ADD appendix for matlab stuff
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@ -13,3 +13,6 @@ figures/*
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!tables
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!tables
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tables/*
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tables/*
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!tables/*.tex
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!tables/*.tex
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!code
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!code/*
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@ -0,0 +1,61 @@
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function diffusion_mab
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% bulk properties
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V_b = 1.075; % volume of bulk solution (cm^3)
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n = 16000; % number of microcarriers in bulk volume
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% ligand properties (IgG)
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m_L0 = 4; % mass of ligand (ug)
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MW_L = 150000; % molecular weight of ligand (g/mol)
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D_LW = 4.8e-7; % ligand diffusion coefficient in water (cm^2/s)
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a = 1; % rxn stoic coeff (L per R)
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% fixed carrier properties
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geometry = 0.190; % geometric diffusion factor of microcarriers
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R = 0.01275; % microcarrier radius (cm)
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C_R0 = 144; % receptor density in carriers (pM/cm^3, nM)
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D_L_app = D_LW*geometry*60; % apparent ligand diffusion coeff. (cm^2/min)
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% init
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t_0 = 0; % initial time (min)
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t_f = 90; % final time (min)
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alpha = 0.999999999999; % fudge factor to prevent MATLAB implosion
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r_i0 = R * alpha; % init interfatial radius in microcarriers (cm)
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C_Lb0 = m_L0 / MW_L * 1e6 / V_b; % init conc. of ligand in bulk (pmol/cm^3, nM)
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% y1 = r_i, y2 = C_Lb
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dydt = @(t,y) [
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((R * D_L_app * y(2) / a))/(C_R0 * y(1) * (y(1) - R));
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(4 * pi * n * D_L_app * y(2) * R * y(1)) / (V_b * (y(1) - R))];
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[t,Y] = ode45(dydt,[t_0 t_f],[r_i0 C_Lb0]);
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r_i = Y(:,1); % interfacial radius (cm)
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C_Lb = Y(:,2); % bulk ligand concentration (pmol/cm^3, nM)
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% fluxes (pmol/cm^2/min)
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N_L = (D_L_app*C_Lb.*r_i)./(R^2*(r_i/R+(-1)));
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% NOTE: flow rates (pmol/min) will be the same at interface and at r=R due
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% to pseudo-steady state assumption
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w_L = -(4*pi*R^2*N_L);
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% concentration profiles (nM) as function of interfacial radius
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C_r = diff(cumtrapz(t,-w_L))./diff(4/3*pi*r_i.^3);
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% total bound mass (ug)
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m_bt_L = trapz(t,w_L)*MW_L/1e6*n;
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disp(['total bound mass at ',num2str(t(end)),' min']);
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disp([num2str(m_bt_L),' ug, ', num2str(m_bt_L/m_L0*100), '% (ligand)']);
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figure;
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plot(t,C_Lb*MW_L/1e6);
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xlabel('t (min)');
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ylabel('concentration (ug/ml)');
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legend('ligand');
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title ('bulk concentration');
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end
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@ -0,0 +1,80 @@
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function diffusion_stp
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% ADJUST THIS TO MINIMIZE SSE
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geometry = 0.190; % geometric diffusion factor of microcarriers
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% bulk parameters
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V_b = 1.008; % volume of bulk solution (cm^3)
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n = 16000; % number of microcarriers in bulk volume
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% ligand properties (streptavidin)
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m_L0 = 40; % mass of ligand (ug)
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MW_L = 55000; % molecular weight of ligand (g/mol)
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D_LW = 6.2e-7; % ligand diffusion coeff. in water (cm^2/s)
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a = 1; % rxn stoic coeff (L per R)
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% fixed carrier properties
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R = 0.01275; % microcarrier radius (cm)
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C_R0 = 3403; % receptor density in carriers (pM/cm^3, nM)
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D_L_app = D_LW * geometry * 60; % effective diffusion coeff (cm^2/min)
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% init
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t_0 = 0; % initial time (min)
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t_f = 65; % final time (min)
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alpha = 0.999999999999; % fudge factor to prevent MATLAB implosion
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r_i0 = R * alpha; % init. interfatial radius in microcarrier (cm)
