Testing of the U_R model
by Evgenii Kovrigin, 05/31/2011
Process A: R + L = RL Process B: R = R*
Contents
close all clear all figures_folder='U_R_testing_figures';
TEST COMPUTATION OF EQUILIBRIUM POPULATIONS
Set some meaningful parameters
Rtotal=1e-3; % Receptor concentration, M LRratio_array=[0 : 0.02 : 1.45]; % Array of L/R K_a_A=1e6; % Binding affinity constant K_a_B=4; % Isomerization constant (to make it similar to the U-R2 test case) % Set appropriate options for the model (see model file for details) model_numeric_solver='analytical' ; model_numeric_options='none';
Compute arrays for populations and plot
concentrations_array=[]; for counter=1:length(LRratio_array) % compute [concentrations species_names] = equilibrium_thermodynamic_equations.U_R_model(... Rtotal, LRratio_array(counter), K_a_A, K_a_B,... model_numeric_solver, model_numeric_options); % collect concentrations_array = [concentrations_array ; concentrations]; end
Plot
Figure_title= 'U-R model'; X_range=[0 max(LRratio_array)+0.1 ]; % extend X just a bit past last point Y_range=[]; % keep automatic scaling for Y % display figure figure_handle=equilibrium_thermodynamic_equations.plot_populations(... LRratio_array, concentrations_array, species_names, Figure_title, X_range, Y_range); % save it results_output.output_figure(figure_handle, figures_folder, 'Concentrations_plot');
Correct behavior: the R is split between R and R* with a constant ratio.
TEST OF NMR LINE SHAPE CALCULATION FOR U-R
We will generate line shapes for U-R in conditions where we can expect specific patterns. We will plot a spectrum at a specific solution conditions (we will not use Series of datasets for simplicity).
1. Ligand-free receptor in slow exchange should show two separate peaks with areas proportional to their equilibrium concentrations. This ratio should be fixed and independent of Rtotal.
2. Fast exchange in the same setting should reveal population-weighted average peak. Similarly, dilution should not have any effect on peak positions.
3. Addition of saturating concentration of ligand should give us a single peak at the RL frequency.
% Create data object for 1D NMR line shapes test1=NMRLineShapes1D('Simulation','Test_1'); test1.set_active_model('U_R-model', model_numeric_solver, model_numeric_options) test1.show_active_model() % to look up necessary parameters of the model
ans = Active model: Model 6: "U_R-model" Model description "1D NMR line shape for the U-R model, no temperature/field dependence, analytical" Model handle: line_shape_equations_1D.U_R_model_1D Current solver: analytical Model parameters: 1: Rtotal 2: LRratio 3: log10(K_A) 4: log10(K_B) 5: k_2_A 6: k_2_B 7: w0_R 8: w0_Rstar 9: w0_RL 10: FWHH_R 11: FWHH_Rstar 12: FWHH_RL 13: ScaleFactor
Set range for X to extend beyond resonances
w_min= -300; w_max= 500; datapoints=100; % Does not matter much because smooth curve is anyway calculated test1.set_X(linspace(w_min, w_max, datapoints)); test1.X_range=[w_min w_max]; % this sets display range in the plots % Set fixed range for Y test1.Y_range=[0 8e-5] ; % Set noise RMSD to 0 for both X and Y X_RMSD=0; Y_RMSD=0;
Slow exchange in free receptor
Use the same thermodynamic parameters as above. Add spectral and kinetic parameters:
LRratio = 0.001 ; Log10_K_A = log10(K_a_A); Log10_K_B = log10(K_a_B); k_2_A = 1; % dissociation rate constant of the complex, 1/s k_2_B = 1; % dissociation rate constant of the receptor dimer, 1/s w0_R = -200; % NMR frequency of the free receptor R, 1/s w0_R2 = 200; % NMR frequency of the dimer R2, 1/s w0_RL = 400; % NMR frequency of the bound complex RL, 1/s FWHH_R = 20; % line width at half height of the peak of R, 1/s FWHH_R2 = 20; % line width at half height of the peak of R2, 1/s FWHH_RL = 20; % line width at half height of the peak of RL, 1/s ScaleFactor = 1; % a multiplier for spectral amplitude (used only when fitting data) % compute equilbrium concentrations [concentrations species_names] = equilibrium_thermodynamic_equations.U_R_model(... Rtotal, LRratio, K_a_A, K_a_B,... model_numeric_solver, model_numeric_options) % plot line shapes parameters=[ Rtotal LRratio Log10_K_A Log10_K_B k_2_A k_2_B ... w0_R w0_R2 w0_RL FWHH_R FWHH_R2 FWHH_RL ScaleFactor ]; test1.simulate_noisy_data(parameters, X_RMSD, Y_RMSD); figure_handle=test1.plot_simulation('Slow: free R'); results_output.output_figure(figure_handle, figures_folder, 'Slow_concentrated_R');
concentrations =
1.0e-03 *
0.1998 0.7992 0.0010 0.0000
species_names =
'Req' 'Rstareq' 'RLeq' 'Leq'
Expected ratio of R* peak to R peak is 4:1.
