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Im having trouble understanding Scatchard plots.

Y Axis = Bound/Free Ligand

X Axis = Bound Ligand

The graph has a negative slope.

Why when there is almost no Bound (Y axis = 0) do we get a high positive number for Bound ligand on the X Axis corresponding to Receptor concentration? if there is no bound how is there a positive bound number?

it looks like the less bound ligand on the Y axis, the more bound on the X axis!

## Scatchard plot

**Scatchard plot** — A method for analysing data for freely reversible ligand/receptor binding interactions. The graphical plot is: (Bound ligand/Free ligand) against (Bound ligand) the slope gives the negative reciprocal of the binding affinity, the intercept on… … Dictionary of molecular biology

**Scatchard plot** — Scat·chard plot (skachґərd) [George Scatchard, American chemist, 1892â€“1973] see under plot … Medical dictionary

**Scatchard-Diagramm** — Der Scatchard Plot ist neben dem Hill Diagramm eine Möglichkeit, Enzymaktivität auf Kooperativität zu überprüfen, also auf Frage, ob das Enzym nach der Aufnahme eines ersten Liganden für die Bindung weiterer Liganden mehr oder weniger Energie… … Deutsch Wikipedia

**Scatchard equation** — The Scatchard equation is an equation for calculating the affinity constant of a ligand with a protein. The Scatchard equation is given by:frac

**plot** — A graphical representation. double reciprocal p. a graphic representation of enzyme kinetic data in which 1/v (on the vertical axis), where v is the initial velocity, is plotted as a function of the reciprocal of the substrate concentration… … Medical dictionary

**Scatchard** — George, U.S. chemist and biochemist, 1892–1973. See S. plot … Medical dictionary

**Hill-Plot** — Die Enzymkinetik ist ein Teilgebiet der biophysikalischen Chemie. Sie beschreibt, wie schnell enzymkatalysierte chemische Reaktionen verlaufen. Die Enzymkinetik findet breite Anwendung in Biologie und Medizin, da auch biologische Substrate… … Deutsch Wikipedia

**Receptor-ligand kinetics** — In biochemistry, receptor ligand kinetics is a branch of chemical kinetics in which the kinetic species are defined by different non covalent bindings and/or conformations of the molecules involved, which are denoted as receptor(s) and ligand(s) … Wikipedia

**Dissociation constant** — Kd redirects here. For other uses, see KD (disambiguation). In chemistry, biochemistry, and pharmacology, a dissociation constant is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate)… … Wikipedia

**Eadie-Hofstee-Diagramm** — Die Enzymkinetik ist ein Teilgebiet der biophysikalischen Chemie. Sie beschreibt, wie schnell enzymkatalysierte chemische Reaktionen verlaufen. Die Enzymkinetik findet breite Anwendung in Biologie und Medizin, da auch biologische Substrate… … Deutsch Wikipedia

## Characterization of a Receptor Using a Radioligand

Saturation radioligand binding experiments measure specific radioligand binding at equilibrium at various concentrations of the radi-oligand. Analyze these data to determine receptor number and affinity. Because this kind of experiment used to be analyzed with Scatchard plots (more accurately attributed to Rosenthal), they are sometimes called "Scatchard experiments".

The analyses depend on the assumption that you have allowed the incubation to proceed to equilibrium. This can take anywhere from a few minutes to many hours, depending on the ligand, receptor, temperature, and other experimental conditions. The lowest concentration of radioligand will take the longest to equilibrate. When testing equilibration time, therefore, use a low concentration of radioligand (perhaps 10-20% of the K D ).