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C_Lb0 = m_L0 / MW_L * 1e6 / V_b; % init. conc. of ligand in bulk (pmol/cm^3, nM)
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% y1 = r_i, y2 = C_Lb
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dydt = @(t,y) [
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((R * D_L_app * y(2) / a)) / (C_R0 * y(1) * (y(1) - R));
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(4 * pi * n * D_L_app * y(2) * R * y(1)) / (V_b * (y(1) - R))];
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[t,Y] = ode45(dydt, [t_0 t_f], [r_i0 C_Lb0]);
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r_i = Y(:, 1); % interfacial radius (cm)
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C_Lb = Y(:, 2); % bulk ligand concentration (pmol/cm^3, nM)
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% fluxes (pmol/cm^2/min)
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N_L = (D_L_app * C_Lb .* r_i) ./ (R^2 * (r_i / R + (-1)));
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% NOTE: flow rates (pmol/min) will be the same at interface and at r=R due
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% to pseudo-steady state assumption
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w_L = -(4 * pi * R^2 * N_L);
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% concentration profiles (nM) as function of interfacial radius
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C_r = diff(cumtrapz(t, -w_L)) ./ diff(4 / 3 * pi * r_i.^3);
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% total bound mass (ug)
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m_bt_L = trapz(t, w_L) * MW_L / 1e6 * n;
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disp(['total bound mass at ', num2str(t(end)),' min']);
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disp([num2str(m_bt_L), ' ug, ', num2str(m_bt_L / m_L0 * 100), '% (ligand)']);
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% experimental data
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exp_time1 = [7.5 15 30 60];
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exp_time2 = [15 20 25 30];
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exp_conc1 = [20.20 17.15 13.78 13.99]; % bulk concentration in ug/ml
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exp_conc2 = [18.73 16.74 14.91 14.75];
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figure;
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plot(exp_time1,exp_conc1, '.');
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hold on;
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plot(exp_time2,exp_conc2, '.');
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plot(t, C_Lb * MW_L / 1e6);
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hold off;
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xlabel('t (min)');
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ylabel('concentration (ug/ml)');
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legend('exp1', 'exp2', 'ligand');
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title ('bulk concentration');
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% fitting to experimental data (least squares algorithm)
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e1 = interp1(t, C_Lb, exp_time1) * MW_L / 1e6 - exp_conc1;
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e2 = interp1(t, C_Lb, exp_time2) * MW_L / 1e6 - exp_conc2;
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% MINIMIZE THIS (SSE)
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disp(['SSE: ', num2str(sum([e1 e2].^2))]);
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end
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@ -0,0 +1,179 @@
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function microcarrier_diffusion_washing()
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% initial/reaction volume for carriers (cm^3)
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V_b0 = 1;
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disp(['###Biotin washing###', sprintf('\n')])
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n_biotin = 10; % nmol biotin remaining after attachment
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D_biotin = 5.0e-6; % diffusion coeff (cm^2/s)
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carrier_stages(V_b0, 3, n_biotin, D_biotin)
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disp(['###Streptavidin washing###', sprintf('\n')])
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m_stp = 15; % ug stp remaining after attachment
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n_stp = m_stp / 55000 * 1000;
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D_stp = 6.2e-7; % diffusion coeff (cm^2/s)
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carrier_stages(V_b0, 2, n_stp, D_stp)
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disp(['###Antibody washing###', sprintf('\n')])
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m_Ab = 1; % ug Ab added
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n_Ab = m_Ab / 150000 * 1000;
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D_Ab = 4.8e-7; % diffusion coeff (cm^2/s)
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carrier_stages(V_b0, 2, n_Ab, D_Ab)
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end
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function carrier_stages(V_b0, stages, n_L, D_L)
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% initial concentration of L (nM)
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C_L = n_L / (V_b0 / 1000);
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% max fill for 15 ml conical tube is about 10ml at vortex speed = 6 without
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% splashing the medium (PBS) onto the cap/rim and possibly compromising
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% sterility and losing carriers
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%
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% wash volume, fill up to this to let biotin out of carriers (cm^3)
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V_wash = 10;
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% dilution volume, fill up to this to reduce total biotin in solution (cm^3)
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V_dilution = 15;
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C_L_bulk = C_L;
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for i = 1:stages
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disp([sprintf('\n'), '---Stage ', num2str(i), '---', sprintf('\n')])
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[C_L, n_L_carrier, n_L_bulk] = carrier_diffusion_out(V_b0, ...
