Signal and binding. II. Converting physico-chemical responses to macromolecule–ligand interactions into thermodynamic binding isotherms

    Signal and binding. II. Converting physico-chemical responses to macromolecule–ligand interactions into thermodynamic binding isotherms Wlodzimierz Bujalowski, Maria J. Jezewska, Paul J. Bujalowski PII: DOI: Reference:

S0301-4622(16)30424-0 doi:10.1016/j.bpc.2016.12.005 BIOCHE 5953

To appear in:

Biophysical Chemistry

Received date: Revised date: Accepted date:

11 November 2016 26 December 2016 26 December 2016

Please cite this article as: Wlodzimierz Bujalowski, Maria J. Jezewska, Paul J. Bujalowski, Signal and binding. II. Converting physico-chemical responses to macromolecule–ligand interactions into thermodynamic binding isotherms, Biophysical Chemistry (2016), doi:10.1016/j.bpc.2016.12.005

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Signal and Binding. II. Converting Physico-Chemical Responses to Macromolecule -

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Ligand Interactions into Thermodynamic Binding Isotherms§

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Wlodzimierz Bujalowski*#, Maria J. Jezewska, and Paul J. Bujalowski

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Department of Biochemistry and Molecular Biology,

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Department of Obstetrics and Gynecology#, The Sealy Center for Structural Biology, Sealy Center for Cancer Cell Biology

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The University of Texas Medical Branch at Galveston 301 University Boulevard

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Galveston, Texas 77555-1053

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Running Title: Quantitative thermodynamic analysis of physico-chemical titrations

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§This work was supported by NIH Grant GM58565 (to W. B.). *Corresponding author: Dr. W. M. Bujalowski Department of Biochemistry and Molecular Biology The University of Texas Medical Branch at Galveston 301 University Boulevard

Galveston, Texas 77555-1053 USA Tel: (409) 772-5634 Fax: (409) 772-1790 Email: [email protected]

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ABSTRACT

Physico-chemical titration techniques are the most commonly used methods in characterizing

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molecular interactions. These methods are mainly based on spectroscopic, calorimetric,

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hydrodynamic, etc., measurements. However, truly quantitative physico-chemical methods are

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absolutely based on the determination of the relationship between the measured signal and the total average degree of binding in order to obtain meaningful interaction parameters. The

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relationship between the observed physico-chemical signal of whatever nature and the degree of binding must be determined and not assumed, based on some ad hoc intuitive

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relationship/model, leading to determination of the true binding isotherm. The quantitative

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methods reviewed and discussed here allow an experimenter to rigorously determine the degree

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of binding and the free ligand concentration, i.e., they lead to the construction of the thermodynamic binding isotherm in a model-independent fashion from physico-chemical titration curves.

Keywords: Physico-Chemical Titrations, Binding Isotherms, Thermodynamics

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INTRODUCTION

Obtaining the equilibrium, binding isotherm is the primary method in analyses of ligand-

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macromolecule interactions. The equilibrium, binding isotherm is the functional dependence of

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the total average degree of binding (number of ligand molecules bound per macromolecule) upon

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the free ligand concentration [1-6]. True thermodynamic isotherm reflects only this functional dependence. In a practical experimental setup, it is the functional dependence of the total average

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degree of binding upon the total ligand concentration, although the free ligand concentration is still the independent variable (see below). In other words, the thermodynamic isotherm cannot be

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dependent upon any models/assumptions, concerning the relationships between the physico-

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chemical parameter used to monitor the binding and the total average degree of binding.

Nevertheless, the thermodynamic isotherm alone, obtained in model-independent fashion, provides the maximum stoichiometry and only approximate estimates of macroscopic affinities of the examined association reaction. The molecular aspects of the interactions, such as discrete character of binding sites, the overlap of potential binding sites, intrinsic binding constants, cooperativity parameters, allosteric equilibrium constants, etc., are obtained through the analysis of the constructed thermodynamic isotherm by statistical thermodynamic models [1-6]. In other words, statistical thermodynamic models are not used to construct the binding isotherm but to analyze it. Application of statistical thermodynamic models to examine the obtained thermodynamic binding isotherm is based on the knowledge about the studied systems, e.g., their structural characteristics, like lattice or subunit structure of the macromolecule, character of 3

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lattice monomers, geometrical arrangement of subunits, etc. This is an extra-thermodynamic

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knowledge, i.e., it reaches beyond the experimental, thermodynamic binding analysis.

