Tutorial on two romantic hearts

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We illustrate the practical advantage of having submodels within a project and an important consequence of understanding biological growth within the GPT-framework.

Conclusion: using a combination of polarity patterns to set local axes for anisotropic growth and patterns of differential specified growth to regulate the growth of shape would be powerful.

Illustrating two independent ways to form shapes and the use of submodels.

The full interaction function is shown at the bottom. The line of code that selects the submodel and the start of each submodel is shown in red.

Uniform specified polariser (no polariser gradient). Creating a shape using a specified pattern of isotropic growth.

Result: simple patterns tend to produce blobby shapes.

Conclusion: Whilst it may be possible to produce complex shapes such as the outgrowth shown in the GFtbox icon, we don't know how.

Pattern of isotropic specified growth (no polariser)
Pattern of isotropic specified growth after growing to 3 times the original area.
Uniform specified growth. Creating a shape using a specified pattern of diffusable polariser.

Result: simple patterns can readily produce sharp shapes.

Conclusion: It is not so easy to produce blobby shapes using patterns of polariser alone.

Pattern of specified polariser levels (green-cyan). Polariser can diffuse and the gradient is arrowed. Uniform specified growth (red).
Patterns and shape after growing to 3 times the original area.
% Section 1
function m = gpt_twowayheart_20110531( m )
%m = gpt_twowayheart_20110531( m )
%   Morphogen interaction function.
%   Written at 2011-05-31 19:51:32.
%   GFtbox revision 3548, 2011-05-31 14:37:10.747930.

% The user may edit any part of this function between delimiters
% of the form "USER CODE..." and "END OF USER CODE...".  The
% delimiters themselves must not be moved, edited, deleted, or added.

    if isempty(m), return; end

    fprintf( 1, '%s found in %s\n', mfilename(), which(mfilename()) );

    try
        m = local_setproperties( m );
    catch
    end

    realtime = m.globalDynamicProps.currenttime;

% Section 2
%%% USER CODE: INITIALISATION

% In this section you may modify the mesh in any way whatsoever.
    if (Steps(m)==0) && m.globalDynamicProps.doinit % First iteration
        % Set up names for variant models.  Useful for running multiple models on a cluster.
        m.userdata.ranges.modelname.range = { 'PolariserBased', 'DifferentialGrowthBased' };  % CLUSTER
         m.userdata.ranges.modelname.index = 1;                       % CLUSTER
    end
    modelname = m.userdata.ranges.modelname.range{m.userdata.ranges.modelname.index};  % CLUSTER
    disp(sprintf('\nRunning %s model %s\n',mfilename, modelname));
    	
    % Set priorities for simultaneous plotting of multiple morphogens, if desired.
    m = leaf_mgen_plotpriority( m, {'ID_PLUSORG', 'ID_MINUSORG'}, [1,2], [0.4,0.4] );

    % Set colour of polariser gradient arrows.
    m = leaf_plotoptions(m,'highgradcolor',[0,0,0],'lowgradcolor',[1,0,0]);
    m = leaf_plotoptions(m,'decorscale',1.5);

    % setup a multiplot of the following morphogens
    m = leaf_plotoptions( m, 'morphogen', {'V_KAREAL','ID_PLUSORG','ID_MINUSORG'});
%%% END OF USER CODE: INITIALISATION

% Section 3
%%% SECTION 1: ACCESSING MORPHOGENS AND TIME.
%%% AUTOMATICALLY GENERATED CODE: DO NOT EDIT.

