NMR Reporter Assays

In the absence of a competitor the fraction of bound reporter is calculated using the following logic.

Where \([P]\) and \([L]\) are the free protein and reporter concentrations and \([PL]\) is the bound concentration.

Using the mass conservation law, the free reporter and protein concentrations can be expressed in terms of the bound concentration and the total reporter and protein concentrations (\([L]_T\) and \([P]_T\), respectively):

Substituting these into the \(K_d\) equation gives

Which can be rearranged into a quadratic form to solve for \([PL]\).

Since the fraction of bound reporter, \(f_0=\frac{[PL]}{[L_T]}\)

The plot below shows the bound fraction of reporter across a typical range of conditions that was tested by Vieira de Castro and Ciulli (2019). It is clear the protein concentration has a more significant effect on the bound fraction than the reporter concentration in this experimental range. Selecting a reporter and protein concentration becomes a trade off between having high \(C_2\), \(Z'\) and signal to noise. In this case, a good assay could be developed by selecting 50 μM reporter and 0.5-1 μM protein.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib
matplotlib.use('Agg')

def f0(Pt: float, Lt: float, Kd: float) -> float:
    """ calculate fraction of bound ligand for 1:1 binding model

        Parameters
        ----------
        Pt : float
            total concentration of protein (M)
        Lt : float
            total concentration of ligand (M)
        Kd : float
            dissociation constant (M)

        Returns
        -------
        f : float
            fraction of ligand bound to protein

    """
    a = 1.0
    b = -1.0*(Pt+Lt+Kd)
    c = Pt*Lt
    return (-b-np.sqrt(b**2-4*a*c))/(2*Lt)

Pt = [0.125,0.250,0.500,1.000]
Lt = [5,10,25,50,100]
Kd = 145.0
fs = [["Protein","Reporter","Fraction"],]
for p in Pt:
    for l in Lt:
        f = f0(p,l,Kd)
        fs.append([p,l,f])
df = pd.DataFrame(fs[1:],columns=fs[0])
for name, group in df.groupby("Protein"):
    plt.plot(group.Reporter, group.Fraction,"o-",label=f"{name:.2f}")

plt.legend(title="[Protein] - μM")
plt.xlabel("[Reporter] - μM")
plt.ylabel(r"fraction of bound reporter - $f_0$")
plt.title("Reporter 19 ($K_d=145\pm29$ μM) from Vieira de Castro and Ciulli (2019)")
fname = "../static/fraction_bound.png"
plt.savefig(fname)
return fname

This is defined as the total competitor concentration (i.e. \([I]+[PI]\) or free competitor + bound competitor, resp) required to reduce the fraction of bound reporter ligand by 50% (\(f=f_0/2\)).

The following graph demonstrates the range of \(IC_{50}\) values that it would be possible to measure using a competition assay for a given reporter \(K_d\) and fraction of bound receptor (\(f_0\)). The point at which the slope starts to tend towards 0 represents the lower limit of \(K_i\) that could be accurately measured by the assay. The smaller the \(f_0\) the greater the range of \(IC_{50}\) values that can be measured. NMR based reporter assays have the advantage of being sensitive to a wide range of \(IC_{50}\) values since these assays can be effectively performed at very low \(f_0\) values (\(f_0\) ~ 0.005). There is a trade off between achieving a low \(f_0\) and a high enough signal to noise (\(S/N\)) ratio and \(C_2\) value to give a robust assay.

from typing import Union
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

def calcIC50(Ki: Union[float, np.array], Kd: float,f0: float, L0: float) -> Union[float, np.array]:
    """ calculate IC50 value

        Parameters
        ----------
        Ki : float | np.array
            dissociation constant of competitor (M)
        Kd : float
            dissociation constant of reporter (M)
        f0 : float
            initial bound fraction of reporter
        L0 : float
            reporter concentration in (M)

        Returns
        -------
        ic50 : float | np.array
            concentration of competitor at which 50% displacement is achieved
            i.e. f = f0/2

    """
    ic50 = ((f0*Kd)/((1.0-f0)*(2.0-f0)) + (f0*L0)/2.0) * ((Ki*(2.0-f0))/(Kd*f0)+1.0)
    return ic50

Kd = 50e-6 # 50 μM
Ki = np.logspace(-9,-3)
L0 = 50e-6 # reporter concentration
f0s = [0.005,0.05,0.5] # fraction of bound ligand
for f0 in f0s:
    ic50 = calcIC50(Ki,Kd,f0,L0)
    plt.loglog(Ki,ic50,"o-",label=f"f0 = {f0}")
plt.grid(linestyle="--",alpha=0.75)
plt.ylabel("IC$_{50}$ (M)")
plt.xlabel("K$_{i}$ (M)")
plt.legend(title=f"Kd = {Kd*1e6:.1f} μM")
fname = "../static/ic50.png"
plt.savefig(fname)
return fname

An exact mathematical expression for competitive binding was defined by Wang (1994). For a simple 1:1 binding systems this involves solving a quadratic expression (see Simple binding model). However, for a competitive binding system a cubic expression must be solved (see https://en.wikipedia.org/wiki/Cubic_equation).

The following reaction scheme describes a competition experiment:

Where \([P]\), \([L]\) and \([I]\) are protein, reporter and competitor concentrations and \([PL]\) and \([PI]\) are the bound reporter and bound competitor concentrations respectively.

and

The solution is found with a cubic expression and Wang (1995) showed that this can be solved using one of the unique and physically meaningful cubic roots shown below.