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Although the direct methods of studying the ligand - macromolecule interactions do

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provide the total average degree of binding and the corresponding free ligand concentration, they

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possess limitations like, e.g., application only for small ligand molecules (equilibrium dialysis), or may perturb the examined equilibrium (filter binding method, gel electrophoresis) [7-10,

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preceding paper]. On the other hand, the indirect methods, based on observing a physicochemical signal, predominantly a spectroscopic signal reflecting saturation of the macromolecule

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or the ligand, require that the observed changes of the monitored physico-chemical parameter

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16, preceding paper].

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correlate with the total average degree of binding and the concentration of the free ligand [6,11-

In most interacting/binding systems subjected to the quantitative analysis, the functional relationship between the observed physico-chemical signal and the total average degree of binding is never a priori known. It has to be determined [6,11,12,16]. Very often, it is assumed that the observed change of the physico-chemical/spectroscopic signal is directly proportional to the degree of saturation of the macromolecule and/or the ligand, i.e., the signal is a linear function of the total average degree of binding. However, in such cases, the obtained interaction parameters are not more accurate than the applied assumption. This is not the problem in the case of single ligand binding processes where indeed the observed relative changes of the signal always reflect the saturation of the macromolecule or the ligand [1,6,16,17, preceding paper]. 4

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But it is already a lot to know about the examined system that only a single ligand molecule

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binds [17]. In more complex situations, even with binding reactions involving only two ligand molecules, the failure of the applied statistical thermodynamic model to "fit" the titration curve

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may be due either to the failure of the model or the failure of the assumption, on which the

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"binding isotherm" is based. Ignoring these facts will particularly be serious if the “isotherm” is

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the basis to decide, which alternative models actually describe the examined binding process

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[6,18, preceding paper].

Why would not the observed spectroscopic signal be a linear function of the total average

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degree of binding? For instance, the macromolecule may possess functionally different sites,

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each characterized by different spectroscopic properties, which are differently affected by the

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bound ligand [18]. If there are cooperative interactions, the physical state of the macromolecule and/or the ligand may change in different sites, as the saturation process progresses [18,19]. The number of cooperative contacts among binding sites and/or bound ligand molecules may not be a linear function of the total average degree of binding. Different binding modes of the ligand may be characterized by different responses of the physico-chemical/spectroscopic signal [12,20,21]. In more complex situations, combinations of all mentioned above cases may occur.

In this second part of our review, we address the fundamental problem of obtaining thermodynamic and physico-chemical/spectroscopic parameters free of assumptions about the relationship between the observed signal and the degree of ligand or macromolecule saturation. We will mainly address quantitative methods as applied to the use of the fluorescence intensity, 5

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which is the most often encountered spectroscopic technique in biochemical studies [6,11,16-

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19,20-30]. Nevertheless, we will also discuss the same analyses for other commonly applied physico-chemical signals (e.g., fluorescence anisotropy, polarization, calorimetry, sedimentation

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velocity). It should be stressed that the obtained relationships are general and applicable to any

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signal used to monitor interactions and proportional to the concentrations of different states of

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macromolecule or ligand (e.g., absorbance, circular dichroism, NMR line width, chemical shift,

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etc.).

Furthermore, we are concerned only with cases where the examined physico-

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chemical/spectroscopic signal originates only from the macromolecule or only from the ligand. This condition is easily experimentally realized and amounts to monitoring only the

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macromolecule or the ligand saturation process. Quantitative examination of one reaction is already complex enough and the discussed methods are only valid in the absence of ligand or macromolecule aggregation, within experimentally applied concentration ranges. The specific cases of binding systems discussed below are selected from our works focused on various protein - nucleotide, protein - nucleic acid interactions [6,13,14,17-19,22-30]. This allows us to achieve the best presentation of applicability and feasibility of the discussed methods, as we possess the hands-on insights how these data were obtained and analyzed.

