Biochemistry Online: An Approach Based on Chemical Logic

Biochemistry Online




Last Update:  3/25/16

  • Learning Goals/Objectives for Chapter 5A:  After class and this reading, students will be able to

    • write equations for the dissociation constant (KD), mass balance of total macromolecule (M0), total ligand (L0), and [ML]  as a function of L or Lo ([ML] = [M0][L]/(KD+ [L])  (when Lo >> Mo or when free [L] is known) and Y = fractional saturation =  Y = ([ML]/[M0] = [L]/(KD+ [L])
    • decide which of two given equations for [ML] should be used under conditions when the above conditions for L0 and L are given
    • based on the equation ([ML] = [M0][L]/(KD+ [L]) draw qualitative graphs for different given L0, L, and Kd values
    • determine fraction saturation given relatives values of Kd and L, assuming L0 >> M0
    • compare relative % bound for covalent binding of protons to an acid and noncovalent binding of a ligand to a macromolecule given pka/pH and Kd/L values
    • describe differences in binding curves for binding of a ligand to a macromolecule and the dimerization of a macromolecule
    • derive an equation which shows the relationships between the rate constant for binding (kon), dissociation (koff) and the thermodynamic dissociation (Kd) or equilibrium constant (Keq).
    • describe the structural and mathematic differences between specific and nonspecific binding
    • given a Kd, estimate t1/2 values for the lifetime of the ML complex.
    • describe techniques used to determine ML for given L or L0 values, including those that do and do not require separation of ML from M , so that Kd values for a M and L interaction can be determined
    • List advantages of isothermal titration calorimetry and surface plasmon resonance in determination of binding interaction parameters

    A2.  Interpretation of Binding Analyzes

    It is important to get a mathematical understanding of the binding equations and graphs. It is equally important to get an intuitive understanding of their properties. Just as we used the +/- 2 rule in determining at a glance the charge state of an acid, you need to be able to determine the extent of binding (how much of M is bound with L) given their relative concentrations and the Kd. The usual situation is that [Mo] is << [Lo]. What happens to the binding curves for M + L <===> ML if the Kd gets progressively lower?  Intuitively, you should expect that binding will increase, especially as L gets greater. The curves below should help you develop the intuition you need with respect to binding equilibria.

    Fig:  ML vs L at Varying Kd's

    Fig:  ML vs L at Even Lower Kd's

    Fig:  ML vs L at a Very Low Kd!

    Note that in the last graph, given the same Mo and Lo concentrations, the "titration curves" for a binding equilibrium characterized by even tighter binding (for example, a Kd = 0.5 pM or 0.05 pM) would be indistinguishable from the graph when Kd = 5 pM.  It should be apparent that for all of these Kd values, all of the added ligand is bound until [Lo] > [Mo].  To differentiate these cases, much lower ligand concentrations would be required such that on addition of ligand, all is not bound.    Also note that this curve is NOT hyperbolic, which makes sense since the graph is of Y vs Lo, not Y vs L, and since Lo is not >> Mo.

    WolframWolfram Mathematica CDF Player - Interactive Graph of Y vs L at two different Kd values (free plugin required)


    SageMathLogoInteractive SageMath Graph: Y vs L at 2 different Kd values

    It is quite interesting to compare graphs of Y (fractional saturation) vs L (free) and Y vs Lo (total L) in the special case when Lo is not >> Mo.    Examples are shown below when Mo = 4 μM, Kd = 0.19 μM .  Under the ligand concentration used, it should be apparent the L can't be approximated by Lo.

    Figure:   Y vs L and Y vs Lo when Lo is not >> Mo


    Two points should be evident from these graphs when  L is not approximated by Lo:

    • a graph of Y vs Lo is not truly hyperbolic, but it does saturate

    • a Kd value (ligand concentration at half-maximal binding) can not be estimated by inspection from the Y vs Lo, but it can be from the Y vs L graph.

    Fig:  Comparison of Covalent Binding of Protons vs Noncovalent Binding of Ligand



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