combined_model

MATLAB Code for Binding Model of RBP4, TTR, and Retinol
Inputs for model
Code implementation for all participants
results table
Sensitivity analysis varying TTR and Retinol concentrations
results table for sensitivity analysis
sensitivity analysis figures
define odes
define odes for sensitivity analysis
System of ODEs

MATLAB Code for Binding Model of RBP4, TTR, and Retinol

clear clc

Inputs for model

define rate constants from literature Kds

k1 = 10e8 ;  % diffusion (M^-1 s^-1)
k2 = 1.46e-7 * k1;  % Kd of 178 nM from Muhilal and Glover 1975, 190 nM Cogan 1976, 70 nM from Noy 1992 (mean = 146 nM)
k3 = 10e8 ;  % diffusion (M^-1 s^-1)
k4 = 2.82e-7  * k3;  % alpha*Kd1 (mean 215 nM from 150 nM from Rostom 1998, 294 nM White and Kelly 2001, 200 nM Malpeli 1996)
k5 = 10e8 ;  % diffusion (M^-1 s^-1)
k6 = 1.2e-6 * k5;  % Kd2  1.2 uM from Malpeli 1996
k7 = 10e8; % diffusion (M-1 s-1)
k8 = 3.44e-8; % alpha*Kd2 (observed 35 nM Noy 1992)

% Time span for the simulation
tspan = [0 1];  %[inital:increment:end]

% Constrain solutions to be positive
options = odeset('MaxStep', 0.0001, 'NonNegative', 1:6);

Code implementation for all participants

% Import measured total concentrations for each particpiant

% Define the parameters for each participant with a table
opts = spreadsheetImportOptions("NumVariables", 4);
% Specify sheet and range
opts.Sheet = "Sheet1";
opts.DataRange = "A2:D32";
% Specify column names and types
opts.VariableNames = ["ParticipantID", "ROL_total", "RBP4_total", "TTR_total"];
opts.VariableTypes = ["string", "double", "double", "double"];
% Specify variable properties
opts = setvaropts(opts, "ParticipantID", "WhitespaceRule", "preserve");
opts = setvaropts(opts, "ParticipantID", "EmptyFieldRule", "auto");
% Import the data
participants = readtable("C:\Users\isolab\UW\isolab - Aprajita Yadav\Manuscripts\Z_RBP4 TTR VAO Paper\Binding_model_git\VAOtable.xlsx", opts, "UseExcel", false);
clear opts

% Create an array for all the final results to go into
ParticipantID = cell(height(participants), 1);
ROLu_end = zeros(height(participants), 1);
RBP4u_end = zeros(height(participants), 1);
TTRu_end = zeros(height(participants), 1);
aRBP4_TTR_end = zeros(height(participants), 1);
hRBP4_end = zeros(height(participants), 1);
hRBP4_TTR_end = zeros(height(participants), 1);
ROL_total_end = zeros(height(participants), 1);
RBP4_total_end = zeros(height(participants), 1);
TTR_total_end = zeros(height(participants), 1);

% loop for all participants
for i = 1:height(participants)
    % Extract participant data
    ParticipantID{i} = participants.ParticipantID{i};  % Participant ID
    ROL_total = participants.ROL_total(i);  % Total ROL for the participant
    RBP4_total = participants.RBP4_total(i);  % Total RBP4 for the participant
    TTR_total = participants.TTR_total(i);  % Total TTR for the participant

    % Initial conc (we assume nothing is complexed to start)
    initial_hRBP4 = 0;
    initial_hRBP4_TTR = 0;
    initial_aRBP4_TTR = 0;
    initial_ROL_u = ROL_total - initial_hRBP4 - initial_hRBP4_TTR;
    initial_RBP4_u = RBP4_total - initial_hRBP4 - initial_hRBP4_TTR - initial_aRBP4_TTR;
    initial_TTR_u = TTR_total - initial_hRBP4_TTR - initial_aRBP4_TTR;
    initial_conditions = [initial_ROL_u, initial_RBP4_u, initial_hRBP4, initial_TTR_u, initial_hRBP4_TTR, initial_aRBP4_TTR];

    %solve ode
    [t, y] = ode89(@(t, y) odesystem(t, y, k1, k2, k3, k4, k5, k6, k7, k8), tspan, initial_conditions, options);

    % Convert y-axis to micromolar (µM) from molar (M)
    y_um = y * 1e6;  % Convert concentration to µM