### Nonspecific binding (NSB)

In addition to binding to the receptors, radioligand also binds to other sites termed nonspecific sites. Nonspecific binding is assessed by measuring radioligand binding in the presence of a saturating concentration of an unlabeled drug that binds to the receptor(s) of interest. The theory is that under those conditions, virtually all the receptors are occupied by the unlabeled drug, so the radioligand can only bind to nonspecific sites. Subtract the nonspecific binding at a particular concentration of radioligand from the total binding at that concentration to calculate the specific radioligand binding to receptors. *[Discussion: what causes NSB and what can one do to minimize it.]*

Two questions should be obvious: 1) what unlabeled drug should you use, and 2) at what concentration? In characterizing an assay system, the rule of thumb is to use a different drug than the radioligand, ideally in a different chemical class. (Once a system is well characterized, it may be acceptable to use the same drug.) You want to use enough to block virtually all the specific radioligand binding, but not so much that you cause more general physical changes to the membrane that might alter specific binding. A useful rule-of-thumb is to use the unlabeled compound at a concentration equal to 100-1000 times its K D for the receptors.

Nonspecific binding is usually linear with the concentration of radioligand (within the range it is used). Add twice as much radioligand, and you'll see twice as much nonspecific binding. The left figure shows a schematic of total and nonspecific binding. The figure on the right shows the difference between total and nonspecific binding, also known as the specific binding.

If only a small fraction of the ligand binds to the receptor, then the free concentration of ligand equals the added concentration in both the tubes used to measure total binding and the tubes used to measure nonspecific binding. In this case, you can subtract the nonspecific from the total to get specific binding. If the assumption is not valid, then the free concentration of ligand will differ in the two sets of tubes. In this case subtracting the two values makes no sense, and determining specific binding is difficult, and other strategies are needed.

### Analysis of data to determine B max and K D

As derived earlier, such experiments often are based on the One Site Binding Equation that we derived earlier:

Again, this analysis is based on several assumptions:

- Only a small fraction of the radioligand binds. The free concentration is almost identical to the concentration you added.
- There is no cooperativity. Binding of a ligand to one binding site does not alter the affinity of another binding site.
- Your experiment has reached equilibrium.
- Binding is reversible and follows the law of mass action.

### Scatchard plots

In the days before nonlinear regression programs were widely available, scientists transformed data into a linear form, and then analyzed the data by linear regression. There are a variety of algebraically equivalent ways to linearize such data, including the Lineweaver-Burk, Eadie-Hofstee, Wolff, and Scatchard-Rosenthal plots. With "perfect" data, all yield identical answers, yet each is affected more by different types of experimental error. The most commonly used of these methods in the pre-computer era was the Scatchard plot, where the X axis is specific binding ( B ) and the Y axis is specific binding divided by free radioligand concentration ( B/F ).

It is possible to estimate the B max and K D from a Scatchard plot (B max is the X intercept K D is the negative reciprocal of the slope). This transformation, however, distorts the experimental error, and thus violates the assumptions of linear regression. The B max and K D values you determine by linear regression of Scatchard transformed data are likely to be far from their true values. You should analyze your data with nonlinear regression. Do not analyze your data with Scatchard plots.

After analyzing your data with nonlinear regression, however, it is often useful (traditional?) to display data as a Scatchard plot. The human retina and visual cortex are wired to detect edges (straight lines), not rectangular hyperbolas. Scatchard plots are often shown as insets to the saturation binding curves. They are especially useful when you want to show a change in B max or K D .

## There was a problem providing the content you requested

How to analyze dissociation experiments Fit the data to this equation using nonlinear regression to determine the rate constant. Because homologous competitive binding experiments use a single concentration of radioligand which can be lowthey consume less radioligand and rosenthsl are more practical when radioligands are expensive or difficult to synthesize.

As derived earlier, such experiments often are based on the One Site Binding Equation that we derived earlier:.

Because data are collected at concentrations of D equally spaced on a log axis, the uncertainty is symmetrical when the equation is written in terms of the log of IC 50but is not symmetrical when written in terms of IC Archived from the original on The receptors do not all bind the unlabeled drug with the same affinity. Despite the theoretical complexities, agonist binding curves often turn out rossnthal fit rectangular hyperbolas or one- or two-site competitive binding curves.

The effector domain may be a transcriptional activator A or repressor R ,[1] a methylation domain M or a nuclease N. The human retina and visual cortex are wired to detect edges straight linesnot rectangular hyperbolas.

If the one-site model is correct you expect to get an F ratio near 1. This equation is based on these assumptions: Nonspecific binding is usually linear with the concentration of radioligand within the range rodenthal is used. Some people are surprised to see that the observed rate of association depends in part on the dissociation rate constant.