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V_wash, C_L, C_L_bulk, D_L);
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C_L_bulk = C_L*V_wash / V_dilution;
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n_L_bulk = n_L_bulk * V_b0 / V_dilution;
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end
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disp([sprintf('\n'), 'final bulk amount = ' , num2str(n_L_bulk), ' nmol']);
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disp(['final total amount = ', num2str(n_L_carrier + n_L_bulk), ' nmol']);
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disp(['overall reduction = ', ...
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num2str((1 - (n_L_carrier + n_L_bulk) / n_L) * 100), ' %', ...
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sprintf('\n')]);
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end
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function [C_carrier_ave_final, n_Lc_f, n_Lb_f] = ...
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carrier_diffusion_out(V_b0, V_bf, C_Lc_pw, C_Lb_pw, D_LW)
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% V_b0: prewash volume of bulk solution (cm^3)
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% V_bf: wash volume of bulk solution (cm^3)
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% C_Lc_pw: prewash concentration of ligand in carriers (pmol/cm^3, nM)
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% C_Lb_pw: prewash concentration of ligand in bulk (pmol/cm^3, nM)
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% D_LW: ligand diffusion coefficient in water (cm^2/s)
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% number of microcarriers in bulk volume
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n = 16000;
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% geometric diffusion factor of microcarriers
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geometry = 0.190;
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% microcarrier radius (cm)
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R = 0.01275;
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% apparent ligand diffusion coeff in microcarrier (cm^2/min)
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D_L_app = D_LW * geometry * 60;
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% void fraction
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void = 0.95;
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% volume occupied by carriers (cm^3)
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V_c = void * 4 / 3 * pi * R^3 * n;
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% prewash amount in carriers (nmol)
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n_Lc_pw = (V_c / 1000) * C_Lc_pw;
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% prewash amount outside carriers (nmol)
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n_Lb_pw = ((V_b0 - V_c) / 1000) * C_Lb_pw;
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t_0 = 0; % initial time (min)
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t_f = 30; % final time (min)
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C_Lb0 = n_Lb_pw / ((V_bf - V_c) / 1e3); % initial conc. in bulk (nM)
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% final conditions (after long time)
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% we use this calculate an average bulk concentration to create a constant
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% boundary in the BVP, otherwise we have a to solve a free BVP,
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% which involves sacrificing a kitten...
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n_L_trans = (V_bf * n_Lc_pw - V_c * n_Lb_pw) / (V_c + V_bf);
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% final concentration of ligand in bulk solution (pmol/cm^3, nM)
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C_Lbf = (n_Lb_pw + n_L_trans) / (V_bf / 1e3);
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disp(['initial bulk concentration = ', num2str(C_Lb0), ' nM']);
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disp(['final bulk concentration = ', num2str(C_Lbf), ' nM']);
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disp(['bulk percent change = ', ...
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num2str((1 - C_Lb0 / C_Lbf) * 100), '%']);
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disp(['initial carrier concentration = ', num2str(C_Lc_pw), ' nM']);
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disp(['initial carrier amount = ', num2str(n_Lc_pw), ' nmol']);
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disp(['initial bulk amount = ', ...
