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A calculator for finding binding parameters from a scatchard plot
ANALYTICAL
BIOCHEMISTRY
56, 306-309
A Calculator
for from
(1973)
Finding
Binding
a Scatchard
Parameters
Plot
The description of the binding or association of small molecules with macromolecules forms an important part of the understanding of many biological processes. The most widely used technique for finding the binding parameters is the Scatchard plot (1) ; a plot of r/u vs T, where r is that fraction of available binding sites of a macromolecule occupied by a ligand and u is the concentration of free ligand. Taking as a model a macromolecule with 5” types of binding sites with each type of site consisting of nj sites with association constant Ki, t,he Scatchard plot is used to extract, from experimental binding studies of this macromolecule, the parameters ni and Ki. When there is only one type of binding site the Scatchard plot is a straight. line corresponding to the equation
r/u=
-Kr+Kn
(11
describing the law of mass action for this simple system. The intercept on the r-axis of the Scatchard plot equals the number of binding sites, n, and the r/u intercept equals the product of n and the association constant, K. When the macromolecule has more than one type of binding site a curved line is obtained in the experimental Scatchard plot. The paramet,ers ni and Ki, for the ith type of sites, are now not so easily obtained. Rosenthal (2) has presented a particularly simple method for the determination and presentation of binding parameters from the Scatchard plot of such complex syst,ems. The graphic analysis for a system with two noninteracting groups of binding sites is summarized in Fig. 1. Line 1 and line 2 represent the “binding lines” for the two types of binding sites and the binding parameters can be extracted from each line as before for the case of a macromolecule with a single type of binding site. The curve in Fig. 1 presents the experimentally determined data for this system. The relation that must be satisfied in order that the two lines represent a proper breakdown of the complex experimental curve is OAj = BiCj, where j = 1 to Copyright All rights
m
for all rr~7z” radial lines.
306 @ 1973 by Academic Press, Inc. of reproduction in any form reserved.
(2)
SHORT
Fx. binding
307
COMMTlNICATIONS
1. Scatchard plot for a model sites on a macromolecule.
consisting
of one
ligand
and
two
types
of
A trial and error procedure for determining the “binding lines” from the experimental curve would be to choose two arbitrary lines and test the criteria represented by Eq. 2. The lines would be readjusted (slopes and intercepts) until a specified degree of match to the experimental curve was found. [See also (3) .] We have devised a particularly simple calculator to perform this trial and error procedure (Fig,. 21. The calculator is made from a l/s X 11 X 11 in. transparent lucite sheet. The ordinate and abscissa are marked on the lucite. A weak, coiled tension spring with a diameter of the order of $& in. is fastened, slightly stretched, along each of these axis. A string held between a slot on the spring on each axis will represent, a “binding
308
SHORT
COMMUNICATIONS
FIG. 2. Drawing of a calculator for determining the “binding lines” that fit ali experimental Scatchard plot. a, string loops; b, coiled tension spring; c, binding lines.
line.” Continuous string loops1 stretched between a hole at the origin of the axis and holes around the periphery of the lucite sheet represent the radial lines of Fig. 1. The knot or crimp which ties the string into a loop is darkened and serves as a marker on the top side of the lucite calculator. The procedure for determining the “binding lines” from an experimentally derived Scatchard plot, assuming that the system can be approximated by two groups of noninteracting sites, is as follows: 1. Place the transparent lucite sheet over an 81/ X 11 graph of the experimental points. 2. Represent a guess of the two “binding lines” on the calculator by placing a string for each of the lines between a slot of the spring from each axis. 3. Pick up the calculator and slide the string loop so that darkened knot is on t.he intersections (furthest from the origin) of the binding lines and radial lines (Points Bi of Fig. 1). 4. Place a spring clamp marker? on the string loop under the lucite at the intersections (closest to the origin) of the “binding lines” and radial lines (Points Ai of Fig. 1). 5. To perform the mathematical operation given by Eq. 2, slide the ‘String may be monofilament nylon fishing line. ‘For example, a miniature alligator clip or E-Z hook Products, Box 105, Covington, KY.).