The dissociation rate constant k off is expressed in units of inverse time, usually min The foundations for which ligand binding assay have been built are a result of Karl Landsteinerinand his work on immunization of animals through the production of antibodies for certain proteins. The dissociation constant is the inverse of the association constant. Is the binding truly reversible? In the simplest case, of a receptor with a single class of noninteracting sites, receptor-ligand binding follows the Scatchard equation:.

### Scatchard Plot (Molecular Biology)

In other projects Wikimedia Commons. Y is the total binding you measure in the presence of various concentrations of the unlabeled drug, and log[D] is the logarithm of the concentration of competitor plotted on the X axis.

The experiment in the figure was designed to determine the B max and the experimenter didn’t care too much about the value of the K d. If not, you can find it using a table of F statistics. A bit of algebra simplifies it: Determine receptor number and affinity by using the same compound as the labeled and unlabeled ligand.

These data can distort extrapolations to the total number of binding sites that will be filled at a saturating ligand concentration 3. This can be faster and easier than other screening methods.

Saturation analysis is used in various types of tissues, such as fractions of partially purified plasma from tissue homogenatescells transfected with cloned receptors, and cells that are either in culture or isolated prior to analysis.

The slope factor is a number that describes the steepness of the curve. The rate of association number of binding events per unit of time equals [Ligand][Receptor]konwhere kon is the association rate constant in units of M-1min This can then be converted to the free concentration in molar.

A linear plot results, as shown in Figure 1. The experiment has reached equilibrium. Binding to one site does not alter affinity at another site. More simply, nearly the entire curve will cover two log units fold change in concentration.

## Competitive binding experiments

Competitive binding experiments measure the binding of a single concentration of labeled ligand in the presence of various concentrations of unlabeled ligand.

The experiment is done with a single concentration of radioligand. How much should you use? There is no clear answer. Higher concentrations of radioligand are more expensive and result in higher nonspecific binding, but also result in higher numbers of cpm bound and thus lower counting error. Lower concentrations save money and reduce nonspecific binding, but result in fewer counts of specific binding and thus more counting error. Many investigators choose a concentration approximately equal to about the K d of the radioligand for binding to the receptor, but this is not universal.

You need to let the incubation occur until equilibrium has been reached. How long does that take? Your first thought might be: "as long as it takes the radioligand to reach equilibrium in the absence of competitor." It turns out that this may not be long enough. You should incubate for 4-5 times the half-life for receptor dissociation as determined in an off-rate experiment (see page 19).

Typically, investigators use 12-24 concentrations of unlabeled compound spanning about six orders of magnitude.

The top of the curve is a plateau at a value equal to radioligand binding in the absence of the competing unlabeled drug. This is total binding. The bottom of the curve is a plateau equal to nonspecific binding (NS). The difference between the top and bottom plateaus is the specific binding. Note that this not the same as B max . When you use a low concentration of radioligand (to save money and avoid nonspecific binding), you have not reached saturation so specific binding will be much lower than the B max .

The Y axis can be expressed as cpm or converted to more useful units like fmol bound per milligram protein or number of binding sites per cell. Some investigators like to normalize the data from 100% (no competitor) to 0% (nonspecific binding at maximal concentrations of competitor).

The concentration of unlabeled drug that results in radioligand binding halfway between the upper and lower plateaus is called the IC 50 (inhibitory concentration 50%) also called the EC 50 (effective concentration 50%). The IC 50 is the concentration of unlabeled drug that blocks half the specific binding.

If the labeled and unlabeled ligand compete for a single binding site, the steepness of the competitive binding curve is determined by the law of mass action. The curve descends from 90% specific binding to 10% specific binding with an 81-fold increase in the concentration of the unlabeled drug. More simply, nearly the entire curve will cover two log units (100-fold change in concentration).

Competitive binding curves are described by this equation:

Y is the total binding you measure in the presence of various concentrations of the unlabeled drug, and log[D] is the logarithm of the concentration of competitor plotted on the X axis. Nonspecific is binding in the presence of a saturating concentration of D, and Total is the binding in the absence of competitor. Y, Total and Nonspecific are all expressed in the same units, such as cpm, fmol/mg, or sites/cell.

