Functions

This page includes the documentation of each function in binding_curve_viewer.py, see the source code at GitHub Page.

Determination of Dissociation Constant

binding_curve_viewer.dissociation_constant(pt_const=100.0, kd_const=100.0, plot_points=1000, shown_points_num=971)

The function dissociation_constant generates the saturation curves of protein-ligand complex. Here, we specify that the protein is the component in fixed concentration. The ligand can be any molecules, e.g., DNA, RNA, or peptides, that binds to the target protein. The dissociation constant, \(K_\mathrm{d}\), is defined by the equation

(1)\[K_\mathrm{d}=\frac{[\mathrm{P}][\mathrm{L}]}{[\mathrm{PL}]}\]

where \([\mathrm{P}]\) is the concentration of the free protein, \([\mathrm{L}]\) is the concentration of the free ligand, and \([\mathrm{PL}]\) is the concentration of the protein-ligand complex.

According to the parameters of \(K_\mathrm{d}\) and concentration of the total protein \([\mathrm{P}]_{\mathrm{total}}\), the following equation derived from equation 1 is used to generate the saturation curve by a step size sequence of \([\mathrm{PL}]\).

(2)\[[\mathrm{L}]_{\mathrm{total}}=\frac{([\mathrm{P}]_{\mathrm{total}}+K_\mathrm{d}-[\mathrm{PL}])[\mathrm{PL}]}{[\mathrm{P}]_{\mathrm{total}}-[\mathrm{PL}]}\]

Parameters:

• pt_const (float) – The concentration of the total protein (\([\mathrm{P}]_{\mathrm{total}}\)) in the unit of nM (default value is 100).

• kd_const (float) – The value of the \(K_\mathrm{d}\) in the unit of nM (default value is 100).

• plot_points (int) – The number of points (i.e., smoothness) of lines (default value is 1000).

• shown_points_num (int) – The number of points in lines shown in the figure (default value is 971). To avoid the long tail before proteins are fully occupied by high concentration of ligand, by default, binding curves end when 97.0% of total protein are bound. shown_points_num should be less than plot_points.

Returns:

A Tuple of widgets and figures, including p1_header, config_col, tab_conc, tab_frac can be used to generate the web application.

Return type:

Tuple

Kinetics of Association and Dissociation

binding_curve_viewer.kinetics_association_dissociation(p0=100, l0=150, koff=0.01, kon=100000.0, plot_points=1000, shown_points_num=991)

The function kinetics_association_dissociation generates the association and dissociation curves of protein-ligand complex. The association kinetics is described by the pseudo-first order process and second-order process. The ligand can be any molecules, e.g., DNA, RNA, or peptides, that binds to the target protein. The dissociation constant, \(K_\mathrm{d}\), is defined by the equation

(3)\[K_\mathrm{d}=\frac{k_{\mathrm{off}}}{k_{\mathrm{on}}}\]

where \(k_{\mathrm{off}}\) is the dissociation rate constant, \(k_{\mathrm{on}}\) is the association rate constant.

In the second-order binding process, the relationship between the concentration of the protein-ligand complex and the time is

(4)\[t=\frac{1}{k_{\mathrm{on}}([\mathrm{PL}]_1-[\mathrm{PL}]_2)}\ln\frac{[\mathrm{PL}]_2([\mathrm{PL}]_1-[\mathrm{PL}])}{[\mathrm{PL}]_1([\mathrm{PL}]_2-[\mathrm{PL}])}\]

where the \([\mathrm{PL}]_1\) and \([\mathrm{PL}]_2\) are the roots of a quadratic equation and they can be calculated from the \([\mathrm{P}]_0\), \([\mathrm{L}]_0\), and \(K_\mathrm{d}\) (\([\mathrm{P}]_0\) and \([\mathrm{L}]_0\) is the initial concentration of the protein and ligand, see Kinetics of the second-order binding process in Appendix 1 for their expressions).

In the pseudo-first-order binding process, the relationship between the concentration of the protein-ligand complex and the time is

(5)\[[\mathrm{PL}]=\frac{k_{\mathrm{on}}[\mathrm{L}]_0[\mathrm{P}]_0}{k_{\mathrm{on}}[\mathrm{L}]_0+k_{\mathrm{off}}}(1-\exp(-(k_{\mathrm{on}}[\mathrm{L}]_0+k_{\mathrm{off}})t))\]

Parameters:

• p0 (float) – The initial concentration of the free protein in the unit of nM (default value is 100).

• l0 (float) – The initial concentration of the free ligand in the unit of nM (default value is 150).

• koff (float) – The value of the dissociation rate constant (\(k_{\mathrm{off}}\)) in the unit of s^-1 (default value is 1e-2).

• kon (float) – The value of the association rate constant (\(k_{\mathrm{on}}\)) in the unit of M^-1 s^-1 (default value is 1e5).