clip
(E-Z
Hook
Test
309
SHORT COMMUNICATIONS
string loops until the bottom marker is at the hole at the origin. (Steps 4 and 5 may be performed sequentially, one loop at a time.j 6. The upper marker knots are now a Scatchard curve representing a system consisting of two groups of binding sites whose characteristics are given by the two “binding lines”. If the match of these points to the experimental curve on the graph paper beneath the calculator is not good, rechoose the lines and begin at step 3 again. Continue until a good match is obtained. If a good match cannot be obtained, the original assumption of two groups of noninteracting binding sites is probably false. It is possible to change the calculating procedure to include a third or more “binding lines”. Three lines, for example, can be matched as follows: After step 5 has been completed for two lines, move the lower markers to the intersections of a “third binding line” and the radial lines. Slide t.he string loops so that the lower marker is again at the origin. The darkened knots will be the new Scatchard plot. The mechanics of the procedure is not difficult but the ‘Lguessing” of the slopes and intercepts of the binding lines gets much more difficult for 3 and more lines. The calculator described can be used as an alternative to paper, pencil, and straight edge (3) for the extraction of binding parameters from an experimentally obtained Scatchard plot. This procedure sacrifices accuracy for the sake of speed and convenience and would t.herefore be the most useful when rapid convergence in finding the binding parameters is desired, as when exploring for qualitative features of a binding system. After the approximate lines are found, the calculator can be used as a simple overlay and more ac.curate determinations of the binding lines made using dividers, instead of movement of t,he string loops, to satisfy the conditions of Eq. 2. REFERENCES 1.
SCATCHARD.
G.
2. ROSENTHAL.
3.
DANCHIN,
H. -4.,
(1949)
N.Y.
Ann.
Acad.
Sci. 51, 660.
E. (1967) Anal. Biochem. 20, 525-532. AND GUERON, M. (1970) &r. J. Biochem.
16, 532.
BERNARD E. Department of Physiology Medical College of Pennsylvania
%OO Henru Avenue Philadelphia, Pennsylvania Received June 18, 1973;
19189 accepted Jztll~ 28, 1973
PENNOCK
BIOCHEMISTRY
56, 306-309
A Calculator
for from
(1973)
Finding
Binding
a Scatchard
Parameters
Plot
The description of the binding or association of small molecules with macromolecules forms an important part of the understanding of many biological processes. The most widely used technique for finding the binding parameters is the Scatchard plot (1) ; a plot of r/u vs T, where r is that fraction of available binding sites of a macromolecule occupied by a ligand and u is the concentration of free ligand. Taking as a model a macromolecule with 5” types of binding sites with each type of site consisting of nj sites with association constant Ki, t,he Scatchard plot is used to extract, from experimental binding studies of this macromolecule, the parameters ni and Ki. When there is only one type of binding site the Scatchard plot is a straight. line corresponding to the equation
r/u=
-Kr+Kn
(11
describing the law of mass action for this simple system. The intercept on the r-axis of the Scatchard plot equals the number of binding sites, n, and the r/u intercept equals the product of n and the association constant, K. When the macromolecule has more than one type of binding site a curved line is obtained in the experimental Scatchard plot. The paramet,ers ni and Ki, for the ith type of sites, are now not so easily obtained. Rosenthal (2) has presented a particularly simple method for the determination and presentation of binding parameters from the Scatchard plot of such complex syst,ems. The graphic analysis for a system with two noninteracting groups of binding sites is summarized in Fig. 1. Line 1 and line 2 represent the “binding lines” for the two types of binding sites and the binding parameters can be extracted from each line as before for the case of a macromolecule with a single type of binding site. The curve in Fig. 1 presents the experimentally determined data for this system. The relation that must be satisfied in order that the two lines represent a proper breakdown of the complex experimental curve is OAj = BiCj, where j = 1 to Copyright All rights
m
for all rr~7z” radial lines.
306 @ 1973 by Academic Press, Inc. of reproduction in any form reserved.