Use nonlinear regression to fit your competitive binding curve to determine the log(IC 50 ).

In order to determine the best-fit value of IC 50 (the concentration of unlabeled drug that blocks 50% of the specific binding of the radioligand), the nonlinear regression problem must be able to determine the 100% (total) and 0% (nonspecific) plateaus. If you have collected data over wide range of concentrations of unlabeled drug, the curve will have clearly defined bottom and top plateaus and the program should have no trouble fitting all three values (both plateaus and the IC 50 ).

With some experiments, the competition data may not define a clear bottom plateau. If you fit the data the usual way, the program might stop with an error message. Or it might find a nonsense value for the nonspecific plateau (it might even be negative). If the bottom plateau (0%) is incorrect, the IC 50 will also be incorrect. To solve this problem, you should define the nonspecific binding from other data. All drugs that bind to the same receptor should compete for all specific radioligand binding and reach the same bottom plateau value. When running the curve fitting program, set the bottom plateau of the curve to a constant equal to binding in the presence of a standard drug known to block all specific binding.

Similarly, if the curve doesn't have a clear top plateau, you should set the total binding to be a constant equal to binding in the absence of any competitor.

## Calculating the K i from the IC 50

Calculate the K i from the IC 50 , using the equation of Cheng and Prusoff (Cheng Y., Prusoff W. H., Biochem. Pharmacol. 22: 3099-3108, 1973).

In thinking about this equation, remember that K i is a property of the receptor and unlabeled drug, while IC 50 is a property of the experiment. By changing your experimental conditions (changing the radioligand used or changing its concentration), you'll change the IC 50 without affecting the K i .

### Why determine log(IC 50 ) rather than IC 50 ?

The equation for a competitive binding curve (page 14) looks a bit strange since it combines logarithms and antilogarithms (10 to the power). A bit of algebra simplifies it :

If you fit the data to this equation, you'll get the same curve and the same IC 50 . Since the equation is simpler, why not use it? The difference appears only when you look at how nonlinear regression programs assess the accuracy of the fit as a confidence interval. Even after converting from a log scale to a linear scale, you'll end up with different confidence intervals for the IC 50 .

Which confidence interval is correct? With nonlinear regression, the standard error of the fit variables are only approximately correct. Since the confidence intervals are calculated from the standard errors, they too are only approximately correct. The problem is that the real confidence interval may not be symmetrical around the best fit value. It may extend further in one direction than the other. However, nonlinear regression programs always calculate symmetrical confidence intervals (unless you use advanced techniques). When writing the equation for nonlinear regression, therefore, you want to arrange the variables so the uncertainty is as symmetrical as possible. Because data are collected at concentrations of D equally spaced on a log axis, the uncertainty is symmetrical when the equation is written in terms of the log of IC 50 , but is not symmetrical when written in terms of IC 50 . You'll get more accurate confidence intervals from fits of competitive binding data when the equation is written in terms of the log(IC 50 ).

A competitive binding experiment is termed homologous when the same compound is used as the hot and cold ligand. The term heterologous is used when the hot and cold ligands differ. Homologous competitive binding experiments can be used to determine the affinity of a ligand for the receptor and the receptor number. In other words, the experiment has the same goals as a saturation binding curve. Because homologous competitive binding experiments use a single concentration of radioligand (which can be low), they consume less radioligand and thus are more practical when radioligands are expensive or difficult to synthesize.

Analyze a homologous competitive binding curve using the same equation used for a one-site heterologous competitive binding to determine the top and bottom plateaus and the IC 50 .

The Cheng and Prussoff equation lets you calculate the K i from the IC 50 (see page 15). In the case of a homologous competitive binding experiment, you assume that the hot and cold ligand have identical affinities so that K d and K i are the same. Knowing that, simple algebra converts the equation to:

You set the concentration of radioligand in the experimental design, and determine the IC 50 from nonlinear regression. The difference between the two is the K d of the ligand (assuming hot and cold ligands bind the same).

