Total ligand equation

To parameterize and solve the total metal and total ligand equations for a given solution, we also need to know the formation constants of the complexes. The stepwise formation constants of mononuclear complexes are usually defined as follows (cf. Beck, 1970)... 

Substituting this expression into the equation above yields a new expression relating the MS response to four variables the total ligand concentration [S]q, which is the known, independent variable in a titration experiment the fCj, which is the dependent variable of interest the total receptor concentration [ ]q and the MS response calibration factor Cms ... 

Therefore, plotting the ALIS MS response from a titration series versus the total ligand concentration yields a saturation binding curve that can be fit to this equation by nonlinear regression analysis to yield the of the ligand of interest. 

The kinetics of a system of competing ligands can be modeled by simultaneous numerical solution of these two equations given initial values for the system parameters, including the total protein concentration [ ]q, the total ligand and total inhibitor concentrations [S]q and [I]q, the rates of association and dissociation for the interacting components of the mixture ks-on, ks-off, fei-on, and ki.off, and initial values for [ S] and [ /]. Note that simultaneous solution of these equations where the initial value of [ S] is not zero allows the behavior of the system to be modeled versus time upon addition of an excess of inhibitor. 

Both binding isotherm (Equation 8.15) and double-reciprocal plots (Equation 8.16) are expressed as a function of free ligand concentration, which is not known. By means of the mass balance on ligand,L can be related to the known total ligand concentration... 

Z.-X. Wang and R.-F. Jiang, A Novel Two-Site Binding Equation Presented in Terms of the Total Ligand Concentration, FEBS Lett., 392, 245 (1996). 

Approximate values of free ligand concentration a may be obtained from equation (3.15), using the values of v and the experimentally obtained values of M, m and A. Another procedure for determining free ligand concentration involves the relationship between a property X and the total ligand concentration A for a number of values of M. Plots of X vs. Ab are made followed by lines parallel to x-axis to obtain the relationship between A... 

Copper is added in nine steps from 0. 5 to I6 jjlM one hour equilibrium is allowed after each addition before a subsample (3 0 ml) is filtered and acidified. Total dissolved copper is measured in the filtrate by d.p.a. s. v. Calibration of Mn02 with Cu is carried out at the same concentration, temperature, ionic strength and pH conditions. The cupric ion concentration can then be calculated using the Langmuir equation. Mass balance from the measured total Cu and the cupric ion gives the complexed copper concentration, CuL the total ligand concentration and the conditional stability constant, K, for the formation of CuL, Cu" + L = CuL,... 

This equation predicts that addition of either cocrystal component to a solution in excess of S decreases the cocrystal solubility when the preceding conditions apply. A plot of the solubility of cocrystal A B as a function of total ligand in solution according to Eq. (9) is shown in Fig. 12. 

A]x is the solubility of cocrystal A B, when measuring total A in solutions under the equilibrium conditions described in Eq. (1). By combining the above equations, the cocrystal solubility can be expressed in terms of the total ligand concentration [B]t according to... 

In order to fully define the titration system it is essential to account for the total metal ion added to solution, and the total ligand concentrations, Cn and Cl2, available to complex metal. This is done via mass balance equations shown in equations 16 thru 18. 

Equation 16 relates the total concentration of metal added to solution to the concentration of both complexes and to the free metal ion concentration. This equation may be redefined to describe the actual number of binding sites involved in quenching for a particular system (i.e. three or more sites). Equations 17 and 18 are mass balance equations characterizing the total ligand concentration for each fluorescent site. 

Simple thermodynamic calculations based on literature data (5-12) support the choice of phosphates as the optimum mineral phases for actinide immobilization. The calculations considered every relevant species reported (5-72) that contained protons, hydroxide, or the ligand in question for each metal ion. Where necessary, equilibrium constants were corrected to 0.1 M ionic strength using the Davies equation. As an example, the calculated solubility of europium, thorium, and uranium in various media at p[H] 7.0 (p[H] = - log of the hydrogen ion concentration), 0.001 M total ligand concentration, 0.1 M ionic strength, and 25 °C are shown in Table I. Within the constraints of the calculation, the solubility of thorium is limited by Th(OH)4, but the lowest europium and uranyl solubilities are observed for phosphates. 

Titration of the ditopic pentadentate ligands with either a kinetically labile d-transition metal (M = Fe°, Co", Zn") or a lanthanide ion often leads to the observation of several different species in equilibrium. Qn the other hand, the RML3 helicate is always the most abimdant species when the titrant is a stoichiometric R"Vm" mixture, when the total ligand concentrations is >10 M, and in presence of poorly coordinating anions the other complexes are often not detected in NMR spectra under these conditions. The spectrophotometric data have been fitted to the following set of equations (A stands for nd (M) or for 4f (R) ion, charges are omitted for clarity) ... 

