Program description
This Mathematica program simulates individual actin-mEos2 trajectories within a 2D geometry representing a typical growth cone. The trajectory length is n= 3200 steps of 50 ms, and the time during which the moelcule is emitting fluorescence is chosen in a exponential distribution with average 1.8 s. The actin molecule alternates between periods of free diffusion as monomers (‘outside’), and periods of flow when incorporated into filaments (‘inside’). When flowing, it can interact transiently with substrate adhesion (‘bound’). The transitions are governed by 4 kinetic rates. The output of the program is a text file (sortie.dat) with 6 columns (simulation number, frame number, x corrdinate (µm), y coordinate (µm), area (10 pixels), intensity (100 arbitrary units)). It is saved by default in the Mathematica folder. Before excecuting the program, please remove the notes in blue.

**********************************************************************************

<<Statistics`ContinuousDistributions`
ndist=NormalDistribution[0,1];

*Dimensions of the simulations
n=3200;
delta=0.05;
nsimul=400;
step=0;

*Photoactivation and photobleaching processes
trajectorylength=1.817;
bleaching=ExponentialDistribution[1/trajectorylength];
emission=Table[0,{nsimul}];
For[h=1,h<nsimul+1,h++,
    emission=ReplacePart[emission,Round[(n-100)*Random[]],h];
    ];
extinction=Table[0,{nsimul}];
For[h=1,h<nsimul+1,h++,
    extinction=ReplacePart[extinction,Round[Random[bleaching]/delta],h];
    ];
extended=Table[0,{nsimul+1}];
For[h=1,h<nsimul+1,h++,
    extended=ReplacePart[extended,Extract[extinction ,h],h+1];
    ];
longueur=Sum[Extract[extinction,k],{k,1,nsimul}];

*Data file definition
fichier=Table[{1,1,7,5,10,0},{longueur}];

*Diffusion and flow coefficients
outside=3;
free=Sqrt[2*outside*delta];
inside=0.001;
bound=Sqrt[2*inside*delta];
lateral=0.003;
flowing=Sqrt[2*lateral*delta];
vitesse=0.125;

**kinetic rates
ka=0.3;
attach=ka*delta;
kd=0.03;
detach=kd*delta;
kc=1; 
coupl=kc*delta;
ku=0.02;
uncoupling=ku*delta;

*Growth cone geometry
front=1;
base=4;
espace=7;
width=5;
rayon=2*width;
theta=N[ArcSin[espace/rayon]];
zoom=10;

*Simulation loop
nombre[0]=Random[];
For[i=1,i<nsimul+1,i++,
    Print[i];
*Initial conditions
    theta0=2*theta*(Random[]-0.5);
    rayon0=base +(rayon-base)*Random[];
    xexo=rayon0*Sin[theta0];
    ygrec0=rayon0*Cos[theta0]-width;
    absc=Table[xexo,{n}];
    ord=Table[ygrec0,{n}];
    
    flow=True;
    For[j=1, j<n+1,j++,
      binding=False;
      x[1]=xexo;
      y[1]=ygrec0;
      t[1]=theta0;
      r[1]=rayon0;
      
      numen=Random[];
      x1=Random[ndist];
      y1=Random[ndist];
      
      If[rayon-front<r[j-1]<rayon\[And]-theta+0.05<t[j-1]<theta-0.05 \[And]
          coupl>numen,flow=True];
      
      If[flow==True,
        
        If[attach>nombre[j-1] ,
          	
          x[j_]=x[j-1]+bound*x1;
          y[j_]=y[j-1]+bound*y1;
          	
          r[j_]=Sqrt[(x[j])^2+(y[j]+width)^2];
          t[j_]=ArcTan[x[j]/(y[j]+width)];
          
          absc=ReplacePart[absc,x[j],j];
          ord=ReplacePart[ord,y[j],j];
          binding=True;
          If[detach<numen,nombre[j_]=nombre[j-1],nombre[j_]=Random[]];
          ];
        
        If[ binding==False,
          x[j_]=(r[j-1]- vitesse*delta)*Sin[t[j-1]]+flowing*x1;
          y[j_]=(r[j-1]- vitesse*delta)*Cos[t[j-1]]-width+flowing*y1;
          r[j_]=Sqrt[(x[j])^2+(y[j]+width)^2];
          t[j_]=ArcTan[x[j]/(y[j]+width)];
          absc=ReplacePart[absc,x[j],j];
          ord=ReplacePart[ord,y[j],j];
          nombre[j_] =Random[];
          ];
        
        If[r[j-1]<base\[Or]uncoupling>numen,flow=False];
        ];
      
      If[flow==False ,
        
        While[
          Sqrt[(x[j-1]+free*x1)^2+(y[j-1]+free*y1+width)^2]>
              rayon \[Or]Abs[ArcTan[(x[j-1]+free*x1)/(y[j-1]+free*y1+width)]]>
              theta\[Or]y[j-1]+free*y1<-width,{x1,y1}={Random[ndist],
              Random[ndist]}];
        x[j_]=x[j-1]+free*x1;
        absc=ReplacePart[absc,x[j],j];
         y[j_]=y[j-1]+free*y1;
        ord=ReplacePart[ord,y[j],j];
        r[j_]=Sqrt[(x[j])^2+(y[j]+width)^2];
        t[j_]=ArcTan[x[j]/(y[j]+width)];
        ];
      
      If[Extract[emission,i]-1<j<Extract[emission,i]+Extract[extinction,i],
        fichier=
            ReplacePart[
              fichier,{i+step,j,zoom*(espace+Extract[absc,j]),
                zoom*(width+Extract[ord,j]),10,100},
              Sum[Extract[extended,h],{h,1,i}]+j-Extract[emission,i]+1];
        ];
      
      ];
    ];

*Data export
Export["sortie.dat",fichier,"Table"];