Prepare TotalFit session for easy display of many graphs
storage_session=TotalFit('All_tests'); % set aside for plotting later storage_session.copy_dataset_into_array(test1);
Dilution of free receptor
Rtotal=1e-4; % compute equilbrium concentrations [concentrations species_names] = equilibrium_thermodynamic_equations.U_R_model(... Rtotal, LRratio, K_a_A, K_a_B,... model_numeric_solver, model_numeric_options) parameters=[ Rtotal LRratio Log10_K_A Log10_K_B k_2_A k_2_B ... w0_R w0_R2 w0_RL FWHH_R FWHH_R2 FWHH_RL ScaleFactor ]; test1.simulate_noisy_data(parameters, X_RMSD, Y_RMSD); figure_handle=test1.plot_simulation('Slow: 1/10 R'); results_output.output_figure(figure_handle, figures_folder, 'Slow_diluted_R'); % set aside for plotting later storage_session.copy_dataset_into_array(test1);
concentrations =
1.0e-04 *
0.1998 0.7992 0.0010 0.0000
species_names =
'Req' 'Rstareq' 'RLeq' 'Leq'
When receptor is diluted there is no change in the ratio between the areas of two peaks reflecting independence of R-R* equilibrium of concentration.
Fast exchange in free receptor
Diluted
k_2_B=1000; parameters=[ Rtotal LRratio Log10_K_A Log10_K_B k_2_A k_2_B ... w0_R w0_R2 w0_RL FWHH_R FWHH_R2 FWHH_RL ScaleFactor ]; test1.simulate_noisy_data(parameters, X_RMSD, Y_RMSD); figure_handle=test1.plot_simulation('Fast: 1/10 R'); results_output.output_figure(figure_handle, figures_folder, 'Fast_diluted_R'); % set aside for plotting later storage_session.copy_dataset_into_array(test1);
The weighted average peak is at 4/5 and 1/4 between resonance frequencies of R* and R.
Concentrated: Return to the original condition
Rtotal=1e-3; % compute equilbrium concentrations [concentrations species_names] = equilibrium_thermodynamic_equations.U_R_model(... Rtotal, LRratio, K_a_A, K_a_B,... model_numeric_solver, model_numeric_options) parameters=[ Rtotal LRratio Log10_K_A Log10_K_B k_2_A k_2_B ... w0_R w0_R2 w0_RL FWHH_R FWHH_R2 FWHH_RL ScaleFactor ]; test1.simulate_noisy_data(parameters, X_RMSD, Y_RMSD); figure_handle=test1.plot_simulation('Fast: 1/1 R'); results_output.output_figure(figure_handle, figures_folder, 'Fast_concentrated_R'); % set aside for plotting later storage_session.copy_dataset_into_array(test1);
concentrations =
1.0e-03 *
0.1998 0.7992 0.0010 0.0000
species_names =
'Req' 'Rstareq' 'RLeq' 'Leq'
The peak is not shifted upon increasing Rtotal as expected.
