%Monolayer Energy and junctional tension on neighbor cells during apoptotic cell extrusion occurs %full description of variables and equations in in the Computational %Supplement. %the energy calculation is derived from the Hamiltonian %E_i=-J.pi+K.pi^2+(ai-ao)^2 %general variables a0=1; %equilibrium cell area eta=1; % deviation of cell density from the equillibiurm one ai=eta*a0; % area of cells pi=sqrt(24*ai/sqrt(3)); %perimeter of cells nu_astk=0.0721; %topology factor for equilibrium condition J=1; K_J_ratio(:,1)=(0.00:0.1:1); %range in whihc the K/J ratio wants to be varied %t is a time like variable that marks the progress of extrusion and rosette solution after it forms %t=0 correspond to the start of the calculation. In here, all cells are in %a perfect hexagonal packing. %t=1 is extrusion complete with neighboour cells forming a rosette %t=2 defines a time where cells disolve the rosette forming new tricellular junctions t(:,1)=(0:0.01:1.50); tsize=size(t,1); numbKJratiovalues=size(K_J_ratio,1); %preallocation of variables NOTE: noext=="no extruding cells"; %ext=="extruding cells". p_noext=zeros(tsize,numbKJratiovalues); %perimeter a_noext=zeros(tsize,numbKJratiovalues); %area nu_noext=zeros(tsize,numbKJratiovalues); %topology factor E_noext_percell=zeros(tsize,numbKJratiovalues); %energy T_noext_percell=zeros(tsize,numbKJratiovalues); %junctional tension p_ext=zeros(tsize,numbKJratiovalues); a_ext=zeros(tsize,numbKJratiovalues); nu_ext=zeros(tsize,numbKJratiovalues); E_ext_percell=zeros(tsize,numbKJratiovalues); T_ext_percell=zeros(tsize,numbKJratiovalues); Etot=zeros(tsize,numbKJratiovalues); %total energy aveE=zeros(tsize,numbKJratiovalues); %average energy Delta_Etot=zeros(tsize,numbKJratiovalues); %Energy minus Energy at time zero aveDeltaE=zeros(tsize,numbKJratiovalues); %Delta_Etot average per cell Atot=zeros(tsize,numbKJratiovalues); %total area Ttot=zeros(tsize,numbKJratiovalues); %sum of tension across all junctions aveT_percell=zeros(tsize,numbKJratiovalues); %average tension Delta_Ttot=zeros(tsize,numbKJratiovalues); %tension minus tension at time ==0 aveDeltaT=zeros(tsize,numbKJratiovalues); for i=1:numbKJratiovalues for j=1:tsize K=J*K_J_ratio(i,1); if t(j,1)<1 %here add the conditions for cells before extrusion %Perimeter, area and energy of non extruding cells p_noext(j,i)=pi.*(1+t(j,1)/6); a_noext(j,i)=ai.*(1+(2-t(j,1)).*t(j,1)/6); nu_noext(j,i)=a_noext(j,i)./(p_noext(j,i).^2); E_noext_percell(j,i)=-J.*p_noext(j,i)+K.*(p_noext(j,i).^2)+(a_noext(j,i)-a0).^2; T_noext_percell(j,i)=-J+2*K.*(p_noext(j,i))+4*a_noext(j,i)*nu_noext(j,i)*p_noext(j,i); %Perimeter, area and energy of extruding cells p_ext(j,i)=pi.*sqrt(1-(2-t(j,1)).*t(j,1)); a_ext(j,i)=ai.*(1-(2-t(j,1)).*t(j,1)); nu_ext(j,i)=a_ext(j,i)./(p_ext(j,i).^2); E_ext_percell(j,i)=-J.*p_ext(j,i)+K.*(p_ext(j,i).^2)+(a_ext(j,i)-a0).^2; T_ext_percell(j,i)=-J+2*K.