    %the following outputs the concentrations of all species at the end
    ROLu_end(i) = y_um(end, 1);
    RBP4u_end(i) =  y_um(end, 2);
    aRBP4_TTR_end(i) =  y_um(end, 3);
    TTRu_end(i) = y_um(end, 4);
    hRBP4_end(i) = y_um(end, 5);
    hRBP4_TTR_end(i) = y_um(end, 6);
    ROL_total_end(i) = ROLu_end(i) + hRBP4_end(i) + hRBP4_TTR_end(i);
    RBP4_total_end(i) = RBP4u_end(i) + aRBP4_TTR_end(i) + hRBP4_end(i) + hRBP4_TTR_end(i);
    TTR_total_end(i) = TTRu_end(i) + aRBP4_TTR_end(i) + hRBP4_TTR_end(i);
end

ParticipantID = cellstr(ParticipantID);

results table

Remove Percent sign to display and write results table to .csv

results_table = table(ParticipantID, ROLu_end, RBP4u_end, TTRu_end, aRBP4_TTR_end, hRBP4_end, hRBP4_TTR_end, ...
    ROL_total_end, RBP4_total_end, TTR_total_end, ...
    'VariableNames', {'ParticipantID', 'ROL_u (µM)', 'RBP4_u (µM)', 'TTR_u (µM)', 'RBP4:TTR (µM)', 'ROL:RBP4 (µM)', ...
    'ROL:RBP4:TTR (µM)', 'ROL_total (µM)', 'RBP4_total (µM)', 'TTR_total (µM)'});
% disp(results_table);
% filename='RBP4_TTR_Retinol_Binding.csv';
% writetable(results_table, filename);

Sensitivity analysis varying TTR and Retinol concentrations

hold RBP4 constant at 2 µM

% ROL and TTR concentrations
TTR_list = [2.5e-6:.25e-6:7.5e-6];
ROL_list = [1e-6:0.05e-6:2e-6];

% empty array for results
ROLu_end_s = zeros(length(TTR_list), 1);
RBP4u_end_s = zeros(length(TTR_list), 1);
TTRu_end_s = zeros(length(TTR_list), 1);
aRBP4_TTR_end_s = zeros(length(TTR_list), 1);
hRBP4_end_s = zeros(length(TTR_list), 1);
hRBP4_TTR_end_s = zeros(length(TTR_list), 1);
ROL_total_end_s = zeros(length(TTR_list), 1);
RBP4_total_end_s = zeros(length(TTR_list), 1);
TTR_total_end_s = zeros(length(TTR_list), 1);
Per_ROLu_end_s = zeros(length(TTR_list), 1);
Per_complex_end_s = zeros(length(TTR_list), 1);
Per_holo_end_s = zeros(length(TTR_list), 1);
Per_RBP4u_end_s = zeros(length(TTR_list), 1);
Per_hRBP4_end_s = zeros(length(TTR_list), 1);
Per_aRBP4TTR_end_s = zeros(length(TTR_list), 1);
Per_hRBP4_TTR_end_s = zeros(length(TTR_list), 1);

for i = 1:length(TTR_list);
    TTR_total_s = TTR_list(i);
for  j = 1:length(ROL_list);
     RBP4_total_s = 2e-6;
    ROL_total_s= ROL_list(j);

% Initial conc (we assume nothing is complexed to start)
 initial_hRBP4_s = 0;
 initial_hRBP4_TTR_s = 0;
 initial_aRBP4_TTR_s = 0;
 initial_ROL_u_s = ROL_total_s - initial_hRBP4_s - initial_hRBP4_TTR_s;
 initial_RBP4_u_s = RBP4_total_s - initial_hRBP4_s - initial_hRBP4_TTR_s - initial_aRBP4_TTR_s;
 initial_TTR_u_s = TTR_total_s - initial_hRBP4_TTR_s - initial_aRBP4_TTR_s;
 initial_conditions_s = [initial_ROL_u_s, initial_RBP4_u_s, initial_hRBP4_s, initial_TTR_u_s, initial_hRBP4_TTR_s, initial_aRBP4_TTR_s];

 % solve odes
 [t, y] = ode89(@(t, y) odesystem_s(t, y, k1, k2, k3, k4, k5, k6, k7, k8), tspan, initial_conditions_s, options);

 % Convert y-axis to micromolar (µM) from molar (M)
  y_um = y * 1e6;  % Convert concentration to µM