Since fraction of bound ligand is \(f=\frac{[EL]}{L_0}\) then

Below is a plot showing simulated IC50 curves and demonstrates the range of \(Ki\) values that it would be possible to accurately measure using a model reporter system with the following experimental parameters:

Weaker reporters allow for a wider range of \(K_i\) measurements. Due to limitations with small molecule solubility it is unlikely that competitor concentrations greater than 1 mM are accessible. For example, this means that for a 5 μM \(K_d\) reporter the upper limit for observing full displacement is around the low μM \(K_i\) range, whereas for a weaker 500 μM \(K_d\) reporter almost 100% displacement can be achieved in the high μM \(K_i\) range upon addition of 1 mM of competitor.

from typing import Union
from cycler import cycler
import numpy as np
import matplotlib
from matplotlib import cm
matplotlib.use('Agg')
import matplotlib.pyplot as plt

def calcF(I: Union[float,np.array], Lt: float, Ki: float, Kd: float, Et: float) -> Union[float, np.array]:
    """ Calculate the fraction of bound reporter

        Wang (1995) defined an exact analytical solution to competitive binding.
        The following is a slight rearrangement of equation 14 to obtain the bound
        fraction of ligand based on the unique physically meaningful root of the
        cubic expression.

        Parameters
        ----------
        I : float | np.array
            concentration of competitor in M
        Lt : float
            total concentration of reporter `L_t` in M
        Ki : float
            `K_i` for the competitor in M
        Kd : float
            `K_d` for the reporter in M
        Et : float
            total concentration of protein `E_t` in M

        Returns
        -------
        f : float | np.array
            fraction of reporter bound to protein in the presence of competitor

    """
    a = Kd + Ki + Lt + I + Et
    b = (I-Et)*Kd + (Lt-Et)*Ki + Kd*Ki
    c = -Kd*Ki*Et
    theta = np.arccos((-2*a**3 + 9*a*b - 27*c)/(2*np.sqrt((a**2-3*b)**3)))
    d = 2*np.sqrt((a**2-3*b))*np.cos(theta/3) - a
    f = d / (3*Kd + d)
    return f

# some fancy colors
cmap = cm.viridis
len_cycle = 7
c = np.linspace(0,1,len_cycle)
custom_cycler = (cycler(color=[cmap(i) for i in c]))# +
                 #cycler(lw=np.arange(len_cycle)+1))

# configure plot
fig = plt.figure(figsize=(8,10))
ax1 = fig.add_subplot(211)
ax2 = fig.add_subplot(212)
ax1.set_prop_cycle(custom_cycler)
ax2.set_prop_cycle(custom_cycler)

# set parameters
Kis = np.logspace(-9,-3,len_cycle)
Lt = 50e-6
Kds = [5e-6,500e-6]
Et = 1e-6
I = np.logspace(-7,-1) # reporter concentration

# calculate and plot
axes = [ax1,ax2]
for Kd,ax in zip(Kds,axes):
    for Ki in Kis:
        Fw = calcF(I,Lt,Ki,Kd,Et)
        ax.semilogx(I,Fw,"o-",label=f"{Ki:.1E}")

    ax.grid(which='both',linestyle="--",alpha=0.75)
    ax.set_ylabel("$f$")
    ax.set_xlabel("[I] (M)")
    ax.set_title(r"$K_d$"+f" = {Kd*1e6:.1f} μM"+r", $L_t$"+ f" = {Lt*1e6:.1f} μM, " + r"$E_t$ " + f"= {Et*1e6:.1f} μM")
    ax.legend(title=r"$K_i$ (M)")
plt.tight_layout()
fname = "../static/fraction_bound_2.png"
plt.savefig(fname)
return fname

Viriera and Ciulli (2019) have described an assay quality metric, \(C_2\), or contrast which is the fractional decrease in signal intensity in the presence of protein. Larger \(C_2\) values are usually preferred as this improves the dynamic range of the assay.

Written in terms of intensity ratios this is equivalent to:

Where \(\frac{ I_{\tau,long,+protein} }{ I_{\tau,short,+protein}}\) is the ratio of reporter signal intensities at the long and short delay time \(\tau\) in the presence of protein and \(\frac{ I_{\tau,long,free} }{ I_{\tau,short,free}}\) is the ratio of reporter signal intensities in the absence of protein at the long and short delay times, respectively.

Another useful metric for defining the quality of the reporter assay is the \(Z'\) score which was first described by Zhang, Chung and Oldenburg (1999) for use in high-throughput screening (HTS) applications.

Where \(\mu_{c+}\) and \(\mu_{c-}\) are the mean value for the positive (with protein) and negative (without protein) controls, respectively, and \(\sigma_{c+}\) and \(\sigma_{c-}\) are the standard deviations of the values for the positive and negative controls respectively. Calculation of a reasonable \(Z'\) score would require that one includes \(>3\) control samples ( ideally more!). \(Z'\) values of greater than 0.5 are considered to be good. However, as reported by Almeida, Panova and Walser (2021) \(Z'\) values greater than 0.8 should be achievable using the ratio of ratios method (see above) for an NMR reporter assay.

The fluorine nucleus has advantages over proton for use as a reporter

1. Background free
2. Highly sensitive to weak binding due to high chemical shift anisotropy (CSA) and large chemical shift difference between free and bound states

This means that high \(C_2\) can be achieved relative to \(^1H\) at a given concentration leading to wider assay dynamic range.

A good starting point is 1 μM protein and 50 μM reporter. However, there is no substitute for testing a few different protein and reporter concentrations to judge which provides the highest \(C_2\), \(Z'\) and \(\frac{S}{N}\) ratios. Vieira de Castro and Ciulli (2019) demonstrated this very clearly. It is also a good idea to test multiple reporters across a range of affinities. As was shown by Vieira de Castro and Ciulli (2019) the optimal reporter affinity can vary from 10s to 100s of μM.

1. T2 filtered experiments
   • CPMG for \(^{19}F\)
   • \(T_{1\rho}\) for \(^1H\)