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num2str(n_Lb_pw), ' nmol', sprintf('\n')]);
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m = 2; % spherical
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r = linspace(0, R, 50);
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t_f = 1;
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tolerance = 0.1; % iterate until center and outside conc are within this
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increment = 1; % length of time increments (min)
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diff = 1000000; % init to some huge number
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while diff > tolerance
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t = linspace(t_0, t_f, 50);
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Y = pdepe(m,@pde,@init,@bound,r,t);
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C = Y(:,:,1);
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% final concentration in center of carrier at final time
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C_f_center = C(end,1);
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% test to see how far off the center is from bulk
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diff = C_f_center / C_Lbf - 1;
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t_f = t_f + increment;
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end
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% average final concentration of carriers
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C_carrier_ave_final = (C_f_center + C_Lbf) / 2;
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% nmol still remaining in carriers
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n_Lc_f = C_carrier_ave_final * (V_c / 1000);
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% final nmol in bulk
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n_Lb_f = n_Lb_pw + (n_Lc_pw - n_Lc_f);
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% percent nmol of ligand removed from carriers
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perc_removed = (1 - n_Lc_f / n_Lc_pw) * 100;
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% amount actually transferred over theoretical
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efficiency = (n_Lc_pw - n_Lc_f) / n_L_trans * 100;
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disp(['final time to equilibrium = ', num2str(t_f), ' min']);
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disp(['Total removed from carriers = ', num2str(n_Lc_pw - n_Lc_f), ' nmol']);
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disp(['Total remaining in carriers = ', num2str(n_Lc_f), ' nmol']);
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disp(['Total remaining in bulk = ', num2str(n_Lb_f), ' nmol']);
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disp(['Percent removed from carriers = ', num2str(perc_removed), ' %']);
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function [c, f, s] = pde(r, t, c, DcDr)
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c = 1;
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f = D_L_app * DcDr;
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s = 0;
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end
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function u0 = init(r)
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u0 = C_Lc_pw;
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end
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function [pl,ql,pr,qr] = bound(rl,cl,rr,cr,t)
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pl = 0;
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ql = 1;
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% assume that the concentration boundary
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% is the average of the initial and
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% theoretical final concentration in bulk
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pr = cr - (C_Lbf + C_Lbf)/2;
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qr = 0;
