},
       v1+{x = 0.07,  y = 1.0,   z = 0.095},
       v1+{x = 1.0,   y = 0.01,  z = 0.098},
       v1+{x = 0.52,  y = 0.53,  z = 0.1},
       v1+{x = 0.01,  y = 0.02,  z = 0.099},
       v0+{x = -0.4,  y = 0.02,  z = 0.099},
       v0+{x = 1.4,  y = 0.02,  z = 0.099}
;;

init := Meca[iter=1000](delaunayfy(D2:(), pre_init)) ;;

//show(outfile, 1, init);;
//system("imoview "+outname);;

// --- Evolution -------------------------------------------------------

fun step(sys) = (
 Evol(Meca(sys))
) ;;

step[iter=100000,interlude=show(outfile,10)](init) ;;

system("./imoview_black "+outname);;
!quit ;;
</pre></code>

=Gillespie Based Simulation=

==Global Variables==
<code><pre>
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
(* Global constants                                                                               *)
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
Na := 6.0221415e23 ;;
VolumeUnit := 1/Na ;;
fun gamma(V) = V ;; // actually Gamma(V)=V*VolumeUnit*Na, but VolumeUnit has been choosen in such a way that Gamma(V)=V waiting for the right volume (mean volume of a germinal bacterium)

K_CreD   := 0.2   ;;  // Cre degradation
K_DAPiD  := 0.2   ;;  // DAP degradation
K_DAPApI := 0.5   ;;  // DAPAp inhibition by DAP
K_DAPApA := 0.1   ;;  // DAPAp desinhibition releasing a DAP molecule
K_CreP1  := 1.0   ;;  // Cre production (DAPAp not inhibited)
K_CreP2  := 0.001 ;;  // Cre production (DAPAp inhibited)
K_Diff   := 0.05  ;;  // Differentiation
K_DAPiP  := 1.0   ;;  // DAP production
K_DAPEx  := 0.5   ;;  // DAP export
K_DAPIm  := 1.0   ;;  // DAP import
K_DAPeD  := 0.1   ;;  // DAP degradation

K_Div    := 0.02   ;; // Mitosis rate
K_Death  := 0.01   ;; // Death rate

nb_dape  := 0 ;; // Number of DAP molecule in the environment (initial condition)

fun globals_help() = (
  stdout << "\n** Help of the \"global\" module **\n\n" ;
  stdout << "Globals\n" ;
  stdout << "\t- Na (float): Avogadro number [6.0221415e23]\n\n" ;
  stdout << "\t- VolumeUnit (float): mean volume of a bacterium [1/Na]\n\n" ;
  stdout << "\t- K_CreD (float): Cre degradation reaction rate in bacteria [0.01]\n\n" ;
  stdout << "\t- K_DAPiD (float): DAP degradation reaction rate in bacteria [0.01]\n\n" ;
  stdout << "\t- K_DAPApI (float): DAPAp inhibition by DAP [2.0]\n\n" ;
  stdout << "\t- K_DAPApA (float): DAPAp re-activation [0.001]\n\n" ;
  stdout << "\t- K_CreP1 (float): Cre production with activated DAPAp [0.01]\n\n" ;
  stdout << "\t- K_CreP2 (float): Cre production with inhibited DAPAp [0.0]\n\n" ;
  stdout << "\t- K_Diff (float): bacteria differentition [0.1]\n\n" ;
  stdout << "\t- K_DAPiP (float): DAP production in bacteria [0.5]\n\n" ;
  stdout << "\t- K_DAPEx (float): DAP exportation [0.1]\n\n" ;
  stdout << "\t- K_DAPIm (float): DAP importation [0.1]\n\n" ;
  stdout << "\t- K_DAPeD (float): DAP degradation reaction rate in the environment [1.0]\n\n" ;

  stdout << "Functions\n" ;
  stdout << "\t- gamma(V) (float -> float):\n\t\tgamma function for reaction to stochastic constants translation\n\n"
) ;;
</pre></code>

