Simulation_main.m

%% PDE simulation bead case

c0= 20; % initial nutrient concentration

etaC = @(x)  0.05 * x./(1.2-x); % function to compute eta_i
etaS2 = etaC(1);

n1_v12_long = zeros(26,8);
n2_v12_long = zeros(26,8);
S1_v12_long = zeros(26,8);
S2_v12_long = zeros(26,8);

k = 1;
x = 40;
for l = 0.3:0.1:1
    etaS1 = etaC(l);
    try
        [n1,n2,S1,S2] = simuV1_2(etaS1,etaS2,c0,x);
        f = numel(n1);
        n1_v12_long(1:f,k) =  n1(:);
        n2_v12_long(1:f,k) =  n2(:);
        S1_v12_long(1:f,k) =  S1(:);
        S2_v12_long(1:f,k) =  S2(:);
    catch ME
        warning(['Problem for l=',num2str(l),' in solving equations. Problem is:\n',ME.message]);
    end
    k=k+1;
end

%% ODE simulation secondary metabolites

etaC =  @(x,c) 1/c * x./(1.2-x);
c0 = 20;

l = 0.1:0.1:1;

maxS2=[];
gr_odeAlph_M = zeros(numel(l),1);
for k=1:numel(l) 
    
    eta1 = etaC(l(k),20);
    eta2 = 0.5;
    alpha =  0.2;

    odeSim = @(t,y) odeOfSimu_Alpha(t,y,eta1,eta2,alpha);

    [t,y] = ode45(odeSim, [0 30], [1e-2;c0;0.0001]);

    figure(5)
    hold on
    plot(t,y(:,1))
    figure(6)
    hold on
    plot(t,y(:,2))
    plot(t,y(:,3))

    maxS2 = [maxS2;max(y(:,3))];

    x1 = eta1*y(:,2);
    x2 = eta2*y(:,3);
    gr_ode = (x1+x2)./(x1+x2+1);
    figure(7)
    hold on
    plot(y(:,1),1.2 * gr_ode)
    
    gr_odeAlph_M(k) = 1.2 * mean(gr_ode(y(:,1)>0.05 & y(:,1)<0.5));
end

%% PDE simulation secondary metabolite case

etaC = @(x,c) 1/c * x./(1.16-x);
tf = [33,25,20,17,14,12,11,10,9]; % simulation lengths

c0 = 20;

nW_v5_12_AlphChi_sl  = zeros(tf(1)+1,10,1);
nY_v5_12_AlphChi_sl  = zeros(tf(1)+1,10,1);

S1_v5_12_AlphChi_sl = zeros(tf(1)+1,10,1);
S2_v5_12_AlphChi_sl = zeros(tf(1)+1,10,1);

etaS2 = 0.5; 
k = 1;
for l = 0.2:0.1:1
    try
        etaS1 = etaC(l,20);
        
        alpha = 0.5;
        chi = 35 * (2 - gr_odeAlph_M(k)/1.2);
        
        tic
        [u,model,res] = simuV5_12_AlphaChi(etaS1,etaS2,c0,tf,alpha,chi);

        toc

        nW_v5_12_AlphChi_sl( 1:(tf(k)+1) ,k) = squeeze(mean(u(:,1,:),1));
        nY_v5_12_AlphChi_sl( 1:(tf(k)+1) ,k) = squeeze(mean(u(:,2,:),1));

        S1_v5_12_AlphChi_sl( 1:(tf(k)+1) ,k) = squeeze(mean(u(:,3,:),1));
        S2_v5_12_AlphChi_sl( 1:(tf(k)+1) ,k) = squeeze(mean(u(:,4,:),1));

        savefig(figure(2),['Maps_v5_12_AlphChi_l-',num2str(l),'_e2-',num2str(etaS2),'.fig']);
        savefig(figure(3),['Ncells_v5_12_AlphChi_l-',num2str(l),'_e2-',num2str(etaS2),'.fig']);
        savefig(figure(4),['CC_v5_12_AlphChi_l-',num2str(l),'_e2-',num2str(etaS2),'.fig']);

        close(figure(1))
        close(figure(2))
        close(figure(3))
        close(figure(4))
    catch ME
        warning(['Problem for l=',num2str(l),' in solving equations. Problem is:\n',ME.message]);
    end
    k = k+1;
end


simuV5_12_AlphaChi.m

function [u,model,res] = simuV5_12_AlphaChi(etaS1,etaS2,c0,tf,alpha,chi)
    
