Making a dose-response curve

A toy example of how pharmacologists determine binding coefficients using saturation binding data.

In pharmacology, the models used to interpret some measurable response to a range of administered concentrations are derived from the Hill equation. The most basic form of such a curve is: \(Y = B_{max}\times \frac{X}{X+EC_{50}}\), where B_max is the maximal response, and the EC₅₀ is the concentration at which half the response is achieved.

Here is one where you can control the B_max and EC₅₀. The axes are fixed so that the effects on the curve can be seen. We have a nice ligand here with an EC₅₀ in the range of 100 picomolar to 10 micromolar. For fun, I’ve also put in some pretend data so you can have an idea of what your data might look like.

viewof simple_b_max = Inputs.range([.1, 1], {value: 0.5, step: 0.1, label: "Bmax"})

viewof simple_k_d = Inputs.range([-10, -5], {value: -8, step: 0.1, label: "log Kd"})

x_arr = {
  const start = -10;
  const end = -5;
  const n_steps = 200;
  const step = (end-start)/n_steps;
  return [...Array(n_steps).keys()]
    .map(n => n*step+start)
    .map(n => 10**n)
}

sim_exp_x_arr = [1, 3, 10, 30, 100, 300, 1000, 3000, 10000].map(x => x*10**-9)

d_r = (x, bmax, ec50) => bmax * (x/(x+ec50))

simple_d_r = x_arr.map(x_val => ({dose: x_val, response: d_r(x_val, simple_b_max, 10**simple_k_d)}))

sim_exp_y = {
  let y_sim = sim_exp_x_arr.map((x_val => d_r(x_val, simple_b_max, 10**simple_k_d)))
  let random = d3.randomNormal(0, simple_b_max/20)
  let y_err = [Float64Array.from({length: 9}, random), Float64Array.from({length: 9}, random), Float64Array.from({length: 9}, random)]
  function addvector(a,b){
    return a.map((e,i) => e + b[i]);
  }
  let response = [
    addvector(y_sim, y_err[0]),
    addvector(y_sim, y_err[1]),
    addvector(y_sim, y_err[2])
    ]
  return [...Array(9).keys()].map(i => ({dose: sim_exp_x_arr[i],
                                         response1: response[0][i],
                                         response2: response[1][i],
                                         response3: response[2][i]}))
}

Plot.plot({
  marks: [
    Plot.line(simple_d_r, {x: "dose", y: "response"}),
    Plot.dot(sim_exp_y, {x: "dose", y: "response1"}),
    Plot.dot(sim_exp_y, {x: "dose", y: "response2"}),
    Plot.dot(sim_exp_y, {x: "dose", y: "response3"})
  ],
  x: {type: "log"},
  y: {domain: [0,1.2]}
})