==Structure==
<code><pre>
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
(* State of the system                                                                            *)
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
(* The system is composed of bacteria of different kinds: germinal and somatic                    *)
(*                                                                                                *)
(* It is modeled by nested multi-sets. The super multi-set corresponds to the environment where   *)
(* chemicals and bacteria are diffusing.                                                          *)
(* A set of functions is added to memorize the quantity of chemicals in the system in order not   *)
(* to evaluate it at each time.                                                                   *)
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)

cptCre   := 0 ;;
cptDAPe  := 0 ;;
cptDAPi  := 0.0 ;;
cptDAP   := 0 ;;
cptBactS := 0 ;;
cptBactG := 0 ;;
cptBact  := 0 ;;
fun incrCre() = (cptCre := cptCre + 1) ;;
fun decrCre() = (cptCre := cptCre - 1) ;;
fun incrDAP() = (cptDAP := cptDAP + 1) ;;
fun decrDAP() = (cptDAP := cptDAP - 1) ;;
fun incrDAPe() = (cptDAPe := cptDAPe + 1) ;;
fun decrDAPe() = (cptDAPe := cptDAPe - 1) ;;
fun incrDAPi() = (cptDAPi := cptDAPi + 1) ;;
fun decrDAPi() = (cptDAPi := cptDAPi - 1) ;;
fun DAPe_DAPi() = (incrDAPi() ; decrDAPe()) ;;
fun DAPi_DAPe() = (incrDAPe() ; decrDAPi()) ;;
fun incrNCre(N) = (cptCre := cptCre + N) ;;
fun decrNCre(N) = (cptCre := cptCre - N) ;;
fun incrNDAP(N) = (cptDAP := cptDAP + N) ;;
fun decrNDAP(N) = (cptDAP := cptDAP - N) ;;
fun incrNDAPe(N) = (cptDAPe := cptDAPe + N) ;;
fun decrNDAPe(N) = (cptDAPe := cptDAPe - N) ;;
fun incrNDAPi(N) = (cptDAPi := cptDAPi + N) ;;
fun decrNDAPi(N) = (cptDAPi := cptDAPi - N) ;;

fun incrBact() = (cptBact := cptBact + 1) ;;
fun decrBact() = (cptBact := cptBact - 1) ;;
fun incrBactS() = (cptBactS := cptBactS + 1) ;;
fun decrBactS() = (cptBactS := cptBactS - 1) ;;
fun incrBactG() = (cptBactG := cptBactG + 1) ;;
fun decrBactG() = (cptBactG := cptBactG - 1) ;;
fun BactS_BactG() = (raise  (`Erreur "Impossible S to G differentiation.")) ;;
fun BactG_BactS() = (incrBactS() ; decrBactG()) ;;
fun incrNBact(N) = (cptBact := cptBact + N) ;;
fun decrNBact(N) = (cptBact := cptBact - N) ;;
fun incrNBactS(N) = (cptBactS := cptBactS + N) ;;
fun decrNBactS(N) = (cptBactS := cptBactS - N) ;;
fun incrNBactG(N) = (cptBactG := cptBactG + N) ;;
fun decrNBactG(N) = (cptBactG := cptBactG - N) ;;

collection Bact      = bag
and constraint BactG = [~`DAP_Box]Bact
and constraint BactS = [~`LOXP_Box && ~`LOXP_Box_Cre]Bact
and collection Env   = bag  ;;

id := 0 ;;
fun new_bactId() = (id := id+1 ; id) ;;
fun new_bact(t)  = (incrBact() ; Bactify({birth=t, date=0.0, id=new_bactId(), next_state=`Unknown}::seq:()) )  ;;
fun new_bactS(t) = (incrBactS() ; `DAPAp :: `DAP_Box  :: new_bact(t)) ;;
fun new_bactG(t) = (incrBactG() ; (*incrDAPi() ; incrDAP() ;*) `DAPAp :: `LOXP_Box :: new_bact(t)) ;;
fun new_env(t, nbG, nbS) = (
  incrNDAP(nb_dape) ;
  incrNDAPe(nb_dape) ;
  let env = fold((\i.\acc.((new_bactG(t))::acc)),Envify(fold((\n.\acc.(`DAP::acc)),seq:(),nb_dape)), nbG) in
    fold((\i.\acc.((new_bactS(t))::acc)),env, nbS)
) ;;
fun map_bact(f,env) = (
  map((\e.(if Bact(e) then f(e) else e fi)),env) //update_bact
) ;;

