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

    • decide which of two given equations for [ML] should be used under conditions when the above conditions for L
    • write equations for the dissociation constant (KD), mass balance of total macromolecule (Mo), total ligand (Lo), and [ML]  as a function of L or Lo: ([ML] = [Mo][L]/(KD+ [L])  (when Lo >> Mo or when free [L] is known) and Y = fractional saturation =  Y = ([ML]/[Mo] = [L]/(KD+ [L])
    • 0 and L are given
    • based on the equation ([ML] = [Mo][L]/(KD+ [L]) draw qualitative graphs for different given Lo, L, and Kd values
    • determine fraction saturation given relatives values of Kd and L, assuming Lo >> Mo
    • 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 Lo 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

    We have studied macromolecule structure. Now it is time to impart function to these molecules. It is simple to imagine that before these molecules can perform a function, they must interact with specific molecule(s) or ligand(s) in their environment. In fact, binding and subsequent release of a ligand might be the sole function of the macromolecule (example myoglobin binding oxygen).   Binding is the first step necessary for a biological response (with the exception of visual transduction in which photo-induced isomerization of rhodopsin initiates the response).  To understand binding, we must consider the equilbria involved, how binding is affected by ligand and macromolecule concentrations, and how to experimentally analyze and produce binding  curves.

    A1.  Reversible Binding of a Ligand to a Macromolecule

    These derivations can be made and interpreted using simple principles from General Chemistry, which you reviewed and strengthened in Analytical Chemistry, with some slight differences. Biochemists rarely talk about equilibrium or association constants, but rather their reciprocals - the dissociation constants, Kd.   For the reactions M + L ↔  ML, where M is free macromolecule, L is free ligand, and ML is macromolecule-ligand complex (which is held together by intermolecular forces, not covalent forces), the Kd is given by

    Figure: M is free macromolecule, L is free ligand, and ML is macromolecule-ligand complex

    Notice the unit of Kd is molarity, M. The lower the Kd (i.e. the higher the [ML] at any given M and L), the tighter the binding. The higher the Kd, the looser the binding. Kd's for biological molecules are finely tuned to their environments. They vary from about 1 mM (weak interactions) for some enzyme-substrate complex, to pM - fM levels. Examples of very tight, non-covalent interactions include  the avidin (an egg protein)-biotin (a vitamin) and thrombin (enzyme initiating clotting)-hirudin (a leech salivary protein) complexes.

    M = macromolecule; L = ligand

    For a simple equilibrium M + L  <--> ML

    where M = free macromolecule, L = free ligand, and ML = bound M and L (a complex)

    3 equations can be written:

    Equation 1 - Dissociation constant: Kd = ([M]eq[L]eq)/[ML]eq = ([M][L])/[ML] (units of molarity)

    Equation 2 - Mass Balance of M: Mo = M + ML

    Equation 3 - Mass Balance of L: Lo = L + ML

    We would like to derive equations which give ML as a function of known or measurable values.  The Kd equations shows that ML depends on free M and free L.  From Equations 1-3, two different and equally valid equations can be derived for two different cases.  

    • Case 1:  used either when you can readily measure free L or when experimental conditions are such the Lo >> Mo, which is often encountered.  Under these latter conditions, free L = Lo, which you know without measuring it, simply by knowing how much total ligand was added to the system.  
    • Case 2 (more general):  used when you don't know free L or haven't measured it, and you just wish to calculate how much ML is present at equilibrium.  These conditions imply that  Lo is not >> Mo.  (If Lo >> Mo, we would know free L = Lo.) 


    Equation 4 - Substitute 2 into 1: Kd = ([M][L])/[ML] = [Mo-ML][L])/[ML]

    (ML)Kd = (Mo)L - (ML)L

    (ML)Kd + (ML)L = (Mo)L

    (ML)(Kd+L) = (Mo)L

    Equation 5: ML = MoL/(Kd + L)

    This equation is ALWAYS TRUE for the chemical equation written above. L is the free ligand concentration at equilibrium.