A fully saturated receptor
LRratio = 10 ; parameters=[ Rtotal LRratio Log10_K_A Log10_K_B k_2_A k_2_B ... w0_R w0_R2 w0_RL FWHH_R FWHH_R2 FWHH_RL ScaleFactor ]; test1.simulate_noisy_data(parameters, X_RMSD, Y_RMSD); figure_handle=test1.plot_simulation('saturated RL'); results_output.output_figure(figure_handle, figures_folder, 'Saturated_RL'); % set aside for plotting later storage_session.copy_dataset_into_array(test1);
We observe a peak at a frequency of bound form
Summary of individual plots
storage_session.define_series('All-tests',1:length(storage_session.dataset_array)); figure_handle=storage_session.plot_1D_series('All-tests', 'All tests', 0); results_output.output_figure(figure_handle, figures_folder, 'All_tests');
This is a comparison graph of all previous plots
Create a titration series
(see Tutorial 3 for details) Choose parameters
ZERO_LRratio= 1e-3; % This number cannot be too small---the model is numeric and will NOT converge! LRratio_vector=[ZERO_LRratio 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ]; k_2_A=2000; % fast exchange in A k_2_B=1; % slow exchange in B % choose the same parameters as above session_name='Series'; session=TotalFit(session_name); series_name='U-R_simulation'; dataset_class_name='NMRLineShapes1D'; model_name='U_R-model'; number_of_datasets=length(LRratio_vector); X_column=test1.X; % use the same range % create TotalFit session session.create_simulation_series(series_name, dataset_class_name, model_name, ... model_numeric_solver, model_numeric_options,... number_of_datasets, X_column); session.initialize_relation_matrix(sprintf('%s.all_datasets.txt', session_name)); % link parameters parameter_number_array=[ 1 3 4 5 6 7 8 9 10 11 12 13]; session.batch_link_parameters_in_series(series_name, parameter_number_array); session.generate_fitting_environment(sprintf('%s.fitting_environment.txt', session_name)) % prepare parameters first_dataset_index=session.dataset_index(series_name,1); FAKE_VALUE=1E-6; % individual (unlinked) parameters: to be assigned later ideal_parameters=[ Rtotal FAKE_VALUE Log10_K_A Log10_K_B k_2_A k_2_B ... w0_R w0_R2 w0_RL FWHH_R FWHH_R2 FWHH_RL ScaleFactor ]; % prepare formal limits fake_limits=zeros(1,length(ideal_parameters)); Monte_Carlo_range_min=fake_limits; Monte_Carlo_range_max=fake_limits; parameter_limits_min=fake_limits; parameter_limits_max=fake_limits; % assign session.assign_parameter_values(first_dataset_index, ideal_parameters, ... Monte_Carlo_range_min, Monte_Carlo_range_max, parameter_limits_min, parameter_limits_max); % Assign unlinked LRratio and ranges parameter_number=2; values_array=LRratio_vector; unlinked_fake_ranges=zeros(1,number_of_datasets); MonteCarlo_range_min_array=unlinked_fake_ranges; MonteCarlo_range_max_array=unlinked_fake_ranges; parameter_limits_min_array=unlinked_fake_ranges; parameter_limits_max_array=unlinked_fake_ranges; session.assign_unlinked_parameter_in_series(... series_name, parameter_number, values_array, ... MonteCarlo_range_min_array, MonteCarlo_range_max_array, ... parameter_limits_min_array, parameter_limits_max_array); % Simulate series X_standard_dev=0; Y_standard_dev=1e-8; % something really small session.simulate_series(series_name, X_standard_dev, Y_standard_dev); % Plot Y_offset_percent=10; % to offset figures vertically for easier viewing % set plotting ranges X_range=test1.X_range; % use old Y_range=[]; % automatic session.set_XY_ranges_series(series_name, X_range, Y_range) figure_handle=session.plot_1D_series(series_name, session.name, Y_offset_percent); results_output.output_figure(figure_handle, figures_folder, 'Series');
