1. In Topic 8, Equation (8.14), for a simple ligand-receptor binding process, us
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1. In Topic 8, Equation (8.14), for a simple ligand-receptor binding process, using a statistical mechanics approach, we found an expression for the fraction of receptors that had a ligand bound. The answer was written in terms of ligand concentration, temperature, and a quantity bigtriangleupG0 = G0RL - G0R - G0L. In a few sentences or equations, explain why in the case being considered it is acceptable to equate an energy of interaction to a standard free energy (G0) rather than to a thermodynamic energy (U). bigtriangleupG0 (NA is Avogadro's number), then this result was equivalent to one obtained earlier (Equation 8.8) written in terms of the standard free energy difference bigtriangleupe to bigtriangleupe = eb- esol is the energy for a ligand bound to the receptor minus the energy for a ligand free in solution. We stated that, if we equated NAbigtriangleupe, whereExplanation / Answer
One fact that should be apparent is that binding studies become much more straightforward if one is working with pure ligand and receptor and under well-defined experimental conditions. Whenever a system is studied in vivo (or in a biological extract containing numerous types of molecules), the number of experimental variables becomes very high, to the point that it may be difficult to obtain unambiguous results (see following sections). The Model for 1:1 Binding In the very simplest type of binding study involving a simple 1:1 association of ligand and receptor to form a complex, a primary goal might be to determine the Kd . The standard definition of Kd can be algebraically manipulated to yield the following equation: [L]free [RL]/[R]total = fraction of sites occupied = fR = --------------------- (6) Kd + [L]free Ka . [L]free = ---------------------- (7) 1 + Ka . [L]free where [L]free is the free ligand concentration. From this equation it is observed that if one could measure the fraction of sites occupied as a function of [L]free, the data would map out a curve that could be fit to yield a value for Kd. Equations 6 and 7 predict hyperbolic (fractionR) vs [L]free plots (see below). Such plots are sometimes referred to as "isotherms" (for relatively obscure thermodynamic reasons). 9 Important implications and considerations for 1:1 binding isotherm: Remember, it is free [L] that is being plotted. However, since total ligand is often much higher than total receptor, this means that the % ligand that forms a complex with R is often going to be small. In this (very common) case [L]free is effectively equal to [L]total. This is fortunate, because the total ligand concentration is often easily determined, but not the free ligand concentration. THE ASSUMPTION THAT Lfree IS EFFECTIVELY EQUAL TO Ltotal CAN OFTEN BE SAFELY MADE, BUT NOT ALWAYS. THIS ASSUMPTION SHOULD ALWAYS BE SCRUTINIZED BEFORE BEING MADE.
A Short Introduction to Binding Kinetics Before proceeding to a more detailed consideration of binding theory and analysis, it is important to first understand basic kinetics, a short review of which is presented here. Binding and other equilibrium constants are fundamentally related to the rates of interchange between the states involved in the equilibrium process. "Rate" is, of course, a description of how frequently something happens. A unimolecular rate is how fast one molecule does something and will have units of "per second" ( = sec-1 = Hz) or "per minute" ( = min-1). Unimolecular rates are sometimes referred to as zero order rate constants where "zero" means that the rate is independent of any concentration. One example of a zero order rate constant is the radioactive decay of a single isotope (which is determined completely by the type of isotope, not by chemical concentrations or compositions). Another example is the enzyme turnover number: kcat. This rate constant tells the maximum rate that a single enzyme molecule can execute a chemical reaction under conditions where it is saturated with substrate. A "first order" reaction rate is a rate that describes a process that is dependent upon the concentration of a single species. It will have units of ?[concentration]/?time The Simplest Case: 1:1 Stoichiometry R + L RL R:
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