• plot_points (int) – The number of points (i.e., smoothness) of lines (default value is 1000).

• shown_points_num (int) – The number of points in lines shown in the figure (default value is 991). To avoid the long tail before equilibrium, the binding curve ends when 99.0% equilibrium is reached, shown_points_num should be less than plot_points.

Returns:

A tuple of widgets and figures, including kin_header, kin_config_col, plot_kinetics_on, plot_kinetics_disso can be used to generate the web application.

Return type:

Tuple

Iso-affinity Plot

binding_curve_viewer.iso_affinity_plot(p0=100, l0=150, plot_points=100, shown_points_num=100)

To further compare different kinetic properties under the condition of \([\mathrm{P}]_0\) = 100 nM and \([\mathrm{L}]_0\) = 150 nM, we used this function to generate the iso-affinity graph of the second-order binding process. The resultant HTML file will show three parts: 1. iso-affinity graph; 2. binding curves; 3. a table of experiment information.

Parameters:

• p0 (float) – The initial concentration of the free protein in the unit of nM (default value is 100).

• l0 (float) – The initial concentration of the free ligand in the unit of nM (default value is 150).

• plot_points (int) – The number of points (i.e., smoothness) of lines (default value is 100).

• shown_points_num (int) – The number of points in lines shown in the figure (default value is 100). To avoid the long tail before equilibrium, the binding curve ends when 99.0% equilibrium is reached, shown_points_num should be less than plot_points.

Returns:

A tuple of widgets and figures, including p3_header, fig_points, fig_curves, kin_div, kin_table can be used to generate the web application.

Return type:

Tuple

Competitive Binding of Association

binding_curve_viewer.competitive_binding_association(p0=100, l0=75, i0=75, pl_koff=0.01, pl_kon=100000.0, pi_koff=0.01, pi_kon=100000.0, diff_calc_resolution=200000, t99Ntimes=10)

The simulation of the competitive binding of association is initiated by mixing an arbitrary volume of protein with an arbitrary volume of the mixture of ligand and inhibitor. The working concentration of each component at the time of mixing can be determined by the p0, l0, and i0.

(6)\[\mathrm{P} + \mathrm{L} \mathrel{\mathop{\rightleftarrows}^{k_{\mathrm{ligand\textnormal{-}off}}}_{k_{\mathrm{ligand\textnormal{-}on}}}} \mathrm{PL}\]

(7)\[\mathrm{P} + \mathrm{I} \mathrel{\mathop{\rightleftarrows}^{k_{\mathrm{inhibitor\textnormal{-}off}}}_{k_{\mathrm{inhibitor\textnormal{-}on}}}} \mathrm{PI}\]

The competitive binding is described by above equations. To describe the kinetic process of the competitive binding, we used the Mahan-Motulsky equation to calculate the relationship of the \([\mathrm{PL}]\) and time in the association process. The Mahan-Motulsky equation can only be applied in the pseudo-first-order binding process. We devised the numerical method to simulate the second-order kinetic process of the competitive binding of both association and dissociation. See the Methods of the paper for detailed expression.

Parameters:

• p0 (float) – The initial concentration of the free protein in the unit of nM (default value is 100).

• l0 (float) – The initial concentration of the free ligand in the unit of nM (default value is 75).

• i0 (float) – The initial concentration of the free inhibitor in the unit of nM (default value is 75).

• pl_koff (float) – The value of the ligand’s dissociation rate constant (\(k_{\mathrm{ligand\textnormal{-}off}}\)) in the unit of s^-1 (default value is 1e-2).

• pl_kon (float) – The value of the ligand’s association rate constant (\(k_{\mathrm{ligand\textnormal{-}on}}\)) in the unit of M^-1 s^-1 (default value is 1e5).

• pi_koff (float) – The value of the inhibitor’s dissociation rate constant (\(k_{\mathrm{inhibitor\textnormal{-}off}}\)) in the unit of s^-1 (default value is 1e-2).

• pi_kon (float) – The value of the inhibitor’s association rate constant (\(k_{\mathrm{inhibitor\textnormal{-}on}}\)) in the unit of M^-1 s^-1 (default value is 1e5).

• diff_calc_resolution (int) – The resolution of the numerical simulation (default value is 200,000).

• t99Ntimes (int) – The value affects the time length of the simulation (default value is 10).

Returns:

A tuple of widgets and figures, including p4_header, pro_div, lig_div, inhib_div, pro_spinner, lig_spinner, inhib_spinner, lig_koff_div, lig_koff_spinner, lig_kon_div, lig_kon_spinner, inhib_koff_div, inhib_koff_spinner, inhib_kon_div, inhib_kon_spinner, info_div, error_div, ic50_div, ref_div, plot_kinetics_comp_asso, ic50_fig can be used to generate the web application.