fun divide_bactG(t,b,id) = (
  let id' = new_bactId() in
  let nbDAP = count(`DAP,b)
  and nbCre = count(`Cre,b) in
  let bact1_core = Bactify({birth=t, date=0.0, id=id,  next_state=`Unknown}::seq:())
  and bact2_core = Bactify({birth=t, date=0.0, id=id', next_state=`Unknown}::seq:()) in
  let bact1_core = fold((\n.\acc.(`DAP::acc)),bact1_core,nbDAP/2)
  and bact2_core = fold((\n.\acc.(`DAP::acc)),bact2_core,nbDAP - nbDAP/2) in
  let bact2_core = fold((\n.\acc.(`Cre::acc)),bact2_core,nbCre/2)
  and bact1_core = fold((\n.\acc.(`Cre::acc)),bact1_core,nbCre - nbCre/2) in
  let bact1_core = if (member(`DAPAp_i,b)) then (`DAPAp_i) else (`DAPAp) fi :: bact1_core
  and bact2_core = `DAPAp :: bact2_core in
  let bact2_core = if (member(`LOXP_Box_Cre,b)) then (`LOXP_Box_Cre) else (`LOXP_Box) fi :: bact2_core
  and bact1_core = `LOXP_Box :: bact1_core in
    bact1_core :: bact2_core :: seq:()
) ;;

fun set_DAPe(N,env) = (
  let dDAP = cptDAPe - N in
    if (dDAP > 0)
    then (
      (* cptDAPe > N : trop de DAP *)
      cptDAPe := N ;
      cptDAP  := cptDAP - dDAP ;
      diff(env, fold((\n.\acc.(`DAP::acc)),Env:(),dDAP))
    )
    else (
      if (dDAP < 0)
      then (
	(* cptDAPe < N : pas assez de DAP *)
	cptDAPe := N ;
	cptDAP  := cptDAP - dDAP ;
	join(env, fold((\n.\acc.(`DAP::acc)),Env:(),-1*dDAP))
      )
      else (env) fi
    ) fi
) ;;









fun static_help() = (
  stdout << "\n** Help of the \"static\" module **\n\n" ;
  stdout << "Globals\n" ;
  stdout << "\t- cptCre (cptCre): `Cre counter\n\n" ;
  stdout << "\t- cptDAP (cptDAP): `DAP counter\n\n" ;
  stdout << "\t- cptDAPe (cptDAPe): `DAP counter (in the environment)\n\n" ;
  stdout << "\t- cptDAPi (cptDAPi): `DAP counter (in bacteria)\n\n" ;
  stdout << "\t- cptBact (cptBact): bacteria counter\n\n" ;
  stdout << "\t- cptBactG (cptBactG): germinal bacteria counter\n\n" ;
  stdout << "\t- cptBactS (cptBactS): somatic bacteria counter\n\n" ;

  stdout << "Functions\n" ;
  stdout << "\t- incrCre() (unit -> int):\n\t\tincrement cptCre\n\n" ;
  stdout << "\t- decrCre() (unit -> int):\n\t\tdecrement cptCre\n\n" ;
  stdout << "\t- incrDAP() (unit -> int):\n\t\tincrement cptDAP\n\n" ;
  stdout << "\t- decrDAP() (unit -> int):\n\t\tdecrement cptDAP\n\n" ;
  stdout << "\t- incrDAPe() (unit -> int):\n\t\tincrement cptDAPe\n\n" ;
  stdout << "\t- decrDAPe() (unit -> int):\n\t\tdecrement cptDAPe\n\n" ;
  stdout << "\t- incrDAPi() (unit -> int):\n\t\tincrement cptDAPi\n\n" ;
  stdout << "\t- decrDAPi() (unit -> int):\n\t\tdecrement cptDAPi\n\n" ;
  stdout << "\t- DAPe_DAPi() (unit -> int):\n\t\tdecrement cptDAPe, increment cptDAPi\n\n" ;
  stdout << "\t- DAPi_DAPe() (unit -> int):\n\t\tdecrement cptDAPi, increment cptDAPe\n\n" ;
  stdout << "\t- incrBact() (unit -> int):\n\t\tincrement cptBact\n\n" ;
  stdout << "\t- decrBact() (unit -> int):\n\t\tdecrement cptBact\n\n" ;
  stdout << "\t- incrBactG() (unit -> int):\n\t\tincrement cptBactG\n\n" ;
  stdout << "\t- decrBactG() (unit -> int):\n\t\tdecrement cptBactG\n\n" ;
  stdout << "\t- incrBactS() (unit -> int):\n\t\tincrement cptBactS\n\n" ;
  stdout << "\t- decrBactS() (unit -> int):\n\t\tdecrement cptBactS\n\n" ;
  stdout << "\t- BactG_BactS() (unit -> int):\n\t\tdecrement cptBactG, increment cptBactS\n\n" ;