    % Units of length = 100 µm, Unit of time = 1 h, unit of cc = 1mM 
    normSize = 50;
    
    %Dc = 9*36; % 900 µm2/s
    %Ds = 5*36; % 500 µm2/s
    %Chi = 40*36; % 4 10^3 µm2/s
    % KE = 1e-3; % in mM from R. Endres model
    % n0 = 2e-3; % initial OD 
    
    % c0 should be about 20 mM according to mat and meth
    
    times = [0.001,1:tf];

    % creating the pde system
    model = createpde(4);
    
    R1 = [2;4;0;0;normSize;normSize;0;normSize;normSize;0];
    geometryFromEdges(model,decsg(R1));
    generateMesh(model,'Hmax',1,'Hmin',0.01,'GeometricOrder','quadratic'); %'quadratic'
    
    figure(1)
    pdegplot(model,'EdgeLabels','on')
    hold on
    pdeplot(model)
    axis equal
    
    CA = specifyCoefficients(model,'m',0,'d',1,'c',@Dcoef,'a',@GrowthCoeff,'f',@fcoeff);
 
    setInitialConditions(model,@initCond);
    
    q2 = [1.8,0,0,0;
          0,1.8,0,0;
          0,0, 0 ,0;
          0,0,0, 0 ];
    
    
    applyBoundaryCondition(model,'neumann','Edge',2,'q',q2); % to ensure no flux under gravity
    applyBoundaryCondition(model,'neumann','Edge',4,'q',-q2);
    
    
    model.SolverOptions.RelativeTolerance=1e-3;
    model.SolverOptions.AbsoluteTolerance=1e-6;
    model.SolverOptions.MaxIterations = 30000;
    model.SolverOptions.ReportStatistics='on';
    model.SolverOptions.MinStep=0;
    
    res=solvepde(model,times);
    u = res.NodalSolution;
    
    figure(2)
    subplot(2,2,1)
    pdeplot(model,'xydata',u(:,1,end)); %.*(u(:,1,end)>0&u(:,1,end)<0.2)
    title('WT')
    subplot(2,2,2)
    pdeplot(model,'xydata',u(:,2,end));
    title('CheY')
    subplot(2,2,3)
    pdeplot(model,'xydata',u(:,3,end));
    title('S1')
    subplot(2,2,4)
    pdeplot(model,'xydata',u(:,4,end));
    title('S2')
    
    n1 = squeeze(mean(u(:,1,:),1)) ;
    n2 = squeeze(mean(u(:,2,:),1)) ;
    figure(3)
    semilogy(times,n1)
    hold on
    semilogy(times,n2)
    S1 = squeeze(mean(u(:,3,:),1));
    S2 = squeeze(mean(u(:,4,:),1));
    figure(4)
    plot(times,S1)
    hold on
    plot(times,S2)
    
    
    function D = Dcoef(region,state)
        
        Dc = 3*36*1; % swimers 300 µm2/s
        Ds = 5*36*1; % nutrient 500 µm2/s
        Chi = -chi*36*1; % negative for positive chemotaxis
        Kn = 1e-3;
    
        nr = length(region.x);
        u1 = state.u(1,:);
        u3 = state.u(3,:);
        u4 = state.u(4,:);
        un = ones(1,nr);
       
        D = zeros(64,nr);
        D( 1,:) = Dc*un;
        D( 4,:) = Dc*un;
        D(21,:) = Dc*un;
        D(24,:) = Dc*un;
        
        D(33,:) = u1.*Chi./(u3+Kn);  %u1 is the chemotactic cells
        D(36,:) = u1.*Chi./(u3+Kn);
        
        D(49,:) = u1.*Chi./(u4+Kn);  %u1 is the chemotactic cells
        D(52,:) = u1.*Chi./(u4+Kn);
        