Return type:

Tuple

Competitive Binding of Dissociation

binding_curve_viewer.competitive_binding_dissociation(p0=200, l0=150, i0=200, pl_koff=0.01, pl_kon=100000.0, pi_koff=0.01, pi_kon=100000.0, diff_calc_resolution=400000, t99Ntimes=10, volume_ratio=1.0)

The simulation of the competitive binding of dissociation is initiated by mixing the equilibrated protein and ligand with the inhibitor in a certain volume ratio. p0, l0, and i0 is respectively the total concentration of the protein, ligand, and inhibitor.

(8)\[\mathrm{P} + \mathrm{L} \mathrel{\mathop{\rightleftarrows}^{k_{\mathrm{ligand\textnormal{-}off}}}_{k_{\mathrm{ligand\textnormal{-}on}}}} \mathrm{PL}\]

(9)\[\mathrm{P} + \mathrm{I} \mathrel{\mathop{\rightleftarrows}^{k_{\mathrm{inhibitor\textnormal{-}off}}}_{k_{\mathrm{inhibitor\textnormal{-}on}}}} \mathrm{PI}\]

The competitive binding is described by above equations. To describe the dissociation process of the competitive binding, we used the the numerical method to simulate the second-order kinetic process of the competitive binding of dissociation. See the Methods of the paper for detailed expression.

Parameters:

• p0 (float) – The total concentration of the protein before mixing in the unit of nM (default value is 200).

• l0 (float) – The total concentration of the ligand before mixing in the unit of nM (default value is 150).

• i0 (float) – The total concentration of the inhibitor before mixing in the unit of nM (default value is 150).

• pl_koff (float) – The value of the ligand’s dissociation rate constant (\(k_{\mathrm{ligand\textnormal{-}off}}\)) in the unit of s^-1 (default value is 1e-2).

• pl_kon (float) – The value of the ligand’s association rate constant (\(k_{\mathrm{ligand\textnormal{-}on}}\)) in the unit of M^-1 s^-1 (default value is 1e5).

• pi_koff (float) – The value of the inhibitor’s dissociation rate constant (\(k_{\mathrm{inhibitor\textnormal{-}off}}\)) in the unit of s^-1 (default value is 1e-2).

• pi_kon (float) – The value of the inhibitor’s association rate constant (\(k_{\mathrm{inhibitor\textnormal{-}on}}\)) in the unit of M^-1 s^-1 (default value is 1e5).

• diff_calc_resolution (int) – The resolution of the numerical simulation (default value is 400,000).

• t99Ntimes (int) – The value affects the time length of the simulation (default value is 10).

• volume_ratio (float) – The volume ratio of the mixture of protein and ligand to the inhibitor. By default, the ratio is 1.0, i.e., the volume of the mixture of protein and ligand is the same as the inhibitor. Its range is from 1.0 to 1000.0.

Returns:

A tuple of widgets and figures, including p5_header, pro_div, volume_ratio_div, lig_div, inhib_div, pro_spinner, volume_ratio_spinner, lig_spinner, inhib_spinner, lig_koff_div`, lig_koff_spinner, lig_kon_div, lig_kon_spinner, inhib_koff_div, inhib_koff_spinner, inhib_kon_div, inhib_kon_spinner, info_div, error_div, ic50_div, ref_div, plot_kinetics_comp_disso, ic50_fig can be used to generate the web application.

Return type:

Tuple

Simulation of High-Throughput Screening

binding_curve_viewer.hts_simulation(l0=10, kd=100, i0=10000.0)

This simulation is used to theoretically access the sensitivity of the competition-based high-throughput screening. The default experimental conditions are as follows. The \(K_\mathrm{d}\) of the protein and labeled ligand is 100 nM. In experimental groups, after mixing, the initial working concentration of the inhibitor (\([\mathrm{I}]_0\)) is 10 μM. After mixing, the concentration of the total ligand (\([\mathrm{L}]_\mathrm{total}\)) is 10 nM. After mixing, the equilibrium concentration of the protein-ligand complex in blank controls (\([\mathrm{PL}]_0\)) ranged from 10% to 90% of the \([\mathrm{L}]_\mathrm{total}\).

Parameters:

• l0 (float) – The total concentration of the labeled ligand after mixing in the unit of nM (default value is 10).

• kd (float) – The binding affinity of the protein and labeled ligand. The value of the \(K_\mathrm{d}\) in the unit of nM (default value is 100).

• i0 (float) – The working concentration of the inhibitor after mixing in the unit of nM (default value is 10000).

Returns:

A tuple of widgets and figures, including p6_header, info_table, div_kd, spinner_kd, div_inh, spinner_inh, div_lig, spinner_lig, plot_rect can be used to generate the web application.

Return type:

Tuple