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1. Quantitative Equilibrium Analysis of Physico-Chemical Titrations Using the Signal

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Generated From the Macromolecule.

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1. a. Fluorescence Intensity.

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Quantitative characterization of ligand - macromolecule interactions can only be achieved if the physico-chemical/spectroscopic titration curve is converted into a thermodynamic binding

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isotherm, i.e., conversion without any assumption about the relationship between the observed signal and the total average degree of binding. As we discuss in the preceding paper, this is not a

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problem in the case of the single ligand molecule binding, but it becomes a problem for anything

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more complex than that, where the maximum stoichiometry of the binding process is unknown.

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Once the binding isotherm is obtained, an appropriate, statistical thermodynamic binding model can be used to extract binding parameters. Titrations, where the saturation of the macromolecule increases in the course of binding experiment, are referred to as “normal” titrations [6,11-14,31]. We will first address the analysis of the normal titrations where the binding is monitored using the fluorescence intensity as the physico-chemical/spectroscopic signal originating from the macromolecule [6,11-14,30,32-38].

The thermodynamic method of analysis is based on the fact that the equilibrium distribution of the different macromolecule states with a different number of bound ligand molecules, [M]i, is a sole function the free ligand concentration, [L]F [6,11,12,17-30,32-38]. This is a thermodynamic fact, which is independent of any binding models. When the observed signal 7

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originates from the macromolecule, for a given [L]F, the signal, (e.g., fluorescence intensity) Fobs,

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is the algebraic sum of the intensive (concentration independent) physico-chemical/spectroscopic properties of each macromolecular state, Fi, weighted by the concentrations of all corresponding

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macromolecular states, [M]i. The observed fluorescence intensity (physico-chemical signal), Fobs,

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at given [L]T, and [M]T, respectively, is given by the model-independent, signal conservation

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equation, as [6,11,12,17-30,32-38]

(1)

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Fobs FF [M]F Fi [M]i

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 molar intensities of the free macromolecule and the complex, M , where FF and Fi are the i

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respectively, which represents the macromolecule with “i” bound ligand molecules (i = 1 to n)

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[6,11,12,17-19,21-30,32-38, preceding paper]. The model-independent mass conservation equation relates [M]F and [M]i to [M]T by

[M]T  [M]F  [M]i

(2)

 The subsequent step is to define the partial degree of binding, i ("i" moles of ligand bound per 1

mole of macromolecule), which characterizes the complex with "i" bound ligand molecules as

i 



i[M]i [M]T

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(3)

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(4)

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  [M]i  i [M]T  i 

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The concentration of the macromolecule complex with "i" ligand molecules bound, [M]i, is then

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Using expressions for  i and [M]i (eqs. 3 and 4), and eq. 1, one obtains the general relationship

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for the observed signal, Fobs, defined in terms of i and [M]T, as

  Fobs  FF [M]T  [ Fi FF  i ][M]T  i 



(5)

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

 accessible quantity, F , i.e., the relative fluorescence change with respect The experimentally obs to the initial fluorescence intensity of the free macromolecule, FF[M]T, is obtained by rearranging eq. 5, as

Fobs 

F F   Fobs  FF [M]T    i F  i  FF [M]T  FF  i 

and



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(6)

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F  Fobs   i (i )  i 

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(7)

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

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For a given total macromolecule concentration, [M]T, Fobs is the measured fractional fluorescence change observed at a given total ligand concentration, [L] T. The parameter, Fi/i, is

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the average molar fluorescence change per bound ligand in the complex containing "i" ligand

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molecules. Notice, Fi/i is an intensive molecular property of the macromolecule-ligand complex with “i” ligand molecules bound, i.e., independent of the ligand and macromolecule

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concentrations [6,11, preceding paper]. In other words, eq. 7 indicates that Fobs is only a

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function of the free ligand concentration, [L]F, i.e., Fobs is the total average degree of binding,

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i, where the partial degrees of binding are weighted by the corresponding Fi/i. Moreover, if Fobs is a monotonously increasing or decreasing function of [L]T, which is the case in practically most of the titration studies (see below), then for a specific value of Fobs, the value of i, and of the free ligand concentration, [L] F, must be the same for any value of [L] T and [M]T. Therefore, the same value of Fobs at different [M]T’s indicates the same physical state of the macromolecule, i.e., the same i and [L]F [6,11,12,17-19,21-30,32-38].