(2)
SHORT
Fx. binding
307
COMMTlNICATIONS
1. Scatchard plot for a model sites on a macromolecule.
consisting
of one
ligand
and
two
types
of
A trial and error procedure for determining the “binding lines” from the experimental curve would be to choose two arbitrary lines and test the criteria represented by Eq. 2. The lines would be readjusted (slopes and intercepts) until a specified degree of match to the experimental curve was found. [See also (3) .] We have devised a particularly simple calculator to perform this trial and error procedure (Fig,. 21. The calculator is made from a l/s X 11 X 11 in. transparent lucite sheet. The ordinate and abscissa are marked on the lucite. A weak, coiled tension spring with a diameter of the order of $& in. is fastened, slightly stretched, along each of these axis. A string held between a slot on the spring on each axis will represent, a “binding
308
SHORT
COMMUNICATIONS
FIG. 2. Drawing of a calculator for determining the “binding lines” that fit ali experimental Scatchard plot. a, string loops; b, coiled tension spring; c, binding lines.
line.” Continuous string loops1 stretched between a hole at the origin of the axis and holes around the periphery of the lucite sheet represent the radial lines of Fig. 1. The knot or crimp which ties the string into a loop is darkened and serves as a marker on the top side of the lucite calculator. The procedure for determining the “binding lines” from an experimentally derived Scatchard plot, assuming that the system can be approximated by two groups of noninteracting sites, is as follows: 1. Place the transparent lucite sheet over an 81/ X 11 graph of the experimental points. 2. Represent a guess of the two “binding lines” on the calculator by placing a string for each of the lines between a slot of the spring from each axis. 3. Pick up the calculator and slide the string loop so that darkened knot is on t.he intersections (furthest from the origin) of the binding lines and radial lines (Points Bi of Fig. 1). 4. Place a spring clamp marker? on the string loop under the lucite at the intersections (closest to the origin) of the “binding lines” and radial lines (Points Ai of Fig. 1). 5. To perform the mathematical operation given by Eq. 2, slide the ‘String may be monofilament nylon fishing line. ‘For example, a miniature alligator clip or E-Z hook Products, Box 105, Covington, KY.).
clip
(E-Z
Hook
Test
309
SHORT COMMUNICATIONS
string loops until the bottom marker is at the hole at the origin. (Steps 4 and 5 may be performed sequentially, one loop at a time.j 6. The upper marker knots are now a Scatchard curve representing a system consisting of two groups of binding sites whose characteristics are given by the two “binding lines”. If the match of these points to the experimental curve on the graph paper beneath the calculator is not good, rechoose the lines and begin at step 3 again. Continue until a good match is obtained. If a good match cannot be obtained, the original assumption of two groups of noninteracting binding sites is probably false. It is possible to change the calculating procedure to include a third or more “binding lines”. Three lines, for example, can be matched as follows: After step 5 has been completed for two lines, move the lower markers to the intersections of a “third binding line” and the radial lines. Slide t.he string loops so that the lower marker is again at the origin. The darkened knots will be the new Scatchard plot. The mechanics of the procedure is not difficult but the ‘Lguessing” of the slopes and intercepts of the binding lines gets much more difficult for 3 and more lines. The calculator described can be used as an alternative to paper, pencil, and straight edge (3) for the extraction of binding parameters from an experimentally obtained Scatchard plot. This procedure sacrifices accuracy for the sake of speed and convenience and would t.herefore be the most useful when rapid convergence in finding the binding parameters is desired, as when exploring for qualitative features of a binding system. After the approximate lines are found, the calculator can be used as a simple overlay and more ac.curate determinations of the binding lines made using dividers, instead of movement of t,he string loops, to satisfy the conditions of Eq. 2. REFERENCES 1.
SCATCHARD.
G.
2. ROSENTHAL.
3.
DANCHIN,
H. -4.,
(1949)
N.Y.
Ann.
Acad.
Sci. 51, 660.
E. (1967) Anal. Biochem. 20, 525-532. AND GUERON, M. (1970) &r. J. Biochem.
16, 532.
BERNARD E. Department of Physiology Medical College of Pennsylvania
%OO Henru Avenue Philadelphia, Pennsylvania Received June 18, 1973;
19189 accepted Jztll~ 28, 1973
PENNOCK