  stdout << "\t- new_Bact(t) (float -> Bact):\n\t\tcreate a new bacterium at time t\n\n" ;
  stdout << "\t- new_BactG(t) (float -> BactG):\n\t\tcreate a new germinal bacterium at time t\n\n" ;
  stdout << "\t- new_BactS(t) (float -> BactS):\n\t\tcreate a new somatic bacterium at time t\n\n" ;
  stdout << "\t- new_env(t,ng,ns) (float * int * int -> Env):\n\t\tcreate a new environment containing ng germinal bacteria and ns somatic bacteria at time t\n\n" ;
  stdout << "\t- map_bact(f,env) ((Bact -> Bact) * Env -> Env):\n\t\tapply the function f on each bacterium of env\n\n" ;
  stdout << "\t- set_DAPe(N,env) (int * Env -> Env):\n\t\tchange env, cptDAP and cptDAPe to N `DAP\n\n"
) ;;
</pre></code>

==Scheduling==
<code><pre>
constraint schedule = `Empty_SC | schedule_entry

and record schedule_entry = { date:float, id:int, sc:schedule, rule:int } ;;

fun schedule_empty() = `Empty_SC ;;

fun schedule_add_entry(id,date,rule,sc) = (
  switch (sc)
    case `Empty_SC : {date=date,id=id,rule=rule,sc=sc}
    default : (
      if (sc.date > date)
      then {date=date,id=id,rule=rule,sc=sc}
      else {date=sc.date,id=sc.id,rule=sc.rule,sc=schedule_add_entry(id,date,rule,sc.sc)} fi
    )
  endswitch
) ;;

fun schedule_remove_entry(id,sc) = (
  switch (sc)
    case `Empty_SC : `Empty_SC
    default : (
      if (sc.id == id)
      then sc.sc
      else {date=sc.date,id=sc.id,rule=sc.rule,sc=schedule_remove_entry(id,sc.sc)} fi
    )
  endswitch
) ;;

fun schedule_size(sc) = (
  switch (sc)
    case `Empty_SC : 0
    default : 1 + schedule_size(sc.sc)
  endswitch
) ;;

fun schedule_replace_entry(id,date,rule,sc) = (
  switch (sc)
    case `Empty_SC : {date=date,id=id,rule=rule,sc=sc}
    default : (
      if (sc.id == id)
      then schedule_add_entry(id,date,rule,sc.sc)
      else (
	if (sc.date > date)
	then {date=date,id=id,rule=rule,sc=schedule_remove_entry(id,sc)}
	else {date=sc.date,id=sc.id,rule=sc.rule,sc=schedule_replace_entry(id,date,rule,sc.sc)} fi
      ) fi
    )
  endswitch
) ;;

fun schedule_remove_first_entry(sc) = sc.sc ;;

fun schedule_print(sc) = (
  switch (sc)
    case `Empty_SC : stdout << "\n"
    default : (
      stdout << sc.id << "\t(" << sc.rule << ", " << sc.date << ") \n" ;
      schedule_print(sc.sc)
    )
  endswitch
) ;;

schedule := schedule_empty() ;;

fun schedule_help() = (
  stdout << "\n** Help of the \"schedule\" module **\n\n" ;
  stdout << "Globals\n" ;
  stdout << "\t- schedule (schedule): the main schedule initilized with value `Empty_SC\n\n" ;
  stdout << "Functions\n" ;
  stdout << "\t- schedule_empty() (unit -> schedule):\n\t\treturns the empty schedule\n\n" ;
  stdout << "\t- schedule_add_entry(id,d,r,sc) (int * float * int * schedule -> schedule):\n\t\tadd an entry in the schedule sc corresponding to the\n\t\tapplication of the rule r on the bacterium id at the date d\n\n" ;
  stdout << "\t- schedule_add_entry(id,d,r,sc) (int * float * int * schedule -> schedule):\n\t\treplace (or add if not existing) an entry in the\n\t\tschedule sc corresponding to the application of the rule r on the bacterium id at the date d\n\n" ;
  stdout << "\t- schedule_remove_entry(id,sc) (int * schedule -> schedule):\n\t\tremove in sc the first occurence of an event in bacterium id\n\n" ;
  stdout << "\t- schedule_remove_first(sc) (schedule -> schedule):\n\t\tremove the first entry of sc\n\n" ;
  stdout << "\t- schedule_print(sc) (schedule -> stdout):\n\t\tprint sc\n\n"
) ;;
</pre></code>