Expression 7 is not based on any assumption about the relationship between the observed physico-chemical/spectroscopic signal and i. In other words, it is thermodynamically rigorous and independent of any binding model [6,11,15]. It also points out to a very effective method of 10

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transforming a titration curve into a quantitative thermodynamic binding isotherm. In our

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practice, for the affinities encountered in the examined systems and available macromolecular concentrations, only two titrations at two different total concentrations of the macromolecule,

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[M]T1 and [M]T2, are enough to obtain a reliable thermodynamic binding isotherm [6,11,12,17-

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19,21-30,32-38]. Two such theoretical fluorescence titration curves for the hypothetical binding

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process where the ligand binds to two independent and different binding sites on the macromolecule are shown in Figure 1a. The binding system is characterized by intrinsic binding

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constants, K1 = 5x106 M-1, and K2 = 5x105 M-1, and spectroscopic parameters, FB1 = 2.5, FB2 = 0.5, and FC = 3.0 [preceding paper]. The quantity, Fobs (eq. 7), is plotted as a function of the

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logarithm of the total ligand concentration, [L]T, for two macromolecule concentrations selected

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as, [M]T1 = 1x10-6 M and [M]T2 = 5x10-6 M, respectively. For the high macromolecule

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concentration, [M]T2, the titration curve, is shifted toward higher ligand concentration range, as more ligand is required to saturate the macromolecule. For the selected “j” value of (Fobs)j, a horizontal line is drawn that intersects both titration curves. The points of intersections of the horizontal line and the titration curves identify two values of the total ligand concentration, ([L]T1)j and ([L]T2)j. Because the state of the macromolecule, defined by the same value of (Fobs)j, is the same for each titration curve, the values of ([L]F)j and (i)j must also be the same (eq. 7). Thus, the total concentrations of the ligand, ([L]T1)j and ([L]T2)j, are described by two mass conservation equations, namely

([L]T1 )  ([L]F )  (i ) [M] j



j

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j

T1

(8)

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and

([L]T2 )  ([L]F )  (i ) [M] j

j

T2

(9)

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j

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

Combining these two mass conservation equations provides the total average degree of binding

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(Σi)j, and the free protein concentration, ([L]F)j, at a given (Fobs)j , as

and

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  ([L]T2 ) ([L]T1 )  j j   (i )  j ([M]T2 [M]T1 )

(10)



([L]F ) ([L]TX ) (i ) ([M]TX ) j

j

j

(11)

where subscript, X, is  1 or 2 [6,11,12,17-19,21-30,32-38]. Identical calculations are performed through the entire set of the titration curves (Figure 1a). The selected intervals of the observed signal change depend on the resolution of the titration curve, i.e., the magnitude of the observed signal. In our practice, selection of the signal changes differing by 5% of the total observed signal change provides the required resolution of the binding isotherm, giving the model12

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independent values of ([L]F)j and (Σi)j at selected “j” value of (Fobs)j [6,11,12,17-19,21-30,32-

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38].

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Moreover, to obtain most accurate estimates of both ([L]F)j and (Σi)j, the concentration of

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the bound ligand, [L]B, must be comparable to its total concentration, [L] T, i.e., [L]B must be at

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least ~10 - 15% of [L]T. The accuracies of the determination of Σi, and [L]F differ, depending

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on the region of the experimental titration curves. The values of Σi, are mainly affected in the region of [L]T, where the binding reaction approaches saturation. The values of [L] F are mostly

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affected in the initial part of the titration curves, where [L] F constitutes a small fraction of [L]T.