*(p_ext(j,i))+4*a_ext(j,i)*nu_ext(j,i)*p_ext(j,i); %total energy Etot(j,i)=6*E_noext_percell(j,i)+E_ext_percell(j,i); aveE(j,i)=Etot(j,i)./7; %total area Atot(j,i)=6*a_noext(j,i)+a_ext(j,i); %total and average tension Ttot(j,i)=6*T_noext_percell(j,i)+T_ext_percell(j,i); aveT_percell(j,i)=Ttot(j,i)./7; %DeltaE (energy barrier) and DeltaT if t(j,1)==0 Eo=Etot(j,i); To=T_noext_percell(j,i); end Delta_Etot(j,i)=Etot(j,i)-Eo; aveDeltaE(j,i)=Delta_Etot(j,i)./7; Delta_Ttot(j,i)=T_noext_percell(j,i)-To; aveDeltaT(j,i)=Delta_Ttot(j,i)./7; elseif t(j,1)>=1 %here add the conditions for cells after extrusion y=t(j,1)-1; %Perimeter, area and energy of extruding cells p_ext(j,i)=0; a_ext(j,i)=0; nu_ext(j,i)=sqrt(3)/24; E_ext_percell(j,i)=0; T_ext_percell(j,i)=0; %Perimeter, area and energy of non extruding cells is divided in %"pentagonal cells" and "heptagonal cells" %Perimeter, area and energy of "pentagonal cells" p_five=pi*(1/2+1/3*sqrt(1+3*(1-y)^2)); a_five=ai*(7/6-2/3*y); nu_five=a_five/(p_five^2); E_five_percell=-J*p_five+K*p_five^2+(a_five-a0).^2; T_five_percell=-J+2*K.*p_five+4*a_five*nu_five*p_five; %Perimeter, area and energy of "heptagonal cells" p_seven=p_five+pi*y*sqrt(3)*y/3; a_seven=(ai/3)*(7/2+2*y); nu_seven=a_seven/p_seven^2; E_seven_percell=-J*p_seven+K*p_seven^2+(a_seven-a0)^2; T_seven_percell=-J+2*K*p_seven+4*a_seven*nu_seven*p_seven; %average values for non extruding cells a_noext(j,i)=1/2*(a_five+a_seven); p_noext(j,i)=1/2*(p_five+p_seven); nu_noext(j,i)=1/2*(nu_five+nu_seven); E_noext_percell(j,i)=1/2*(E_five_percell+E_seven_percell); T_noext_percell(j,i)=1/2*(T_five_percell+T_seven_percell); %total energy Etot(j,i)=6*E_noext_percell(j,i); aveE(j,i)=Etot(j,i)./6; %total area Atot(j,i)=6*a_noext(j,i)+a_ext(j,i); %total and average tension Ttot(j,i)=6*T_noext_percell(j,i); aveT_percell(j,i)=Ttot(j,i)./6; %DeltaE (energy barrier) and DeltaT Delta_Etot(j,i)=Etot(j,i)-Eo; aveDeltaE(j,i)=Delta_Etot(j,i)./6; Delta_Ttot(j,i)=T_noext_percell(j,i)-To; aveDeltaT(j,i)=Delta_Ttot(j,i)./6; end end end figure, hold on for i=1:1 plot(t(:,1),Atot(:,i),'DisplayName','Total Area') plot(t(:,1),a_ext(:,i),'DisplayName','Area of extruding cell') plot(t(:,1),6.*a_noext(:,i),'DisplayName','Area of "first neighbours"') end legend('show') figure, hold on for i=1:numbKJratiovalues plot(t(:,1),aveDeltaE(:,i),'DisplayName',['K/J ratio=' num2str(K_J_ratio(i))]) end title(['Delta E']) legend('show') figure, hold on for i=1:numbKJratiovalues %plot(t(:,1), T_ext_percell(:,i),'DisplayName','cortical tension in extruding cells "non purse string"') plot(t(:,1), Delta_Ttot(:,i),'DisplayName',['K/J ratio=' num2str(K_J_ratio(i))]) end title(['change in cortical tension in first neighbours']) legend('show') figure, hold on for i=1:numbKJratiovalues %plot(t(:,1), T_ext_percell(:,i),'DisplayName','cortical tension in extruding cells "non purse string"') plot(t(:,1), nu_noext(:,i),'DisplayName',['K/J ratio=' num2str(K_J_ratio(i))]) end title(['nu on first neighbours']) legend('show')