    WolframWolfram Mathematica CDF Player - Interactive Graph of ML vs L at different Mo and Kd values (free plugin required)


    SageMathLogoInteractive SageMath Graph: ML vs L at different Mo and Kd values


    If Lo >> Mo, then the equations simplifies to:

    Equation 6: ML = MoLo/(Kd + L)..

    Dividing Equation 5 by Mo gives the fractional saturation of the macromolecule M, where

    Equation 7: Y = θ= [ML]/Mo = L/(Kd + L)

    where Y can vary from 0 (when L = 0) to 1 (when L >> Kd)

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




    Graphs of ML vs L (equation 5) and ML vs Lo (equation 6), when Lo >> Mo, and Y vs L (equation 7) are all HYPERBOLAs

    Equations 5.  ML = MoL/(Kd + L)   (and by analogy 6 and 7) can be understood best by examining three cases:

    Case 1: L = 0, ML = 0

    Case 2: L = Kd,  ML = MoL/(L + L)= MoL/2L = Mo/2

    which indicates that M is half saturated. In fact the operational definition of Kd is the ligand concentration at which the M is half saturated.

    Case 3: L >> Kd,  ML = Mo


    Equation 8 - Substitute 2 AND 3 into 1: K d = ([M][L])/[ML] = [Mo-ML][Lo-ML]/[ML]

    (ML)Kd = (Mo - ML)(Lo - ML)

    (ML)Kd = (Mo)(Lo) - (ML)(Lo) - (ML)(Mo) + (ML)2 or

    Equation 9: (ML)2 - (Lo + Mo +Kd)(ML) + (Mo)(Lo) = 0, which is of the form

    ax2 + bx + c = 0, where

    • a = 1

    • b = - (Lo + Mo +Kd)

    • c = (Mo)(Lo)

    which are all constants, and

    x = {-b +/- (b2 - 4ac)1/2}/2a or

    Equation 10: ML = {(Lo+Mo+Kd) - ((Lo+Mo+Kd)2 - 4MoLo)1/2}/2


    WolframWolfram Mathematica CDF Player - Interactive Graph of ML at various Lo, Mo, and Kd values (free plugin required)


    A graph of ML calculated from this formula vs free L (or Lo if  Lo >> Mo) give a  A HYPERBOLA.

    Play around with the sliders.  If you set Kd to a very low number and vary Mo, you will see a curve very much like a titration curve with a sharp rise and abrupt plateau that occurs when Mo is approximately equal to Lo. 


    SageMathLogoInteractive SageMath Graph: ML at various Lo, Mo and Kd values

    In the derivations, we came up with two equations for ML:

    • one (Equation 5) using mass conservation on M, which gave:    ML = MoL/[Kd +L]
    • one (Equation 10) using mass conservation on M and L, which gave ML = quadratic equation as function of Mo, Lo, and Kd:  ML = {-(Lo+Mo+Kd) +/- ((Lo+Mo+Kd)2 - 4MoLo)1/2}/2

    Both equations are valid. In the first you must known free L which is often Lo if Mo << Lo. In the second, you don't need to know free M or L at all.  At a given Lo, Mo, and Kd, you can calculate ML, which should be the same ML you get from the first equation if you know free L. 

    Equations 5 and 10 are useful in several circumstances.  They can be used to

    • calculate the concentration of ML if Kd, Mo, and L (for equation 5) or if Kd, Mo, and Lo (for equation 10) are known.  This is analogous to the use of the Henderson-Hasselbach equation to calculate the protonation state (HA) and hence charge state of an acid at various pH values.  In the former case we are measuring the concentration of bound ligand (ML) and in the later case, the concentration of bound protons (HA). 
    • calculate Kd if ML, Mo, and L (for equation 5) or if ML, Mo, and Lo (for equation 10) are known.  Techniques to extract the Kd from binding data will be discussed in the next chapter section.



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