    if isempty(m), return; end

    setGlobals();
    global gNEW_KA_PAR gNEW_KA_PER gNEW_KB_PAR gNEW_KB_PER
    global gNEW_K_NOR gNEW_POLARISER gNEW_STRAINRET gNEW_ARREST
    dt = m.globalProps.timestep;
    polariser_i = gNEW_POLARISER;
    P = m.morphogens(:,polariser_i);
    [kapar_i,kapar_p,kapar_a,kapar_l] = getMgenLevels( m, 'KAPAR' );
    [kaper_i,kaper_p,kaper_a,kaper_l] = getMgenLevels( m, 'KAPER' );
    [kbpar_i,kbpar_p,kbpar_a,kbpar_l] = getMgenLevels( m, 'KBPAR' );
    [kbper_i,kbper_p,kbper_a,kbper_l] = getMgenLevels( m, 'KBPER' );
    [knor_i,knor_p,knor_a,knor_l] = getMgenLevels( m, 'KNOR' );
    [strainret_i,strainret_p,strainret_a,strainret_l] = getMgenLevels( m, 'STRAINRET' );
    [arrest_i,arrest_p,arrest_a,arrest_l] = getMgenLevels( m, 'ARREST' );
    [id_plusorg_i,id_plusorg_p,id_plusorg_a,id_plusorg_l] = getMgenLevels( m, 'ID_PLUSORG' );
    [id_minusorg_i,id_minusorg_p,id_minusorg_a,id_minusorg_l] = getMgenLevels( m, 'ID_MINUSORG' );
    [v_kareal_i,v_kareal_p,v_kareal_a,v_kareal_l] = getMgenLevels( m, 'V_KAREAL' );
    [id_tip_i,id_tip_p,id_tip_a,id_tip_l] = getMgenLevels( m, 'ID_TIP' );
    [id_top_i,id_top_p,id_top_a,id_top_l] = getMgenLevels( m, 'ID_TOP' );
    [s_growth_i,s_growth_p,s_growth_a,s_growth_l] = getMgenLevels( m, 'S_GROWTH' );
    [id_mid_i,id_mid_p,id_mid_a,id_mid_l] = getMgenLevels( m, 'ID_MID' );

% Mesh type: circle
%          centre: 0
%       circumpts: 24
%       coneangle: 0
%         dealign: 0
%          height: 0
%        innerpts: 0
%      randomness: 0.1
%           rings: 4
%         version: 1
%          xwidth: 2
%          ywidth: 2

%            Morphogen   Diffusion   Decay   Dilution   Mutant
%            -------------------------------------------------
%                KAPAR        ----    ----       ----     ----
%                KAPER        ----    ----       ----     ----
%                KBPAR        ----    ----       ----     ----
%                KBPER        ----    ----       ----     ----
%                 KNOR        ----    ----       ----     ----
%            POLARISER         0.1    ----       ----     ----
%            STRAINRET        ----    ----       ----     ----
%               ARREST        ----    ----       ----     ----
%           ID_PLUSORG        ----    ----       ----     ----
%          ID_MINUSORG        ----    ----       ----     ----
%             V_KAREAL        ----    ----       ----     ----
%               ID_TIP        ----    ----       ----     ----
%               ID_TOP        ----    ----       ----     ----
%             S_GROWTH        0.01    ----       ----     ----
%               ID_MID        ----    ----       ----     ----


%%% USER CODE: MORPHOGEN INTERACTIONS

% In this section you may modify the mesh in any way that does not
% Section 4
% alter the set of nodes.