The difference between the top and bottom plateaus of the curve represents the specific binding of radioligand at the concentration you used. Depending on how much radioligand you used, this value may be close to the B max or far from it. To determine the B max , divide the specific binding by the fractional occupancy, calculated from the K d and the concentration of radioligand.

The Scatchard eqn can be written as $displaystyle frac

(Your plot seems to plot bound concentration not fraction bound so needs correcting with total protein to make $Y$ and this is presumably why $n$ comes out to be a small value)

## Scatchard equation

A second type of average affinity, which we call K avis a weighted mean of the affinities, each affinity weighted by its proportional representation in the antibody population. An alternative version that is normalized for antibody concentration is especially useful if the data were obtained at different values of total antibody concentration, [ A ] tinstead of constant [ A ] t.

S2B and S2Calthough for very strong base positive cooperativity, it may not be possible to reverse but rather only modulate cooperativity. Thus, detailed balance equations may be inconsistent with many relevant biological situations of interest in our study.

However, in Equation 8we have substituted B, the concentration of bound ligand.

### Dimerization-based Control of Cooperativity

The simplest case is that of the interaction of antibody with monovalent ligand. The total binding T includes a component of non-specific binding NSBwhich is non-saturable, and the remainder is specific binding SB which saturates at Bmax. Kinetics of binding The simplest assumption about the scaatchard of the binding of drugs to receptors is that one molecule of drug D binds reversibly to one receptor molecule R to form a drug-receptor negativee DR:.

From a pharmacological perspective, drugs that act by inhibiting dimerization, such as trastuzumab or pertuzumab for ErbB2 positive breast cancer 45may not only inhibit signaling directly, but may also alter the cooperativitu and dose response of the system. These results suggest that the dimerization motif provides a novel mechanism for both generating and quantitatively tuning cooperativity that, due to the ubiquity of dimerization motifs in biochemical systems, may play a major role in a host of biological functions.

Results and Discussion Different Dimerization Schemes Cause Widely Varying Cooperativity Behavior To begin exploring how the dimerization motif affects cooperativity and ultrasensitivity, we consider the situation where a signal, S, binds to a downstream protein, P, and the downstream protein dimerizes Fig. A Scatchard plot is a plot of the ratio of concentrations cooperativitty bound ligand to unbound ligand versus the bound ligand concentration.

From Wikipedia, the free encyclopedia.

## Receptor-ligand interaction: a new method for determining binding parameters without a priori assumptions on non-specific binding.

Analysis of receptor-ligand binding characteristics can be greatly hampered by the presence of non-specific binding, defined as low-affinity binding to non-receptor domains which is not saturable within the range of ligand concentrations used. Conventional binding analyses, e.g. according to the methods described by Scatchard or Klotz, relate the amount of specific receptor-ligand binding to the concentration of free ligand, and therefore require assumptions on the amount of non-specific binding. In this paper a method is described for determining the parameters of specific receptor-ligand interaction which does not require any assumption or separate determination of the amount of non-specific binding. If the concentration of labelled free ligand is constant, a plot of Fu/(B0*-B*) versus Fu yields a linear relationship, in the case of a single receptor class, in which Fu is the concentration of unlabelled free ligand, B0* is the total amount of labelled bound ligand in the absence of unlabelled ligand and B* is the total amount of labelled bound ligand in the presence of an unlabelled ligand concentration Fu all of these data are readily obtained from binding studies. This linear relationship holds irrespective of the amount of non-specific binding, and the values for receptor density, ligand dissociation constant and a constant for non-specific binding can be readily obtained from it. If the concentration of labelled free ligand is not a constant for all data points, data are first converted according to a straightforward normalization procedure to permit the use of this relationship. The presence of multiple receptor classes with dissociation constants in the range of the ligand concentrations used results in a negative deviation from this linearity, and therefore the presence of multiple receptor classes can be discriminated unequivocally from non-specific binding. Both theoretical and practical advantages of the present method are described. The method, which will be referred to as the linear subtraction method, is illustrated using the binding of tumour promoters and polypeptide growth factors to their specific cellular receptors.