 % the following outputs the concentrations of all species at the end
    ROLu_end_s(j, i) = y_um(end, 1);
    RBP4u_end_s(j, i) =  y_um(end, 2);
    aRBP4_TTR_end_s(j, i) =  y_um(end, 3);
    TTRu_end_s(j, i) = y_um(end, 4);
    hRBP4_end_s(j, i) = y_um(end, 5);
    hRBP4_TTR_end_s(j, i) = y_um(end, 6);

 % total concentrations
    ROL_total_end_s(j, i) = ROLu_end_s(j, i) + hRBP4_end_s(j, i) + hRBP4_TTR_end_s(j, i);
    RBP4_total_end_s(j, i) = RBP4u_end_s(j, i) + aRBP4_TTR_end_s(j, i) + hRBP4_end_s(j, i) + hRBP4_TTR_end_s(j, i);
    TTR_total_end_s(j, i) = TTRu_end_s(j, i) + aRBP4_TTR_end_s(j, i) + hRBP4_TTR_end_s(j, i);

  % calculate percentages
  Per_ROLu_end_s(j, i) = (ROLu_end_s(j, i) / ROL_total_end_s(j, i))*100 ;
  Per_complex_end_s(j, i) = (hRBP4_TTR_end_s(j, i)/ROL_total_end_s(j, i))*100;
  Per_holo_end_s(j, i) = (hRBP4_end_s(j, i)/ROL_total_end_s(j, i))*100;
  Per_RBP4u_end_s(j, i) = (RBP4u_end_s(j,i) / RBP4_total_end_s(j,i))*100;
  Per_hRBP4_end_s(j, i) = (hRBP4_end_s(j,i) / RBP4_total_end_s(j,i))*100;
  Per_aRBP4TTR_end_s(j, i) = (aRBP4_TTR_end_s(j,i) / RBP4_total_end_s(j,i))*100;
  Per_hRBP4_TTR_end_s(j, i) = (hRBP4_TTR_end_s(j,i)/RBP4_total_end_s(j, i))*100;
end
end

results table for sensitivity analysis

results_table_s = table(ROLu_end_s, RBP4u_end_s, TTRu_end_s, ...
    aRBP4_TTR_end_s, hRBP4_end_s, hRBP4_TTR_end_s, ROL_total_end_s, ...
    RBP4_total_end_s, TTR_total_end_s, Per_ROLu_end_s, Per_complex_end_s, ...
    Per_holo_end_s, Per_RBP4u_end_s, Per_hRBP4_end_s, Per_aRBP4TTR_end_s, ...
    Per_hRBP4_TTR_end_s, 'VariableNames', {'ROL (µM)', 'RBP4 (µM)', ...
    'TTR (µM)', 'RBP4:TTR (µM)', 'ROL:RBP4 (µM)', 'ROL:RBP4:TTR (µM)', ...
    'ROL_total (µM)', 'RBP4_total (µM)', 'TTR_total (µM)', ...
    'Percent Retinol unbound','Percent ROL:RBP4:TTR', 'Percent ROL:RBP4', ...
    'Percent apo-RBP4', 'Percent holoRBP4', 'Percent RBP4:TTR', ...
    'Percent complex RBP4'});
%disp(results_table); %remove percent sign to display results table

sensitivity analysis figures

screen_size = get(0, 'ScreenSize');
fig_width = screen_size(3) /5;
fig_height = screen_size(4) / 3;

% figure 3
figure3 = figure('Position', [0, screen_size(4)/4, fig_width, fig_height]);

axes3 = axes('Parent', figure3);
hold(axes3, 'on');

% Create mesh
mesh(ROL_total_end_s,TTR_total_end_s,RBP4u_end_s, 'Parent',axes3);
mesh(ROL_total_end_s,TTR_total_end_s,hRBP4_end_s, 'Parent',axes3);
mesh(ROL_total_end_s,TTR_total_end_s,aRBP4_TTR_end_s, 'Parent',axes3);
mesh(ROL_total_end_s,TTR_total_end_s,hRBP4_TTR_end_s, 'Parent',axes3);

% Axis labels
zlabel(axes3, {'RBP4 species (µM)'});
ylabel(axes3, {'TTR Tetramer (µM)'});
xlabel(axes3, {'Retinol (µM)'});

% Viewing angle
view(axes3,[-52.5 12.4]);
grid(axes3, 'on');
set(axes3,'XTick',[1 1.25 1.5 1.75 2],'YTick',[2 4 6 8],'YTickLabel',...
    {'2','4','6','8'},'ZTick', [0 0.5 1 1.5 2]);
axes3.XTickLabelRotation = 0;

