Single molecule ligand/substrate reactions?

[Cocea]

Hello, bionetters!

I don't know much about chemical kinetics. 
I can understand, though, the logical coherence
of the mathematical model of the reversible 
interaction between a ligand and a substrate.
However, this model is based on a statistical 
assumption: there are many molecules
interacting and thus one can use (mathematical) 
derivatives (dm/dt) and other such tools
to describe this phenomenon. 

How about the situation when you have only one 
or two molecules of substrate and many
molecules of ligand? This corresponds to the 
binding of a promoter of a particular gene in
the genome of a cell to its regulatory proteins; 
these proteins come from the cytosol and,
as far as I know, there is no reason to believe 
that there are not many of them entering
the nucleus. Actual data show that proteins 
bearing a nuclear localisation sequence are -
as expected - located primarily in the nucleus. 

Reversible reactions involving many molecules are 
described in terms of reaction rates and
reaction constants at equilibrium. Can reactions 
where only one or two molecules of one of
the reaction components are available can be 
described by using analogous parameters? I see
such a reaction more like a 'quantum' reaction,
because there are only two or three possible
'states': state one: no binding; state two: one
DNA-factor complex; state three: two DNA-factor
complexes. 

There is a connection between the questions 
above and gel-shift interpretation. Please
correct me if I'm wrong: in preparing his or 
her gel-shift experiment, one naturally
supposes that if the nuclear extract contains 
a factor that binds with high affinity to
the DNA substrate (provided in large enough 
amounts, so that a read-out will be available
at the end), then the same thing happens in vivo 
(where only one or two molecules of
substrate are actually available). This kind of 
experiment involves, most often,
regulatory DNA sequences, and the reasoning 
is simple: high affinity binding in vitro ->
high affinity binding of a single molecule 
in vivo (by the way, beyond its initial
'alchemical' definition, affinity is defined 
in reactions using many molecules - can one
speak of affinity in a one-molecule reaction?) -> 
high frequency of binding/long halflife
in the interaction of the regulatory factors 
with their cognate DNA sequences. But how
does one know that this is actually true?

Is there a proven relation between binding 
affinity and the frequency of binding (or the
frequency of reaction)? Or is it that the 
frequency of reaction depends on the molar
concentrations of the reactants only? I have read 
the chapters on transcription regulation in
Lewin's Genes V and, indeed, increased affinity 
seems to be associated with an increased
frequency of binding (reaction), at least 
in connection with transcription - but has this
been mathematically modelled or proven in any 
quantitative way? 

I would appreciate any information or reference 
to a model describing such interactions,
as well as any opinions on the items above.

I post these questions to this newsgroup only,
so that all (virtual) answers can be gathered here.
one single place.

Laurentiu COCEA
cocea at citi2.fr