I quation 14-10 is the basis for several computer-based methods for determining the formation constant K,. In the usual econstant concentration of metal is used and the total ligand concentration C , is varied. The change in absorbance i/1 is then measured as a function of total ligand concentration and the results statistically analyzed to obtain Ki. Unfortunately, the relationship shown in Equation 14-1 (Its nonlinear, and thus nonlinear regression must be used unless the eqitation is transformed to a linear form.t We can linearize the equation by taking the reciprocal of both sides to obtain... 

Unfortunately, since we do not know the free ligand concentration ([L]), but only the total (Cl), it is necessary to use a fairly complicated expression (such as equation 5.3 from Section 5.4) to estimate K from this. However, in cases of relatively weak binding, when [PL] is small compared to the total ligand concentration, then we might make the approximation that [L] = Cl. 

A general titration curve equation can be derived from the mass balance equation for the ligand, i.e., to state that the total ligand concentration is the sum of the bound and free ligand concentrations. By the bound ligand is meant the ligand incorporated into the metal complex(es). Thus, for Ag ... 

This is the concentration [L]free at 40°C. Note that the other solution of the quadratic equation, X2 = (—0.962 — 1.039)70.154 = —12.99 is a meaningless value so we ignore it. Now it is easy to find the concentration of the receptor-ligand complex at 40.0°C it is the difference between the total ligand and free ligand ... 

The only measurable parameter is X (the increment of total ligand concentration in Vq) as soon as Vq and are known. The insertion of value of V fr equation (5) to equation (4) gives It is clear that if Vq is large compared to Vi for a given X will be small and, correspondingly, the accuracy of the measurement will be low. 

The calculation of stability constants is based on the fact that measurement of total ligand concentration at any point within column allows a determination of the amount bound to protein, since the cross-sectional areas of Vq and V, can be known frcm Independent experiments (ref. 34). When M is totally excluded the free ligand concentration can be calculated frcm equation... 

The data for binding constants of the systan are directly available without knowing molar absorptivity of complexed L or possible absorption by M, since the total ligand concentration in the original sample is independently known. By repeating frontal analysis in different concentrations of L, the binding ratio and stoichiometry is obtained (see equation 2). 

The concentrations of the free and bound ligand are related to the total ligand concentration by the conservation of mass equation... 

Nonlinear regression analysis of a plot of fx against [L] (8) provides the association constant, and the electrophoretic mobility of the formed complex SL derived fi om A/z ax-[L] can be approximated in the equation by the total ligand concentration under the condition that the ligand concentration is much higher than the concentration of the solute or that the association constant is small). 

The equilibrium expressions are given by equations (7) and (8). The total ligand concentration, [Land total calcium ion concentration, [Ca" ],... 

If both [M] and [L] can be measured, keeping [M], constant, but changing the total ligand concentration for each measurement, this relationship leads to a set of simultaneous equation in the ps (one equation for each measurement) which provide a quick way of determining p values. Alternatively, if [L] is made large, so that [L] [M], it is essentially constant. By varying [M], and measuring [M], the ps may similarly be found. 

On the other hand, bulk concentrations are required for estimation of the respective surface concentrations that are the terms of kinetic equations. To obtain the data for the solution layer adjacent to the electrode surface, mass transport of chemically interacting species should be considered. Quantitative formulation of this problem is based on differential equations representing Pick s second law and supplemented with the respective kinetic terms. It turns out that some linear combinations of these equations make it possible to eliminate kinetic terms. So produced common diffusion equations involve total concentrations of metal, ligand and proton donors (cj j, c, and Cj4, respectively) as functions of time and space coordinates. It follows from the relationships obtained that the total metal concentration varies in the same manner as the concentration of free metal ions in the absence of ligand. Simultaneously, the total ligand concentration remains constant within the whole region of the diffusion layer. This proposition also remains valid for proton donors and acceptors. 

Equations (13,14) show that stability constants may be calculated through the experimental determination of at least N different n and [L] data pairs N is the maximum number of ligands that can be coordinated to the metal ion). The average number n depends only on the equilibrium concentration of the free ligand [L] and is independent of both the total ligand and metal concentrations. Corresponding solutions are defined as having different T and Tr values, but the same values of [L] and, consequently, n. 

---