## Scatchard equation

Create a new XY data table, with no subcolumns. Annals of the New York Academy of Sciences. They are especially useful when you want to show a change in Bmax or Kd.

The intercept on the X axis is B max. Scatchard plots are often shown as insets to the saturation binding curves. The human retina and visual cortex evolved to detect edges straight linesnot rectangular hyperbolas, and so it can help to display data this way. To create a Scatchard plot from your specific binding data, use Prism’s Transform analysis, and choose the Scatchard transform from the panel of biochemistry and pharmacology transforms. The plot yields a straight line of slope – Kwhere K is the affinity constant for ligand binding.

If you create a Scatcahrd plot, use it only to display your data. In other projects Wikimedia Commons. Generally, Scatchard and Lineweaver-Burk plots are outdated. The goal is to determine the Kd ligand concentration that binds to half the receptor sites at equilibrium and Bmax maximum number of binding sites.

This biochemistry article is a stub. Then use the Remove Baseline analysis to subtract column B from column A, creating a new results table with the specific binding. If you do this, you won’t get the most accurate values for Bmax and Kd. An alternative approach would be to enter total binding into column A, and nonspecific into column B.

Create a Scatchard plot Before nonlinear regression was available, investigators had to transform curved data into straight lines, so they could analyze with linear regression.

It is named after the American chemist George Scatchard [2] and is sometimes referred to as the Rosenthal-Scatchard equation. Note the name of this data table. From Wikipedia, the free encyclopedia. Their original intention was to transform the data into linear representations of the original data such that linear regression methods could be applied.

Choose to plot no symbols, but to connect with a line. Go scaatchard the Scatchard graph. One way to do this is with a Scatchard plot, which plots specific binding scatdhard. One site — Specific binding Feedback on: It is the specific binding extrapolated to very high concentrations of radioligand, and so its value is almost always higher than any specific binding measured in your experiment.

Double-click on one of the new symbols for that data set to bring up the Format Graph dialog. In a Scatchard plot, assumptions of independence in linear regression model is violated because B bound ligand is used in the X and Y axes.

### Scatchard equation – Wikipedia

The affinity constant is the inverse of the dissociation constant. There are three approaches to dealing with nonspecific binding. It is better to globally fit total and nonspecific bindingwithout subtracting to compute specific binding. You need ppr do the calculation manually, and enter a number.

It is the radioligand concentration needed to achieve a half-maximum binding at equilibrium. Don’t use the slope and sccatchard of a linear regression line to determine values for Bmax and Kd. Archived from the original on Such is the case when ligand bound to substrate is not allowed to achieve equilibrium before the binding is measured or binding is cooperative.

URL of this page: The Scatchard equation is an equation used in molecular biology for calculating the affinity constant of a ligand with a protein. Applications in the Life Sciences. Bmax is the maximum specific binding in the same units as Y.

Perhaps rename it to something appropriate. Introduction In a saturation binding experiment, you vary the concentration of radioligand and measure binding.

One site — Specific binding. Views Read Edit View history. One site — Specific binding? In a saturation binding experiment, you vary the concentration of radioligand and measure binding. Create an XY data table. From the scatcharrd of specific binding, click Analyze, choose nonlinear regression, choose the panel of Saturation Binding equations, and choose One site specific binding.

A Scatchard plot is a plot of the ratio of concentrations of bound ligand to unbound ligand versus polt bound ligand concentration.

Drag the new table from the navigator and drop onto the graph. Biochemistry methods Proteins Biochemistry stubs.

