You can choose a commonly-used model without having to understand how it was derived. But if you want to derive your own models, you'll find it useful to understand the origins of the commonly-used models. Here are three examples. You don't have to be mathematically sophisticated to follow these derivations. They require only basic algebra and simple logic (one step of example 2 requires basic calculus, but you can accept that step on faith and follow the rest.)   

Example model 1. Optical density as a function of concentration
Example model 2. Exponential decay
Example model 3. Equilibrium binding  

Example model 1. Optical density as a function of concentration

Colorimetric chemical assays are based on a simple principle. Add appropriate reactants to your samples to initiate a chemical reaction whose product is colored. When you terminate the reaction, the concentration of colored product is proportional to the initial concentration of the substance you want to assay. Since optical density is proportional to the concentration of colored substances, the optical density will also be proportional to the concentration of the substance you are assaying.

Mathematically, the equation works for any value of X. However, the results only make sense with certain values. Negative X values are meaningless, as concentrations cannot be negative. The model may fail at high concentrations of substance where the reaction is no longer limited by the concentration of substance. The model may also fail at high concentrations if the solution becomes so dark (the optical density is so high) that little light reaches the detector. At that point, the noise of the instrument may exceed the signal. It is not unusual that a model works only for a certain range of values. You just have to be aware of the limitations, and not try to use the model outside of its useful range.  

Example model 2. Exponential decay

Exponential equations are used to model many processes. They are used whenever the rate at which something happens is proportional to the amount which is left. Here are three examples:

	•	When ligands dissociate from receptors, the number of molecules that dissociate in any short time interval is proportional to the number that were bound at the beginning of that interval. Equivalently, each individual molecule of ligand bound to a receptor has a certain probability of dissociating from the receptor in any small time interval. That probability does not get higher as the ligand stays on the receptor longer.

	•	When radioactive isotopes decay, the number of atoms that decay in any short interval is proportional to the number of undecayed atoms that were present at the beginning of the interval. This means that each individual atom has a certain probability of decaying in a small time interval, and that probability is constant. The probability that any particular atom will decay does not change over time. The total decay of the sample decreases with time because there are fewer and fewer undecayed atoms.

	•	When drugs are metabolized by the liver or excreted by the kidney, the rate of metabolism or excretion is often proportional to the concentration of drug in the blood plasma. Each drug molecule has a certain probability of being metabolized or secreted in a small time interval. As the drug concentration goes down, the rate of its metabolism or excretion goes down as well.

Define Y to be the number of ligand-receptor complexes still present (or the number of radioactive atoms that have not yet decayed, or the concentration of drug in the plasma) at any given time X. The rate of change of Y is proportional to Y. Expressed as a differential equation:

Shown as a graph:

ike most nonlinear regression programs, Prism doesn't let you enter a model expressed as a differential equation. Instead, you must enter the equation defining Y as a function of X. To do this, you need to remember a bit of calculus. There is only one function whose derivative is proportional to Y, the exponential function. Integrate both sides of the equation to obtain a new exponential equation that defines Y as a function of X, the rate constant k, and the value of Y at time zero, Y0.

The half-life is the time it takes for Y to drop by 50%. To find the half-life, set Y equal to one-half of Y0 and solve the above equation for X.  It equals the natural logarithm of 2 divided by the rate constant.

In the case of radioactive decay, this model describes exactly what is going on physically. In the case of ligand binding, the model may be a simplification. The model considers that all receptors are either free or bound to ligand. In fact, binding is a complicated process with multiple points of contact between ligand and receptor, so there must be some states of partial binding. Even though the model is simplified, it predicts experimental data very well. Even very simple models can adequately predict the behavior of very complicated systems, and can yield constants (dissociation rate constant in this example) that have a physical meaning. Few models describe a physical process exactly. Models that simplify the true molecular or physiological mechanisms can be very useful, so long as they are not too simple.  

Example model 3. Equilibrium binding

This example derives a very common model that describes equilibrium binding (or enzyme kinetics). Deriving this model does not require any calculus!When a ligand interacts with a receptor, or when a substrate interacts with an enzyme, the binding follows the law of mass action.

In this equation R is the concentration of free receptor, L is the concentration of free ligand, and RL is the concentration of receptor ligand complex. In the case of enzyme kinetics, R is the enzyme and L is the substrate.The association rate constant kon is expressed in units of M^-1min^-1. The rate of RL formation equals R.L.kon. The dissociation constant koff is expressed in units of min^-1. The rate of RL dissociation equals RL.koff. At equilibrium, the backward (dissociation) reaction equals the forward (association) reaction so,Binding studies measure specific binding, which is a measure of RL. Enzyme kinetic assays assess enzyme velocity, which is proportional to RL, the concentration of enzyme-substrate complexes. So you want to arrange the equation to obtain RL on the left.

Define the equilibrium dissociation constant, Kd to equal koff/kon, which is in molar units. In enzyme kinetics, this is called the Michaelis-Menten constant, KM. Rearrange the previous equation to define the concentration of receptor-ligand complexes at equilibrium:

Since you usually can't measure the free concentration of receptor, R, the equation won't be useful until that variable is removed. Fortunately, that's easy to do. Since all receptors are either free or bound, we can express R as the total number of receptors minus the number bound: R= Rtot - RL.Substitute this definitions of R into the previous equation.

RL now appears on both sides of the equation. Rearrange to keep RL on the left.

Since we vary L and measure RL, define Y to be RL (amount of specific binding, or enzyme activity) and X to be L (concentration of ligand or substrate). Finally call the total amount of binding Bmax (instead of Rtot). The equation for equilibrium binding now becomes:

The graph of this equation (left panel below) is sometimes called a rectangular hyperbola or a binding isotherm. If you plot the same data on a semilog plot (the X-axis is log of ligand concentration) it becomes sigmoidal. The only difference between the left and right panel of the graph is whether the X-axis is linear or logarithmic.

The equation of enzyme velocity as a function of substrate concentration is identical except for the names of the variables.