        D(41,:) = Ds*un;
        D(44,:) = Ds*un;
        D(61,:) = Ds*un;
        D(64,:) = Ds*un;
    end
    %         D=[Dc,0,0,0,u1Chi/u5;
%             0,Dc,0,0,0;
%             0,0,Dn,0,0;
%             0,0,0,Dn,0;
%             0,0,0,0,Ds];
    
    function a = GrowthCoeff(region,state)
        
        beta = 13; % in mM/OD 
        lc = 1.2*1;
        
        nr = numel(region.x);
        
        u3 = state.u(3,:);
        u4 = state.u(4,:);
        nS1 = etaS1 .* u3;
        nS2 = etaS2 .* u4;
        
        Ktot =(1 + nS1+nS2);
        nS1 = lc* nS1./Ktot;
        nS2 = lc* nS2./Ktot;
        
        l12= - (nS1 +nS2);
        
        bl1 = beta *  nS1/alpha;
        bl2 = beta *  nS2 - (1-alpha) * bl1; 
        
        a=[     l12    ;zeros(1,nr);    bl1    ;    bl2; % column1
            zeros(1,nr);    l12    ;    bl1    ;    bl2; % column2
            zeros(1,nr);zeros(1,nr);zeros(1,nr);zeros(1,nr); % column3
            zeros(1,nr);zeros(1,nr);zeros(1,nr);zeros(1,nr) % column4
            ];
    end
%         a = [l12 (1), 0  , 0,0;
%             0    (2), l12, 0,0;
%             bl1  (3), bl1, 0,0;
%             bl2  (4), bl2, 0,0
%             ];

    function f = fcoeff(region,state)
        nr = numel(region.x);
        vsed = 1.80; % 180 µm/h
        f = zeros(4,nr);
        f(1,:) = vsed*state.uy(1,:);
        f(2,:) = vsed*state.uy(2,:);
%         f(3,:) = vsed*state.uy(3,:);  % no sedimentation of the nutrient
%         f(4,:) = vsed*state.uy(4,:);
    end
    
    function ci = initCond(locations)
        
        n0 = 2e-3;
        dn0 = 2*1e-4;
        
        nr = numel(locations.x);
        un = ones(1,nr);
        
        ci = [n0 + dn0*(rand(1,nr)-0.5); n0 + dn0*(rand(1,nr)-0.5); c0*un; 0.0001*un];
        % very small s2 initial cc to avoid numerical error negative values
    end
    
    
end

simuV1_2.m

function [n1,n2,S1,S2] = simuV1_2(etaS1,etaS2,c0,x)
    
    
    % Units of length = 100 µm, Unit of time = 1 h, unit of cc = 1mM 
    normSize = 50;
    
    %Dc = 9*36; % 900 µm2/s
    %Ds = 5*36; % 500 µm2/s
    
    n0 = 2e-3; % initial OD //previous value 0.1 area fraction = 10^3 /100*100
    
   % c0 should be about 20 mM according to mat and meth
    
    ci = [n0;n0;c0;0.001];
    
    GR = 20*etaS1/(1+20*etaS1) * 1.2;
    tf = floor(7.5/GR); % simulation duration
    
    times = [0.001,1:tf]; 

    % creating the pde system
    model = createpde(4);
    
    R1 = [2;4;0;0;normSize;normSize;0;normSize;normSize;0];
    geometryFromEdges(model,decsg(R1));
    
    generateMesh(model,'Hmax',0.5,'Hmin',0.01,'GeometricOrder','quadratic'); %'quadratic'
    
    figure(1)
    pdegplot(model,'EdgeLabels','on')
    hold on
    pdeplot(model)
    axis equal
    
    CA = specifyCoefficients(model,'m',0,'d',1,'c',@Dcoef,'a',@GrowthCoeff,'f',[0;0;0;0]);
 
    setInitialConditions(model,ci);
    