Dataset listing is written to "Series.all_datasets.txt". Fitting environment saved in Series.fitting_environment.txt. ans = New parameter relations (indexed as in all_parameters vector): [1] 1:Rtotal | 2:Rtotal | 3:Rtotal | 4:Rtotal | 5:Rtotal | 6:Rtotal | 7:Rtotal | 8:Rtotal | 9:Rtotal [2] 1:LRratio [3] 1:log10(K_A) | 2:log10(K_A) | 3:log10(K_A) | 4:log10(K_A) | 5:log10(K_A) | 6:log10(K_A) | 7:log10(K_A) | 8:log10(K_A) | 9:log10(K_A) [4] 1:log10(K_B) | 2:log10(K_B) | 3:log10(K_B) | 4:log10(K_B) | 5:log10(K_B) | 6:log10(K_B) | 7:log10(K_B) | 8:log10(K_B) | 9:log10(K_B) [5] 1:k_2_A | 2:k_2_A | 3:k_2_A | 4:k_2_A | 5:k_2_A | 6:k_2_A | 7:k_2_A | 8:k_2_A | 9:k_2_A [6] 1:k_2_B | 2:k_2_B | 3:k_2_B | 4:k_2_B | 5:k_2_B | 6:k_2_B | 7:k_2_B | 8:k_2_B | 9:k_2_B [7] 1:w0_R | 2:w0_R | 3:w0_R | 4:w0_R | 5:w0_R | 6:w0_R | 7:w0_R | 8:w0_R | 9:w0_R [8] 1:w0_Rstar | 2:w0_Rstar | 3:w0_Rstar | 4:w0_Rstar | 5:w0_Rstar | 6:w0_Rstar | 7:w0_Rstar | 8:w0_Rstar | 9:w0_Rstar [9] 1:w0_RL | 2:w0_RL | 3:w0_RL | 4:w0_RL | 5:w0_RL | 6:w0_RL | 7:w0_RL | 8:w0_RL | 9:w0_RL [10] 1:FWHH_R | 2:FWHH_R | 3:FWHH_R | 4:FWHH_R | 5:FWHH_R | 6:FWHH_R | 7:FWHH_R | 8:FWHH_R | 9:FWHH_R [11] 1:FWHH_Rstar | 2:FWHH_Rstar | 3:FWHH_Rstar | 4:FWHH_Rstar | 5:FWHH_Rstar | 6:FWHH_Rstar | 7:FWHH_Rstar | 8:FWHH_Rstar | 9:FWHH_Rstar [12] 1:FWHH_RL | 2:FWHH_RL | 3:FWHH_RL | 4:FWHH_RL | 5:FWHH_RL | 6:FWHH_RL | 7:FWHH_RL | 8:FWHH_RL | 9:FWHH_RL [13] 1:ScaleFactor | 2:ScaleFactor | 3:ScaleFactor | 4:ScaleFactor | 5:ScaleFactor | 6:ScaleFactor | 7:ScaleFactor | 8:ScaleFactor | 9:ScaleFactor [14] n/a [15] 2:LRratio [16] n/a [17] n/a [18] n/a [19] n/a [20] n/a [21] n/a [22] n/a [23] n/a [24] n/a [25] n/a [26] n/a [27] n/a [28] 3:LRratio [29] n/a [30] n/a [31] n/a [32] n/a [33] n/a [34] n/a [35] n/a [36] n/a [37] n/a [38] n/a [39] n/a [40] n/a [41] 4:LRratio [42] n/a [43] n/a [44] n/a [45] n/a [46] n/a [47] n/a [48] n/a [49] n/a [50] n/a [51] n/a [52] n/a [53] n/a [54] 5:LRratio [55] n/a [56] n/a [57] n/a [58] n/a [59] n/a [60] n/a [61] n/a [62] n/a [63] n/a [64] n/a [65] n/a [66] n/a [67] 6:LRratio [68] n/a [69] n/a [70] n/a [71] n/a [72] n/a [73] n/a [74] n/a [75] n/a [76] n/a [77] n/a [78] n/a [79] n/a [80] 7:LRratio [81] n/a [82] n/a [83] n/a [84] n/a [85] n/a [86] n/a [87] n/a [88] n/a [89] n/a [90] n/a [91] n/a [92] n/a [93] 8:LRratio [94] n/a [95] n/a [96] n/a [97] n/a [98] n/a [99] n/a [100] n/a [101] n/a [102] n/a [103] n/a [104] n/a [105] n/a [106] 9:LRratio [107] n/a [108] n/a [109] n/a [110] n/a [111] n/a [112] n/a [113] n/a [114] n/a [115] n/a [116] n/a [117] n/a Summary of datasets with lookup vectors (position of each parameter in all_parameter vector) -------- Dataset 1/U-R_simulation-1, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 2 3 4 5 6 7 8 9 10 11 12 13 Dataset 2/U-R_simulation-2, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 15 3 4 5 6 7 8 9 10 11 12 13 Dataset 3/U-R_simulation-3, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 28 3 4 5 6 7 8 9 10 11 12 13 Dataset 4/U-R_simulation-4, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 41 3 4 5 6 7 8 9 10 11 12 13 Dataset 5/U-R_simulation-5, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 54 3 4 5 6 7 8 9 10 11 12 13 Dataset 6/U-R_simulation-6, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 67 3 4 5 6 7 8 9 10 11 12 13 Dataset 7/U-R_simulation-7, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 80 3 4 5 6 7 8 9 10 11 12 13 Dataset 8/U-R_simulation-8, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 93 3 4 5 6 7 8 9 10 11 12 13 Dataset 9/U-R_simulation-9, model "U_R-model" Parameters : Rtotal LRratio log10(K_A) log10(K_B) k_2_A k_2_B w0_R w0_Rstar w0_RL FWHH_R FWHH_Rstar FWHH_RL ScaleFactor Lookup vector : 1 106 3 4 5 6 7 8 9 10 11 12 13
This is an expected result. RL resonance frequency averages with R frequency creating a shifting peak that crosses position of R* peak. R* disappears in-place in slow exchange with R.
CONCLUSIONS
The U-R model for NMR 1D line shapes works as expected.