==Dynamics==
<code><pre>
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
(* Dynamics of the system                                                                         *)
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
(* It exists two kinds of dynamics. The first is related to the chemical reactions that occur     *)
(* everywhere in the system (inside and outside the bacteria + transport of proteins between the  *)
(* outside and inside). This parts will be simulated using a sotchastic process based on the      *)
(* Gillespie's algorithm. The second part of the dynamics to be modeled corresponds to the        *)
(* bacteria behavior like divisions and cells death. In fact, a lot of processes appearing in     *)
(* bacteria and belonging to their normal dynamics are skipped/abstracted as functions of the     *)
(* chemical state of bacteria. For example, a too high concentration of DAP is lethal ; we        *)
(* reprensent this dependance by a function that deletes all bacteria where the threshold is      *)
(* exceeded. These behavior functions have to be evaluated after each chemical update.            *)
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)

constraint ruleId = `CreD | `DAPiD | `DAPApI | `DAPApA | `CreP1 | `CreP2 | `Diff1 | `Diff2 | `DAPiP | `DAPEx | `DAPIm | `DAPeD | `Div | `Death | `Unknown ;;

fun fstOrder(k,v) = k ;;
fun sndOrder(k,v) = k/gamma(v) ;;

//fun count(p,c) = fold((\e.\acc.(acc + if p(e) then 1 else 0 fi)),0,c) ;;
//fun countAll(p1,p2,c) = fold((\e.\acc.(acc + if p2(e) then count(p1,e) else 0 fi)),0,c) ;;


appliedBactRule := `Unkown  ;;
trans inBact = {

  (* Chemicals degradation *)
  `Cre ={ C = fstOrder(K_CreD,1.0)  }=> (appliedBactRule := `CreD  ; <undef>) ;
  `DAP ={ C = fstOrder(K_DAPiD,1.0) }=> (appliedBactRule := `DAPiD ; <undef>) ;

  (* Promotor activity *)
  `DAPAp, `DAP ={ C = sndOrder(K_DAPApI,1.0) }=> (appliedBactRule := `DAPApI ; `DAPAp_i) ;
  `DAPAp_i     ={ C = fstOrder(K_DAPApA,1.0) }=> (appliedBactRule := `DAPApA ; `DAPAp, `DAP) ;
  `DAPAp       ={ C = \c.(fstOrder(K_CreP1* if (member(`DAP,c)) then 1.0 else 1.0 fi,1.0))  }=> (appliedBactRule := `CreP1  ; `DAPAp, `Cre) ;
  `DAPAp_i     ={ C = fstOrder(K_CreP2,1.0)  }=> (appliedBactRule := `CreP2  ; `DAPAp_i, `Cre) ;

  (* Differentiation *)
  `LOXP_Box, `Cre     ={ C = sndOrder(2.0*K_Diff,1.0) }=> (appliedBactRule := `Diff1 ; `LOXP_Box_Cre) ;
  `LOXP_Box_Cre, `Cre ={ C = sndOrder(K_Diff,1.0)     }=> (appliedBactRule := `Diff2 ; `DAP_Box) ;

  (* DAP production (exclusively for somatic cells) *)
  `DAP_Box ={ C = fstOrder(K_DAPiP,1.0) }=> (appliedBactRule := `DAPiP ; `DAP_Box, `DAP) ;

  (* DAP externalization *)
  `DAP ={ C = fstOrder(K_DAPEx,1.0) }=> (appliedBactRule := `DAPEx  ; return(`DAP :: (diff(self,`DAP)) :: seq:())) ;
  
} ;;

appliedEnvRule := `Unkown ;;
nbReleasedDAP  := 0 ;;
deadBactId     := -1 ;;
trans inEnv[upBact,current_t,current_sc] = {

  (* DAP internalization *)
  `DAP, b:Bact   ={ A = (\c.(sndOrder(K_DAPIm,1.0) * cptDAPe * cptBact)) }=> (appliedEnvRule := `DAPIm ; (upBact(current_sc,(`DAP :: b))) ) ;
  
