Hill plot

Peter Gegenheimer peterg at rnaworld.bio.ukans.edu
Sat Jan 21 18:44:59 EST 1995

In <19950116110551.deits at readingrm.bch.msu.edu>, deits at deits@pilot.msu.edu  (Tom Deits) writes:
>In Article <maga-120195170139 at> "maga at vetbio.unizh.ch (Giovanni Maga)" says:
>> , if you obtain a non
>> Mich.-Menten curve, but a sigmoidal one, can you use Hill plot as a test
>> for real cooperativity? 
>Perhaps the best way to answer this is to consider what we mean by
>cooperativity.  I would be inclined to define it as an observed deviation
>from Michaelis-Menten or classical binding curves - a strictly operational
>definition.  In this case, the answer to your question is that any Hill
>plot with n not = 1 exhibits 'real cooperativity.'  However, your question
>probably is better phrased 'can you use a Hill plot as a test for allosteric
>interactions'  which is one physical mechanism which may result in a 
>cooperative effect.  In that case, the answer is no- there is no way to
>prove the underlying mechanism from the kinetic phenomenon, which is of 
>course the classical dilemma in kinetic analysis.
>       Other mechanisms which might lead to cooperativity without invoking
>an allosteric response and which I haven't seen mentioned in this thread
>include hysteresis- a slow, substrate-induced conformation change in a 
>enzyme that means that the steady state is not attained rapidly at all
>substrate concentrations.  This causes a violation of the steady state
>assumption, and can manifest itself as a cooperative interaction.  A good 
>review of this topic can be found in Annuual Review of Biochemistry by
>Carl Frieden in the mid-80's.  
>                               Tom Deits       
>                               deits at pilot.msu.e

This is totally accurate -- all allosteric kinetics give sigmoidal V vs [S]
plots, but not all sigmoidal plots result from allosteric kinetics.

However, the way to tell whether you have a true sigmoidal curve is not the 
"Hill equation" ( a linear double-log plot of the Hill equation), but by 
direct non-linear curvefitting to the Hill equation. The Hill equation is:

  V = Vmax * ([S]0.5)^nH / ( ([S]0.5)^nH + [S]^nH )
  where [S]0.5 is the apparent Km and nH is the Hill coefficient.
This equation will give a precise fit to any allosteric data.

It is important to note the limitations of this curve-fitting: the Hill 
equation is a simplified form of an exact model of sequential allosteric 
kinetics for an enzyme having n active sites.  In the Hill equation, the 
contributions of intermediate forms of the enzyme have been ignored.  As a 
result, the value n (# of sites) is replaced with nH (apparent # of 
fully-cooperative sites.  (Curve-fitting to exact sequential allosteric models
generally involves too many unknowns to get accurate results.)

This information is based on Segal's "Biochemical Calculations", and has been 
verified for kinetics of glycogen phosphorylase b.

|  Peter Gegenheimer            |  pgegen at kuhub.cc.ukans.edu             |
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