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end
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end
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@ -0,0 +1,127 @@
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\begin{verbatim}
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###Biotin washing###
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---Stage 1---
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initial bulk concentration = 879.6421 nM
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final bulk concentration = 986.9753 nM
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bulk percent change = 10.875%
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initial carrier concentration = 10000 nM
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initial carrier amount = 1.3197 nmol
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initial bulk amount = 8.6803 nmol
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final time to equilibrium = 3 min
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||||||
|
Total removed from carriers = 1.1882 nmol
|
||||||
|
Total remaining in carriers = 0.13145 nmol
|
||||||
|
Total remaining in bulk = 9.8686 nmol
|
||||||
|
Percent removed from carriers = 90.0394 %
|
||||||
|
|
||||||
|
---Stage 2---
|
||||||
|
|
||||||
|
initial bulk concentration = 58.4119 nM
|
||||||
|
final bulk concentration = 69.8638 nM
|
||||||
|
bulk percent change = 16.3917%
|
||||||
|
initial carrier concentration = 996.0629 nM
|
||||||
|
initial carrier amount = 0.13145 nmol
|
||||||
|
initial bulk amount = 0.57641 nmol
|
||||||
|
|
||||||
|
final time to equilibrium = 3 min
|
||||||
|
Total removed from carriers = 0.1221 nmol
|
||||||
|
Total remaining in carriers = 0.0093421 nmol
|
||||||
|
Total remaining in bulk = 0.69852 nmol
|
||||||
|
Percent removed from carriers = 92.8928 %
|
||||||
|
|
||||||
|
---Stage 3---
|
||||||
|
|
||||||
|
initial bulk concentration = 4.1514 nM
|
||||||
|
final bulk concentration = 4.9653 nM
|
||||||
|
bulk percent change = 16.3917%
|
||||||
|
initial carrier concentration = 70.7919 nM
|
||||||
|
initial carrier amount = 0.0093421 nmol
|
||||||
|
initial bulk amount = 0.040967 nmol
|
||||||
|
|
||||||
|
final time to equilibrium = 3 min
|
||||||
|
Total removed from carriers = 0.0086782 nmol
|
||||||
|
Total remaining in carriers = 0.00066396 nmol
|
||||||
|
Total remaining in bulk = 0.049645 nmol
|
||||||
|
Percent removed from carriers = 92.8928 %
|
||||||
|
|
||||||
|
final bulk amount = 0.0033096 nmol
|
||||||
|
final total amount = 0.0039736 nmol
|
||||||
|
overall reduction = 99.9603 %
|
||||||
|
|
||||||
|
###Streptavidin washing###
|
||||||
|
|
||||||
|
|
||||||
|
---Stage 1---
|
||||||
|
|
||||||
|
initial bulk concentration = 23.9902 nM
|
||||||
|
final bulk concentration = 26.9175 nM
|
||||||
|
bulk percent change = 10.875%
|
||||||
|
initial carrier concentration = 272.7273 nM
|
||||||
|
initial carrier amount = 0.035991 nmol
|
||||||
|
initial bulk amount = 0.23674 nmol
|
||||||
|
|
||||||
|
final time to equilibrium = 14 min
|
||||||
|
Total removed from carriers = 0.032315 nmol
|
||||||
|
Total remaining in carriers = 0.003676 nmol
|
||||||
|
Total remaining in bulk = 0.26905 nmol
|
||||||
|
Percent removed from carriers = 89.7862 %
|
||||||
|
|
||||||
|
---Stage 2---
|
||||||
|
|
||||||
|
initial bulk concentration = 1.6335 nM
|
||||||
|
final bulk concentration = 1.9538 nM
|
||||||
|
bulk percent change = 16.3917%
|
||||||
|
initial carrier concentration = 27.8557 nM
|
||||||
|
initial carrier amount = 0.003676 nmol
|
||||||
|
initial bulk amount = 0.01612 nmol
|
||||||
|
|
||||||
|
final time to equilibrium = 15 min
|
||||||
|
Total removed from carriers = 0.0034097 nmol
|
||||||
|
Total remaining in carriers = 0.00026636 nmol
|
||||||
|
Total remaining in bulk = 0.019529 nmol
|
||||||
|
Percent removed from carriers = 92.7541 %
|
||||||
|
|
||||||
|
final bulk amount = 0.001302 nmol
|
||||||
|
final total amount = 0.0015683 nmol
|
||||||
|
overall reduction = 99.4249 %
|
||||||
|
|
||||||
|
###Antibody washing###
|
||||||
|
|
||||||
|
|
||||||
|
---Stage 1---
|
||||||
|
|
||||||
|