    % Influx of nutrient 2
    applyBoundaryCondition(model,'neumann','Edge',1, 'g',[0;0;0;normSize*0.2],'q',0);
    % dc/dt = dN/dt/hL2 = hLg/hL2 = g/L => g = L*2mM/10h
    
    model.SolverOptions.RelativeTolerance=1e-3;
    model.SolverOptions.AbsoluteTolerance=1e-6;
    model.SolverOptions.MaxIterations = 30000;
    model.SolverOptions.ReportStatistics='on';
    model.SolverOptions.MinStep=0;
    
    res=solvepde(model,times);
    u = res.NodalSolution;
    
    figure(2)
    subplot(2,2,1)
    pdeplot(model,'xydata',u(:,1,end)); 
    subplot(2,2,2)
    pdeplot(model,'xydata',u(:,2,end));
    subplot(2,2,3)
    pdeplot(model,'xydata',u(:,3,end));
    subplot(2,2,4)
    pdeplot(model,'xydata',u(:,4,end));
    
    n1 = squeeze(mean(u(:,1,:),1));
    n2 = squeeze(mean(u(:,2,:),1));
    figure(3)
    semilogy(times,n1)
    hold on
    semilogy(times,n2)
    S1 = squeeze(mean(u(:,3,:),1));
    S2 = squeeze(mean(u(:,4,:),1));
    figure(4)
    plot(times,S1)
    hold on
    plot(times,S2)
    
    function D = Dcoef(region,state)
        
        Dc = 3*36*1; % 900 µm2/s
        Ds = 5*36*1; % 500 µm2/s
        Chi = -x*36*1; % 4 10^3 µm2/s negative for positive chemotaxis
        Kn = 1e-3;
    
        nr = length(region.x);
        u2 = state.u(2,:);
        u3 = state.u(3,:);
        u4 = state.u(4,:);
        un = ones(1,nr);
       
        D = zeros(64,nr);
        D( 1,:) = Dc*un;
        D( 4,:) = Dc*un;
        D(21,:) = Dc*un;
        D(24,:) = Dc*un;
        D(37,:) = u2.*Chi./(u3+Kn);  % u2 is the chemotactic cells
        D(40,:) = u2.*Chi./(u3+Kn);
        D(41,:) = Ds*un;
        D(44,:) = Ds*un;
        D(53,:) = u2.*Chi./(u4+Kn);
        D(56,:) = u2.*Chi./(u4+Kn);
        D(61,:) = Ds*un;
        D(64,:) = Ds*un;
    end
%         D=[Dc,0,0,0;
%            0,Dc,u2Chi/u3,u2Chi/u4;
%            0,0,Ds,0;
%            0,0,0,Ds];
    
    function a = GrowthCoeff(region,state)
        
		beta = 13; % in mM/OD 
        lc = 1.2*1;
        
        nr = numel(region.x);
        u3 = state.u(3,:);
        u4 = state.u(4,:);
        nS1 = etaS1 .* u3;
        nS2 = etaS2 .* u4;
        
        Ktot =(1 + nS1+nS2);
        nS1 = nS1./Ktot;
        nS2 = nS2./Ktot;
		
        l12= -lc* (nS1+nS2);
        
        bl1 = beta * lc* (nS1);
        bl2 = beta * lc* (nS2);
        
        a=[ l12; zeros(1,nr); bl1; bl2;
            zeros(1,nr); l12; bl1; bl2;
            zeros(1,nr);zeros(1,nr);zeros(1,nr);zeros(1,nr); % column3
            zeros(1,nr);zeros(1,nr);zeros(1,nr);zeros(1,nr) % column4
            ];
    end
%        a = [l12 (1), 0  , 0,0;
%             0    (2), l12, 0,0;
%             bl1  (3), bl1, 0,0;
%             bl2  (4), bl2, 0,0
%             ];
    
end

odeOfSimu_Alpha.m

function dy = odeOfSimu_Alpha(t,y,eta1,eta2,alpha)

    dy = zeros(3,1);

    beta = 13;
    x1 = eta1*y(2);
    x2 = eta2*y(3);
    r = 1+x1+x2;
    l1 = x1./r;
    l2 = x2./r;

    dy(1) = (l1+l2).*y(1);
    dy(2) = -beta.*        l1./alpha          .*y(1);
    dy(3) = -beta.* (l2-l1.*(1-alpha)/alpha)  .*y(1);

end