% Create textboxes
annotation(figure3,'textbox',[0.15 0.32 0.19 0.06],'String',{'RBP4:TTR'},'LineStyle','none');
annotation(figure3,'textbox',[0.15 0.26 0.14 0.06],'String',{'RBP4'},'LineStyle','none');
annotation(figure3,'textbox',[0.64 0.33 0.26 0.06],'String',{'ROL:RBP4'},'LineStyle','none');
annotation(figure3,'textbox',[0.15 0.77 0.64 0.06],'String',{'ROL:RBP4:TTR'},'LineStyle','none');

hold(axes3, 'off')

% figure 2
figure2 = figure('Position', [fig_width, screen_size(4)/4, fig_width, fig_height]); % Position on the right side

axes2 = axes('Parent', figure2);
hold(axes2, 'on');

% Create mesh
mesh(ROL_total_end_s,TTR_total_end_s,RBP4u_end_s, 'Parent', axes2);
mesh(ROL_total_end_s,TTR_total_end_s,hRBP4_end_s, 'Parent', axes2);
mesh(ROL_total_end_s,TTR_total_end_s,aRBP4_TTR_end_s, 'Parent', axes2);

% Create axis labels
zlabel(axes2, {'RBP4 species (µM)'});
ylabel(axes2, {'TTR Tetramer (µM)'});
xlabel(axes2, {'Retinol (µM)'});

% Viewing angle
view(axes2, [-52.5 12.4]);
grid(axes2, 'on');
set(axes2,'XTick',[1 1.25 1.5 1.75 2], 'YTick',[2 4 6 8],'YTickLabel',...
    {'2','4','6','8'},'ZTick',[0 0.25 0.5 0.75 1]);
axes2.XTickLabelRotation = 0;

% Create textboxes
annotation(figure2,'textbox',[0.15 0.71 0.25 0.08],'String',{'RBP4:TTR'},'LineStyle','none');
annotation(figure2,'textbox',[0.15 0.31 0.19 0.08],'String',{'RBP4'},'LineStyle','none');
annotation(figure2,'textbox',[0.69 0.46 0.26 0.08],'String',{'ROL:RBP4'},'LineStyle','none');

hold(axes2, 'off')

define odes

    function dydt = odesystem(t, y, k1, k2, k3, k4, k5, k6, k7, k8)
    ROL_u = y(1);
    RBP4_u = y(2);
    aRBP4_TTR = y(3);
    TTR_u = y(4);
    hRBP4 = y(5);
    hRBP4_TTR = y(6);

    % Differential equations
    dROL_u = k2 * hRBP4 - k1 * ROL_u * RBP4_u +k8*hRBP4_TTR - k7*ROL_u*aRBP4_TTR;
    dRBP4_u = k2 * hRBP4 + k6 * aRBP4_TTR - k1 * ROL_u * RBP4_u - k5 * RBP4_u * TTR_u;
    daRBP4_TTR = k5 * RBP4_u * TTR_u - k6 * aRBP4_TTR - k7 * aRBP4_TTR * ROL_u + k8 * hRBP4_TTR;
    dTTR_u = k6 * aRBP4_TTR + k4 * hRBP4_TTR - k5 * RBP4_u * TTR_u - k3 * hRBP4 * TTR_u;
    dhRBP4 = k1 * ROL_u * RBP4_u + k4 * hRBP4_TTR - k2 * hRBP4 - k3 * hRBP4 * TTR_u;
    dhRBP4_TTR = k3 * hRBP4 * TTR_u - k4 * hRBP4_TTR - k8* hRBP4_TTR + k7* aRBP4_TTR * ROL_u;

    % Results as vector
    dydt = [dROL_u; dRBP4_u; daRBP4_TTR; dTTR_u; dhRBP4; dhRBP4_TTR];
    end

define odes for sensitivity analysis

    function dydt = odesystem_s(t, y, k1, k2, k3, k4, k5, k6, k7, k8)
    ROL_u_s = y(1);
    RBP4_u_s = y(2);
    aRBP4_TTR_s = y(3);
    TTR_u_s = y(4);
    hRBP4_s = y(5);
    hRBP4_TTR_s = y(6);