## Introduction

Insulin and insulin-like growth factors (IGF) 1 and 2 have similar structures and exert their action by activating two closely related receptor tyrosine kinases—the insulin receptor and the IGF1 type I receptor (IGF1 receptor), which share largely overlapping signalling pathways ( Adams *et al*, 2000 De Meyts and Whittaker, 2002 De Meyts, 2004 Denley *et al*, 2005 ). Despite this similarity, the two hormones produce different responses: mostly metabolic for insulin and mitogenic for IGF1 ( Kim and Accili, 2002 ). Dysregulation of their signalling may lead to two different life-threatening diseases: type II diabetes and cancer, which are among the largest global health challenges in the world. So far, there is poor understanding of how these hormones produce such different biological effects using similar signalling networks ( Kim and Accili, 2002 ). It has become clear that a systems biology approach is required to understand the combinatorial nature of signalling specificity ( Shymko *et al*, 1997 Kholodenko, 2007 ). Insulin analogues with altered kinetic properties show enhanced mitogenic potencies, although they bind to the same insulin receptor they also cross-react to a variable extent with the IGF-I receptor ( Shymko *et al*, 1997 Kurtzhals *et al*, 2000 ). Differences in kinetics of ligand binding and receptor activation by the insulin and IGF1 receptors may be one of the factors determining their specificity ( Shymko *et al*, 1997 ). The two receptors’ mechanism of ligand binding and activation displays complex allosteric properties (i.e. negative cooperativity and ligand dependence of the receptor dissociation rate), which no mathematical model has been able to fully account for ( Jeffrey, 1982 Kohanski and Lane, 1983 Hammond *et al*, 1997 Wanant and Quon, 2000 Sedaghat *et al*, 2002 ). Thus, the development of a reliable mathematical model describing the two receptors’ binding kinetics and activation is a critical first step in a systems biology approach to understand the function and specificity of these receptors.

The insulin and IGF1 receptors exist in the membrane as pre-formed covalent dimers of two identical moieties. Their extracellular domains comprise two leucine-rich repeat-containing large domains (L_{1} and L_{2}) separated by a cystein-rich (CR) domain, followed by three fibronectin type III (Fn_{1–3}) domains ( Adams *et al*, 2000 De Meyts and Whittaker, 2002 De Meyts, 2004 ). A crystal structure of this extracellular (unliganded) insulin receptor dimer has recently been solved ( McKern *et al*, 2006 Lawrence *et al*, 2007 Ward *et al*, 2007 , 2008 ). The intracellular portion of the two receptors consists of a kinase domain flanked by regulatory regions ( Hubbard and Miller, 2007 ).

The insulin and IGF1 receptors exhibit complex binding properties. The Scatchard plots for both receptors are concave up, indicating the presence of high- and low-affinity binding sites and/or negative cooperativity ( De Meyts *et al*, 1973 , 1976 De Meyts, 1994 ). The receptors bind only one ligand molecule with high affinity and at least another one with lower affinity. The ligand dissociation rate is dependent on its concentration ( De Meyts *et al*, 1973 , 1976 De Meyts, 1994 ). Furthermore, this dependence is bell-shaped for the insulin receptor, whereas for the IGF1 receptor, it is sigmoid ( Christoffersen *et al*, 1994 ). When the insulin receptor is in a monomeric form, its affinity is reduced 30-fold, the Scatchard plot becomes linear and the dissociation rate of the ligand becomes independent of its concentration ( De Meyts, 1994 , 2004 De Meyts and Whittaker, 2002 ).