% Use the same pattern for both submodels
     RangeTip=(m.nodes(:,1)<-0.8)&...
         (abs(m.nodes(:,2))<0.2);
     RangeMid=(m.nodes(:,1)<=0.5)&...
         (m.nodes(:,1)>-0.5)&...
         (abs(m.nodes(:,2))<0.3);
     RangeTops=(m.nodes(:,1)<=max(m.nodes(:,1))&...
         (m.nodes(:,1)>0.5)&...
         (abs(m.nodes(:,2))>0.3));
    if (Steps(m)==0) && m.globalDynamicProps.doinit  % Initialisation code.
        switch modelname
             case 'PolariserBased'  % 
                % One way to set up a morphogen gradient is by ...
                % Setting up a gradient by clamping the ends (execute only once)
                P(RangeTip)=0;
                P(RangeMid)=0.5;
                P(RangeTops)=1;
                id_plusorg_p=P;
                id_minusorg_p(RangeTip)=1;
                m.morphogenclamp( RangeTops|RangeTip|RangeMid, polariser_i ) = 1;
                m = leaf_mgen_conductivity( m, 'POLARISER', 0.1 );  %specifies the diffusion rate of polariser
                m = leaf_mgen_absorption( m, 'POLARISER', 0.0 );     % specifies degradation rate of polariser
             case 'DifferentialGrowthBased'  % 
                P(:)=0;
                % One way to set up a morphogen gradient is by ...
                % Setting up a gradient by clamping the ends (execute only once)
                s_growth_p(RangeTip)=1;
                s_growth_p(RangeMid)=0.05;
                s_growth_p(RangeTops)=0.8;
                m.morphogenclamp( RangeTops|RangeTip|RangeMid, s_growth_i ) = 1;
                m = leaf_mgen_conductivity( m, 's_growth', 0.001 );  %specifies the diffusion rate of polariser
                m = leaf_mgen_absorption( m, 's_growth', 0.0 );     % specifies degradation rate of polariser
        end
    end
    BasicGrowth=0.01;
    switch modelname
        case 'PolariserBased'  %
            % Every equation to be formatted should end with an at-at Eqn N comment.
            kapar_p(:) = BasicGrowth;  % when isotropic this will be 0.005
            kaper_p(:) = 0.0;   % when isotropic this will be 0.005
            kbpar_p(:) = BasicGrowth;  % when isotropic this will be 0.005
            kbper_p(:) = 0.0;   % when isotropic this will be 0.005
            knor_p(:)  = 0;  % thickness
         case 'DifferentialGrowthBased'  %  
            % Every equation to be formatted should end with an at-at Eqn N comment.
            kapar_p(:) = BasicGrowth*s_growth_p; %0.01;  % when isotropic this will be 0.005
            kaper_p(:) = BasicGrowth*s_growth_p; %0.0;   % when isotropic this will be 0.005
            kbpar_p(:) = BasicGrowth*s_growth_p; %0.01;  % when isotropic this will be 0.005
            kbper_p(:) = BasicGrowth*s_growth_p; %0.0;   % when isotropic this will be 0.005
            knor_p(:)  = 0;  % thickness
    end
    v_kareal_p=kapar_p+kaper_p; % total specified areal growth
% Section 5
%%% END OF USER CODE: MORPHOGEN INTERACTIONS

%%% SECTION 3: INSTALLING MODIFIED VALUES BACK INTO MESH STRUCTURE
%%% AUTOMATICALLY GENERATED CODE: DO NOT EDIT.
    m.morphogens(:,polariser_i) = P;
    m.morphogens(:,kapar_i) = kapar_p;
    m.morphogens(:,kaper_i) = kaper_p;
    m.morphogens(:,kbpar_i) = kbpar_p;
    m.morphogens(:,kbper_i) = kbper_p;
    m.morphogens(:,knor_i) = knor_p;
    m.morphogens(:,strainret_i) = strainret_p;
    m.morphogens(:,arrest_i) = arrest_p;
    m.morphogens(:,id_plusorg_i) = id_plusorg_p;
    m.morphogens(:,id_minusorg_i) = id_minusorg_p;
    m.morphogens(:,v_kareal_i) = v_kareal_p;
    m.morphogens(:,id_tip_i) = id_tip_p;
    m.morphogens(:,id_top_i) = id_top_p;
    m.morphogens(:,s_growth_i) = s_growth_p;
    m.morphogens(:,id_mid_i) = id_mid_p;

%%% USER CODE: FINALISATION

%%% END OF USER CODE: FINALISATION

end


%%% USER CODE: SUBFUNCTIONS

% Section 6
function m = local_setproperties( m )
end