    % Differential equations
    dROL_u_s = k2 * hRBP4_s - k1 * ROL_u_s * RBP4_u_s +k8*hRBP4_TTR_s - k7*ROL_u_s*aRBP4_TTR_s;
    dRBP4_u_s = k2 * hRBP4_s + k6 * aRBP4_TTR_s - k1 * ROL_u_s * RBP4_u_s - k5 * RBP4_u_s * TTR_u_s;
    daRBP4_TTR_s = k5 * RBP4_u_s * TTR_u_s - k6 * aRBP4_TTR_s - k7 * aRBP4_TTR_s * ROL_u_s + k8 * hRBP4_TTR_s;
    dTTR_u_s = k6 * aRBP4_TTR_s + k4 * hRBP4_TTR_s - k5 * RBP4_u_s * TTR_u_s - k3 * hRBP4_s * TTR_u_s;
    dhRBP4_s = k1 * ROL_u_s * RBP4_u_s + k4 * hRBP4_TTR_s - k2 * hRBP4_s - k3 * hRBP4_s * TTR_u_s;
    dhRBP4_TTR_s = k3 * hRBP4_s * TTR_u_s - k4 * hRBP4_TTR_s - k8* hRBP4_TTR_s + k7* aRBP4_TTR_s * ROL_u_s;

    % Results as vector
    dydt = [dROL_u_s; dRBP4_u_s; daRBP4_TTR_s; dTTR_u_s; dhRBP4_s; dhRBP4_TTR_s];
    end

System of ODEs

figure1=figure('Color','w', 'Position', [100, 100, 1066, 700]);
axis off
hold on

y_start = 0.8;
dy = 0.1;
fs = 14;

% Define each equation
eqns = {
    '$$\frac{d\mathrm{ROL}}{dt} = k_2 \cdot \mathrm{ROL:RBP4} - k_1 \cdot \mathrm{ROL} \cdot \mathrm{RBP4} + k_8 \cdot \mathrm{ROL:RBP4:TTR} - k_7 \cdot \mathrm{ROL} \cdot \mathrm{RBP4:TTR}$$'
    '$$\frac{d\mathrm{RBP4}}{dt} = k_2 \cdot \mathrm{ROL:RBP4} + k_6 \cdot \mathrm{RBP4:TTR} - k_1 \cdot \mathrm{ROL} \cdot \mathrm{RBP4} - k_5 \cdot \mathrm{RBP4} \cdot \mathrm{TTR}$$'
    '$$\frac{d\mathrm{RBP4:TTR}}{dt} = k_5 \cdot \mathrm{RBP4} \cdot \mathrm{TTR} - k_6 \cdot \mathrm{RBP4:TTR} - k_7 \cdot \mathrm{RBP4:TTR} \cdot \mathrm{ROL} + k_8 \cdot \mathrm{ROL:RBP4:TTR}$$'
    '$$\frac{d\mathrm{TTR}}{dt} = k_6 \cdot \mathrm{RBP4:TTR} + k_4 \cdot \mathrm{ROL:RBP4:TTR} - k_5 \cdot \mathrm{RBP4} \cdot \mathrm{TTR} - k_3 \cdot \mathrm{ROL:RBP4} \cdot \mathrm{TTR}$$'
    '$$\frac{d\mathrm{ROL:RBP4}}{dt} = k_1 \cdot \mathrm{ROL} \cdot \mathrm{RBP4} + k_4 \cdot \mathrm{ROL:RBP4:TTR} - k_2 \cdot \mathrm{ROL:RBP4} - k_3 \cdot \mathrm{ROL:RBP4} \cdot \mathrm{TTR}$$'
    '$$\frac{d\mathrm{ROL:RBP4:TTR}}{dt} = k_3 \cdot \mathrm{ROL:RBP4} \cdot \mathrm{TTR} - k_4 \cdot \mathrm{ROL:RBP4:TTR} - k_8 \cdot \mathrm{ROL:RBP4:TTR} + k_7 \cdot \mathrm{RBP4:TTR} \cdot \mathrm{ROL}$$'
    };

% Loop through and add each equation
for i = 1:length(eqns)
    text(0.01, y_start - (i-1)*dy, eqns{i}, 'Interpreter', 'latex', 'FontSize', fs)
end

% Add a title at the top
text(0.01, 0.9, '\textbf{Differential Equations for Retinol, RBP4, TTR Binding Model}', 'Interpreter', 'latex', 'FontSize', fs+1)

hold off
set(gca,'Position',[0 0 1 1])

Contents

MATLAB Code for Binding Model of RBP4, TTR, and Retinol

Inputs for model

Code implementation for all participants

results table

Sensitivity analysis varying TTR and Retinol concentrations

results table for sensitivity analysis

sensitivity analysis figures

define odes

define odes for sensitivity analysis

System of ODEs