Insulin has two receptor-binding surfaces on the ‘opposite’ sides of the molecule. The first ‘classical’ binding surface, also involved in insulin dimerization, was defined in the early 1970s ( Pullen *et al*, 1976 De Meyts *et al*, 1978 ), and later validated by alanine-scanning mutagenesis ( Kristensen *et al*, 1997 ), whereas the second surface, also involved in insulin hexamerization, was mapped by alanine-scanning mutagenesis more recently ( De Meyts, 2004 Gauguin *et al*, 2008 ). The dimerization surface interacts with a site located in the L_{1} module of the insulin receptor, as well as a 12 amino-acid peptide from the insert in Fn_{2}, which combine to form ‘site 1’ ( Wedekind *et al*, 1989 Kurose *et al*, 1994 Williams *et al*, 1995 Mynarcik *et al*, 1996 De Meyts and Whittaker, 2002 Kristensen *et al*, 2002 Huang *et al*, 2004 ), whereas the hexamerization surface interacts with a site consisting of residues located in the C-terminal portion of L2 and in the Fn_{1} and Fn_{2} modules (site 2) ( Fabry *et al*, 1992 De Meyts and Whittaker, 2002 Hao *et al*, 2006 Benyoucef *et al*, 2007 Whittaker *et al*, 2008 ). Schäffer (1994) suggested that the high-affinity binding could result from insulin crosslinking site 1 of one receptor half and site 2 of the other half of the receptor dimer, thus leaving the other two sites free for interaction with the ligand. However, this model had difficulty in explaining the ligand dependence of the ligand dissociation rate. De Meyts (1994) suggested that this problem could be solved by assuming that the four sites of the receptor dimer are arranged in a symmetrical antiparallel way, a postulate that was supported by the recent structure determination of the insulin receptor extracellular domain ( McKern *et al*, 2006 ). Despite its seeming simplicity, this model turned out to be notoriously difficult for a quantitative analysis and some researchers even concluded that it did not explain the ligand dependence of the dissociation rate ( Hammond *et al*, 1997 ). The problem is that the qualitative crosslinking models suggested by Schäffer (1994) and De Meyts (1994) are not detailed enough from a biochemical and structural viewpoint for mathematical modelling (especially concerning the precise mechanism that leads to receptor crosslinking). The modelling problem is further aggravated by combinatorial complexity arising from multivalent ligand binding to the receptor in multiple possible conformations. We have now solved this problem.

Here, we present the first mathematical model that accurately reproduces all the kinetic properties of the insulin receptor such as negative cooperativity and the bell-shaped ligand dependence of the receptor dissociation rate. On the basis of the available structural information, we develop a physically plausible model of the receptor activation, which is based on the concept of a harmonic oscillator. We justify thermodynamically that the symmetrically arranged subunits of the insulin receptor dimer experience harmonic oscillatory movements. Analysis of the behaviour of an ensemble of such harmonic oscillators in thermal equilibrium with the surrounding milieu allows to model the receptor activation in a simple way and to substantially reduce the combinatorial complexity. This model is the first one that gives a description of the insulin receptor binding and activation mechanism in terms of interactions between the molecular components and fully takes into account the combinatorial binding complexity (employing 35 insulin receptor intermediaries). Fitting of the model to experimental data provides unique and robust estimates of the kinetic parameters. With a small modification, the model can also be used for the IGF1 receptor. Furthermore, the harmonic oscillator model may be adaptable for many other dimeric or dimerizing receptor tyrosine kinases, cytokine receptors and G-protein-coupled receptors where ligand crosslinking occurs ( De Meyts, 2008 ).

## Abstract

The theoretical background and practical approaches for studying ligand−receptor (protein) binding by solid phase microextraction (SPME) are investigated, along with methods for simultaneous calculation of receptor, free, and total ligand concentrations. With the introduction of new extraction phases (restricted access materials, molecularly imprinted polymers, and immobilized antibodies), SPME allows better separation of small molecules of ligand from larger molecules of receptor, and improved accuracy. This sample preparation method based on nonexhaustive extraction is well suited as a general method to study and quantify systems involving multiple equilibriums, with significant advantages over currently used methods. SPME was used previously for the determination of protein binding constants, but only with conventional extraction phases and in simple cases, with a 1:1 combination ratio between the ligand and the receptor or when negligible depletion conditions were met. The new theoretical approach presented in this study allows the quantification of any binding equilibrium, regardless of the extent of depletion. Restricted-access particles are used as extraction phase, and if the amount of receptor is limited, selected regions of the binding curve may be obtained using a single sample, with a volume as low as 10 μL. The equations developed here are simple and independent of the analytical method used for the quantification of the amount of ligand. Three different practical approaches are presented: the method of multiple standard solutions, the method of successive extractions from the same sample and the method of successive additions to the same sample. The usefulness of this novel approach is demonstrated by using it to determine the binding parameters of some selected drugs to human serum albumin. These parameters are subsequently used to calculate albumin, free drug, and total drug concentrations from unknown mixtures. The results are in good agreement with previously published data. Quantification of the amount of ligand extracted by SPME is done by liquid chromatography coupled with tandem mass spectrometry.

Keywords: solid-phase microextraction • ligand−receptor binding • binding constants • number of binding sites • free concentration • total concentration

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