initial bulk concentration = 0.58643 nM
|
||||||
|
final bulk concentration = 0.65798 nM
|
||||||
|
bulk percent change = 10.875%
|
||||||
|
initial carrier concentration = 6.6667 nM
|
||||||
|
initial carrier amount = 0.00087977 nmol
|
||||||
|
initial bulk amount = 0.0057869 nmol
|
||||||
|
|
||||||
|
final time to equilibrium = 17 min
|
||||||
|
Total removed from carriers = 0.00078901 nmol
|
||||||
|
Total remaining in carriers = 9.0766e-05 nmol
|
||||||
|
Total remaining in bulk = 0.0065759 nmol
|
||||||
|
Percent removed from carriers = 89.683 %
|
||||||
|
|
||||||
|
---Stage 2---
|
||||||
|
|
||||||
|
initial bulk concentration = 0.040335 nM
|
||||||
|
final bulk concentration = 0.048242 nM
|
||||||
|
bulk percent change = 16.3917%
|
||||||
|
initial carrier concentration = 0.6878 nM
|
||||||
|
initial carrier amount = 9.0766e-05 nmol
|
||||||
|
initial bulk amount = 0.00039802 nmol
|
||||||
|
|
||||||
|
final time to equilibrium = 18 min
|
||||||
|
Total removed from carriers = 8.4099e-05 nmol
|
||||||
|
Total remaining in carriers = 6.6675e-06 nmol
|
||||||
|
Total remaining in bulk = 0.00048212 nmol
|
||||||
|
Percent removed from carriers = 92.6543 %
|
||||||
|
|
||||||
|
final bulk amount = 3.2141e-05 nmol
|
||||||
|
final total amount = 3.8809e-05 nmol
|
||||||
|
overall reduction = 99.4179 %
|
||||||
|
\end{verbatim}
|
||||||
|
|
@ -19,6 +19,31 @@
|
||||||
\usepackage[version=4]{mhchem}
|
\usepackage[version=4]{mhchem}
|
||||||
\usepackage{pgfgantt}
|
\usepackage{pgfgantt}
|
||||||
\usepackage{setspace}
|
\usepackage{setspace}
|
||||||
|
\usepackage{listings}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
% my attempt to make MATLAB code look pretty
|
||||||
|
|
||||||
|
\definecolor{dkgreen}{rgb}{0,0.6,0}
|
||||||
|
\definecolor{gray}{rgb}{0.5,0.5,0.5}
|
||||||
|
\definecolor{mauve}{rgb}{0.58,0,0.82}
|
||||||
|
|
||||||
|
\lstset{frame=tb,
|
||||||
|
language=Matlab,
|
||||||
|
aboveskip=3mm,
|
||||||
|
belowskip=3mm,
|
||||||
|
showstringspaces=false,
|
||||||
|
columns=flexible,
|
||||||
|
basicstyle={\small\ttfamily},
|
||||||
|
numbers=none,
|
||||||
|
numberstyle=\tiny\color{gray},
|
||||||
|
keywordstyle=\color{blue},
|
||||||
|
commentstyle=\color{dkgreen},
|
||||||
|
stringstyle=\color{mauve},
|
||||||
|
breaklines=true,
|
||||||
|
breakatwhitespace=true,
|
||||||
|
tabsize=3
|
||||||
|
}
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
% benevolently force figures stay in their own subsection
|
% benevolently force figures stay in their own subsection
|
||||||
|
|
@ -4446,7 +4471,30 @@ hosted using \gls{aws} using their proprietary Aurora implementation.
|
||||||
|
|
||||||
The code is available here: \url{https://github.gatech.edu/ndwarshuis3/mdma}.
|
The code is available here: \url{https://github.gatech.edu/ndwarshuis3/mdma}.
|
||||||
|
|
||||||
\chapter{reaction kinetics code}
|
\chapter{binding kinetics code}
|
||||||
|
|
||||||
|
The \gls{stp} binding kinetic profile was fit and calculated using the following
|
||||||
|
MATLAB code. Note that the \inlinecode{geometry} parameter was varied so as to
|
||||||
|
minimize the \inlinecode{SSE} output.
|
||||||
|
|
||||||
|
\lstinputlisting{../code/diffusion_stp.m}
|
||||||
|
|
||||||
|
The geometric diffusivity from above (the \inlinecode{geometry} variable) was
|
||||||
|
used in the below code to obtain the reaction profile for the \gls{mab} binding
|
||||||
|
step. The model is the same except for the parameters which were changes to
|
||||||
|
reflect the \gls{mab} coating process.
|
||||||
|
|
||||||
|
\lstinputlisting{../code/diffusion_mab.m}
|
||||||
|
|
||||||
|
\chapter{washing kinetics code}
|
||||||
|
|
||||||
|
The wash steps for the \gls{dms} were modeled using the following code:
|
||||||
|
|
||||||
|
\lstinputlisting{../code/microcarrier_diffusion_washing.m}
|
||||||
|
|
||||||
|
Complete output from this code is shown below:
|
||||||
|
|
||||||
|
\input{../code/washing_out.tex}
|
||||||
|
|
||||||
\chapter{references}
|
\chapter{references}
|
||||||
\renewcommand{\chapter}[2]{} % noop the original bib section header
|
\renewcommand{\chapter}[2]{} % noop the original bib section header
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue