[RASMB] still 2 ligands and even more equations...

[Holger Strauss]

Hello,

thanks a lot to Walter Stafford, Dan Hallen and Pierre Bonneau for
their great input - great help and great inspirations.
There are also two Excel-files with the messages below, for which you
might contact Dan Hallen or Pierre Bonneau directly. 
(I hope that's OK with them?)

Thanks again, Holger


---------- Forwarded message ----------

There is a straight forward way of solving your problem analytically.
                  As already said you need to define your system 

                  P + A = PA  ; K(A)=[PA]/([P][A])

                  P + B = PB  ; K(B)=[PB]/([P][B])

                  Total concentrations of A, B an P
                  Pt=[PA]+ [PB] + [P]
                  At=[PA]+ [A]
                  Bt=[PB]+ [B]

                   Rewrite:
                  [PA]= [P]At/(1/K(A)+[P])
                  [PB]= [P]Bt/(1/K(B)+[P])

                  Introduce new relations

                  r(A)=At/Pt ; r(B)=Bt/Pt
                  c(A)=K(A)Pt   ;  c(B)=K(B)Pt

                  The sum of all fractions of protein species must be one
                  (x(P)=[P]/Pt;
                  x(PA)=[PA]/Pt; x(PB)=[PB]/Pt):

                  x(P)+x(PA)+x(PB)=1 

                  The fraction of protein species can be written as

                  x(PA)= r(A)x(P)/(1/c(A)+x(P))
                  x(PB)= r(B)x(P)/(1/c(B)+x(P))

                  So the sum of protein species can be written as:

                  x(P)^3 + ax(P)^2 + bx(P) +c=0

                  where
                  a=1/c(A)+1/c(B)+r(A)+r(B)-1
                  b=(r(A)-1)/c(B) + (r(B)-1)/c(A) +1/(c(A)c(B))
                  c= - 1/(c(A)c(B))

                  x(P)=(2sqrt(a^2-3b) cos(teta/3)-a)/3
                  teta=arccos(-2a^3+9ab-27c)/(2sqrt((a^2-3b)^3)) 

You can read about this in for instance 
Bent W. Sigurskjold, Analytical Biochemistry 277, 260-266
(2000)"Exact Analysis of Competition Ligand Binding by Displacement
Isothermal Titration Calorimetry"

****************************************************************************

It occurred to me that I may have a solution for your inquest about a
system of equations for A+B+C=AB+AC (two ligands+receptor).  The
attachment contains an excel spreadsheet in which you just have to enter
the initial concentrations of A, B,
and C as well as the two KDs (in cells B1..B5, no other cells should be
modify).  Since the system involves cubic forms, three solutions exist
(which appear at the bottom of the sheet)but my experience is that only
one ususally makes sense (the others often involve negative numbers).


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Holger Strauss 

Forschungsinstitut fuer Molekulare Pharmakologie (FMP)
Robert-Roessle Strasse 10

13125 Berlin/Germany

Tel: +49 (0)30 94793 - 223 (office)
                     - 316 (lab)

Fax: +49 (0)30 94793 - 169 


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Science is spectrum analysis; art is photosynthesis.

                                                    Karl Kraus