  (* DAP external degradation *)
  `DAP ={ C = fstOrder(K_DAPeD,1.0) }=> (appliedEnvRule := `DAPeD ; <undef>) ;

  (* BactS death *)
  (x:BactS \/ {id=id}) as b ={ A = (\c.(fstOrder(K_Death,1.0)*cptBactS)) }=> (
    let nbDAP = count(`DAP,b) in
      nbReleasedDAP := nbDAP ;
      appliedEnvRule := `Death ;
      deadBactId := id ;
      map((\e.(`DAP)),iota(nbDAP,seq:()))
  ) ;

  (* BactG mitosis *)
  (_:BactG \/ ({id=id})) as b ={ A = (\c.(fstOrder(K_Div,1.0)*cptBactG)) }=> (
    appliedEnvRule := `Div ;
    let b1b2 = divide_bactG(current_t+'tau,b,id) in
    let b1 = upBact(current_sc,b1b2.(0)) in
    let b2 = upBact(schedule,b1b2.(1)) in
      b1 :: (b2 :: seq:())
  ) ;

} ;;

fun cptUpdate[nbRD](r) = (  // Stochiometric update for the output
  switch (r)
    case `CreD:   decrCre()
    case `DAPiD:  (decrDAPi() ; decrDAP())
    case `DAPApI: (decrDAPi() ; decrDAP())
    case `DAPApA: (incrDAPi() ; incrDAP())
    case `CreP1:  incrCre()
    case `CreP2:  incrCre()
    case `Diff1:  decrCre()
    case `Diff2:  (decrCre(); BactG_BactS())
    case `DAPiP:  (incrDAPi() ; incrDAP())
    case `DAPEx:  DAPi_DAPe()
    case `DAPIm:  DAPe_DAPi()
    case `DAPeD:  (decrDAPe() ; decrDAP())
    case `Death:  (decrNDAPi(nbRD) ; incrNDAPe(nbRD) ; decrBactS() ; decrBact())
    case `Div:    (incrBactG() ; incrBact())
    default: raise (`Erreur ("Diet_coli: Not yet implemented: rule "+r))
  endswitch
) ;;





fun dynamic_help() = (
  stdout << "\n** Help of the \"dynamic\" module **\n\n" ;
  stdout << "Requirements\n" ;
  stdout << "\t- module \"globals\"\n" ;
  stdout << "\t- module \"static\"\n\n" ;
  stdout << "Globals\n" ;
  stdout << "\t- appliedBactRule (ruleId): takes the id of a rule after the application of inBact [`Ùnknown]\n\n" ;
  stdout << "\t- appliedEnvRule (ruleId): takes the id of a rule after the application of inEnv [`Ùnknown]\n\n" ;
  stdout << "Functions\n" ;
  stdout << "\t- fstOrder(k,v) (float * float -> float):\n\t\tkinetics to stochastic conversion\n\n" ;
  stdout << "\t- sndOrder(k,v) (float * float -> float):\n\t\tkinetics to stochastic conversion\n\n" ;
  stdout << "\t- sndOrder(k,v) (float * float -> float):\n\t\tkinetics to stochastic conversion\n\n" ;
  stdout << "\t- inBact(b) (Bact -> Bact | `DAP*Bact):\n\t\tchemical reactions in bacteria ; to be applied with strategy `gillespie\n\n" ;
  stdout << "\t- inEnv[update_bact(Bact -> Bact)](e) (Env -> Env):\n\t\tchemical reactions in environment ; the option update_bact is to update the state of a bacterium modified by the application of the rule\n\n" ;
  stdout << "\t- cptUpdate(r) (ruleId -> unit):\n\t\tupdate of the global counters from \"static\" w.r.t. the applied rule r\n\n"
) ;;
</pre></code>

==ExtSSA==
<code><pre>
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
(* System evolution algorithm                                                                     *)
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)
(* The model is a one depth level P system. Our alogrithm consists in applying the gillespie SSA  *)
(* on each bacterium. A bacterium contains the current state together with the next step and the  *)
(* corresponding reaction delay. The SSA is applied on the environment providing a reaction time. *)
(* The evolution is done where the delay is the smallest.                                         *)
(* ********************************************************************************************** *)
(* ********************************************************************************************** *)

trans no_next_state = { { next_state } as x => x + { next_state = `Unknown, date=0.0 } } ;;

trans update_bact[t,sc] = {
  { id=id } as x => (
    let date     = 0.0 in
    let nx_state = inBact[strategy=`gillespie,
			  postlude=(\c.(date := t+'tau ; c)),
			  iter=1](no_next_state(self))
    in
      schedule        := schedule_replace_entry(id,date,appliedBactRule,sc) ;
      appliedBactRule := `Unkown ;
      x + { date=date, next_state = nx_state }
  )
} ;;

trans evolve_bact[t,sc,idEv] = {
  ((b:Bact \/ ({id=x,next_state=nxs}))) as bb / (idEv == x) => (
    if (Bact(nxs))
    then update_bact[t=t,sc=sc](nxs)
    else (nxs.(0)::(update_bact[t=t,sc=sc](nxs.(1)) :: seq:())) fi
  )
} ;;

fun step[get_global_time,set_global_time,sc](env) = (
  t := get_global_time() ;
  let date' = 0.0 in
  let env'  = inEnv[upBact=(\sc.\x.(let t' = t+'tau in update_bact[t=t',sc=sc,strategy=`default](x))),
		    current_t = t,
		    current_sc = sc,
		    strategy=`gillespie,
		    postlude=(\c.(date' := t + 'tau ; c)),
		    iter=1](env)
  in
    //!! (stdout << "schedule.date=" << schedule.date << " > date'=" << date' << "\n" ; (date' == infinity) || (date' < schedule.date)) ;
    if (date' < sc.date)
    then (
      //stdout << "envionment evolution (applied rule = " << appliedEnvRule << "(" << nbReleasedDAP << "), date = " << date' << "(" << t << "))\n" ;
      cptUpdate[nbRD=nbReleasedDAP](appliedEnvRule) ;
      schedule       := if (appliedEnvRule==`Death) then schedule_remove_entry(deadBactId,schedule) else schedule fi ;
      nbReleasedDAP  := 0 ;
      deadBactId     := -1 ;
      appliedEnvRule := `Unkown
    )
    else (
      //stdout << "bacterium evolution (id = " << sc.id << ", applied rule = " << sc.rule << ", date = " << sc.date << "(" << t << "))\n" ;
      cptUpdate(sc.rule) ;
      nbReleasedDAP  := 0 ;
      deadBactId     := -1 ;
      appliedEnvRule := `Unkown ;
      date' := sc.date ;
      env'  := evolve_bact[strategy=`asynchronous,sc=sc,idEv=sc.id,t=date'](env)
    ) fi ;
    //stdout << env' << "\n" ;
    sc := schedule ;
    set_global_time(date') ;
    env'
) ;;


fun evol_algo_help() = (
  stdout << "\n** Help of the \"evol_algo\" module **\n\n" ;
  stdout << "Requirements\n" ;
  stdout << "\t- module \"schedule\"\n" ;
  stdout << "\t- module \"dynamic\"\n\n" ;
  stdout << "Functions\n" ;
  stdout << "\t- update_bact[t=t(float),sc=sc(schedule)](b) (Bact -> Bact):\n\t\tcompute the next Gillespie step of a bacterium b. The schedule sc is updated and the new value of sc is returned in the global variable schedule. The option t has to be set to the global date.\n\n" ;
  stdout << "\t- evolve_bact[t=t(float),sc=sc(schedule),idEv=id(int)](env) (Env -> Env):\n\t\tmake the bacterium with identifiant id evolve. The schedule sc is updated and the new value of sc is returned in the global variable schedule. The option t has to be set to the global date.\n\n" ;
  stdout << "\t- step[t=t(float),sc=sc(schedule)](env) (Env -> Env):\n\t\tOne step evolution of the whole system. The earliest event is applied in the environment env w.r.t. the schedule sc. The option t has to be set to the global date.\n\n"

) ;;
</pre></code>

==Output for Gnuplot==
<code><pre>
  then (
    let tg = get_global_time() in
      GPoutput << get_global_time() << "\t" <<
      cptCre << "\t" <<
      cptDAP << "\t" <<
      cptDAPe << "\t" <<
      (cptDAPi/cptBact) << "\t" <<
      cptBact << "\t" <<
      cptBactS << "\t" <<
      cptBactG << "\n" ;
      stdout << "Iteration: " << 'iteration << " and Global time: " << tg << " and bacteria population: " << cptBact << "\n"
  )
  else <undef> fi ;
  env
) ;;

fun dataToPng() = (
  gnuplot << "set key outside\n";
  gnuplot << "set xlabel \"Time (arbitrary unit)\"\n";
  gnuplot << "set ylabel \"Number of molecules or structures\"\n";
  gnuplot << "set output \""+ PNGoutput + "\"\n";
  gnuplot << "set term png\n";
  
  gnuplot << "plot " +
  "\""+ GPoutput +"\" using 1:4 title 'DAPe' with lines 3," +
  "\""+ GPoutput +"\" using 1:5 title 'DAPi' with lines 4," +
  "\""+ GPoutput +"\" using 1:6 title 'Bact' with lines 5," +
  "\""+ GPoutput +"\" using 1:7 title 'BactS' with lines 6," +
  "\""+ GPoutput +"\" using 1:8 title 'BactG' with lines 7" +
  "" ;
  gnuplot << "\nquit" ;
  gnuplot << "\n"

) ;;

fun dataToPngDAP() = (
  gnuplot << "set key outside\n";
  gnuplot << "set xlabel \"Time (arbitrary unit)\"\n";
  gnuplot << "set ylabel \"Number of molecules or structures\"\n";
  gnuplot << "set output \""+ PNGoutput + "\"\n";
  gnuplot << "set term png\n";
  
  gnuplot << "plot " +
  "\""+ GPoutput +"\" using 1:4 title 'DAPe' with lines 3," +
  "\""+ GPoutput +"\" using 1:5 title 'DAPi' with lines 4" +
  "" ;
  gnuplot << "\nquit" ;
  gnuplot << "\n"

) ;;

fun dataToPngBact() = (
  gnuplot << "set key outside\n";
  gnuplot << "set xlabel \"Time (arbitrary unit)\"\n";
  gnuplot << "set ylabel \"Number of molecules or structures\"\n";
  gnuplot << "set output \""+ PNGoutput + "\"\n";
  gnuplot << "set term png\n";
  
  gnuplot << "plot " +
  "\""+ GPoutput +"\" using 1:6 title 'Bact' with lines 5," +
  "\""+ GPoutput +"\" using 1:7 title 'BactS' with lines 6," +
  "\""+ GPoutput +"\" using 1:8 title 'BactG' with lines 7" +
  "" ;
  gnuplot << "\nquit" ;
  gnuplot << "\n"

) ;;




fun output_help() = (
  stdout << "\n** Help of the \"output\" module **\n\n" ;
  stdout << "Requirements\n" ;
  stdout << "\t- module \"static\"\n" ;
  stdout << "Globals\n" ;
  stdout << "\t- GPoutput (string): Gnuplot formatted output file [\"/tmp/sortie.dat\"]\n\n" ;
  stdout << "\t- PNGoutput (string): PNG file for Gnuplot exporting (see function dataToPng) [\"/tmp/sortie.png\"]\n\n" ;
  stdout << "\t- gnuplot (string): absolute path to Gnuplot executable [\"/usr/bin/gnuplot\"]\n\n" ;
  stdout << "Functions\n" ;
  stdout << "\t- export_cpt[get_global_time](freq, env) (int * Env -> Env):\n\t\tExport in GPoutput the state of global module counters (to be used as interlude of an iterated function)\n\n" ;
  stdout << "\t- dataToPng() (unit -> unit):\n\t\texport GPoutput to the PNG file PNGoutput using Gnuplot (located at path gnuplot)\n\n"
) ;;
</pre></code>

==Main program==
<code><pre>
!set match_random := 100 ;;

duration  := 1500.0 ;;
gt        := 0.0      ;; //global time
fun ggt() = gt ;; //global time set and get function... an assert is used to check that time only increases
fun sgt(x) = (!! (gt <= x) ; gt := x) ;;

!include "globals.mgs"   ;;
!include "static.mgs"    ;;
!include "schedule.mgs"  ;;
!include "dynamic.mgs"   ;;
!include "evol_algo.mgs" ;;
!include "output.mgs"    ;;

schedule := schedule_empty() ;;

env := map_bact((\b.(update_bact[t=gt,sc=schedule](b))),new_env(gt,10,0)) ;;

step[fixpoint=(\c1.\c2.(gt > duration || cptBact == 0)), interlude=B(export_cpt[get_global_time=ggt](10),I(*set_DAPe(nb_dape)*)), sc=schedule, get_global_time=ggt, set_global_time=sgt](env) ;;

dataToPng() ;;
</pre></code>