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As a consequence, selection of the macromolecule concentrations is critical for obtaining

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accurate estimates of ([L]F)j and (Σi)j over the largest possible region of the titration curves. The selection process is guided by preliminary titrations, behavior of the system (possible aggregation) and also by the precision of the fluorescence intensity measurement. If too high concentrations are used, reliable values of i can be obtained but not of [L]F. For too low concentrations, the small separation of the titration curves on the total ligand concentration scale will result in the low resolution of the monitored signal. Discussion above constitutes only guidelines. If nothing is known about the behavior of the examined system (usually, nothing is known about interesting new systems, but not always), then the preliminary titrations are the empirical method that provides the initial estimates of the ligand affinity and suitable total macromolecule concentrations to perform quantitative analysis. As an example, if the affinity is in the range of ~1x107 M-1 than total macromolecule concentrations from ~5x10-7 M to ~5x10-6 13

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M should provide required resolution, with macromolecular concentrations differing by factors

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~2 – 3.

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The dependence of the observed fluorescence intensity upon the total average degree of

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binding for the considered model (Figure 1a) is shown in Figure 1b. The plot is nonlinear (as

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assumed by our hypothetical binding model) and shows two binding phases, the high-affinity and low-affinity phase. It is clear that any postulation about the linear character of the plot would

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introduce significant error in the determined binding parameters. Notice, the functional dependence of Fobs upon i, as shown in Figure 1b, can be used to estimate the maximum

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value of i, i.e., the maximum stoichiometry of the binding process. In experimental practice,

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usually the maximum value of the observed signal, Fmax, can be determined with adequate

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precision, at the plateau of the titration curve, while values of i can be obtained only in a certain range of ligand concentration and up to a certain value of Fobs (see below). As a result, a plot of Fobs as a function of i, does not reach Fmax, although Fmax is known. In such cases, an extrapolation of the low affinity phase of the function Fobs(i) to Fmax provides the estimate of the maximum value of i, i.e., the maximum stoichiometry of the formed complex, as depicted in Figure 1c [6,11,12,17-19,21-30,32-38].

We mentioned above that the quantitative method can be used if Fobs is the monotonously increasing or decreasing function of [L]T. More precisely, it can be used for any parts of the titration curves which behave like a monotonous increasing or decreasing function of [L] T. We illustrate the problem using a simple model of the ligand binding to a fluorescing macromolecule 14

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possessing two independent and different binding sites, characterized by different relative

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fluorescence changes, induced by the ligand binding to a particular site [preceding paper]. The dependence of the relative fluorescence intensity, Fobs, as a function of the total ligand

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concentration, [L]T, for the selected binding model is shown in Figure 2a (solid line). The curve

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has been obtained using K1 = 5x106 M-1, K2 = 5x105 M-1, FB1 = 1.5, FB2 = 1.5, and FC = 3.0.

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The entire titration curve is clearly a monotonous increasing function of [L] T and we can apply the general quantitative method to obtain the binding isotherm over the whole spectroscopic

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titration curve (see above). Titrations for the same system with the same binding constants, but different values of the relative fluorescence changes, characterizing the association with each

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particular binding site, are also included in the Figure 2a. The final value of the plateau is

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selected to be the same for all titration curves. It is obvious that already for, FB1 = 4 and FB2 =

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-1, the titration curve contains two binding phases corresponding the association with the highand low-affinity binding sites. The negative value of FB2 indicates that there is a fluorescence quenching accompanying the binding to the weak-affinity site. Such a behavior would already indicated a complex association process, which must include at least two binding sites, but, in practice, the experimenter would not a priori know the exact number of the sites.

The general quantitative analysis can be applied to each part of the titration curve where Fobs is the monotonous increasing or decreasing function of [L] T. The dependence of the relative fluorescence intensity, Fobs, as a function of the total ligand concentration, [L]T, for the same binding system as in Figure 2a, with K1 = 5x106 M-1, K2 = 5x105 M-1, FB1= 5.0, FB2 = 2.0, and FC = 3.0, obtained for two different macromolecule concentrations is shown in Figure 15

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2b. There is a high-affinity binding phase, where the relative fluorescence change is a

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monotonously increasing function of [L] T and the low-affinity phase, where Fobs is a monotonously decreasing function of [L]T. There is also an intersection point of the titration

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curves, resulting from the complex signal behavior. The general method (see above) applies to

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the first binding phase, up to [L]T corresponding to the intersection point. For each selected value

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of (Fobs)j, one obtains the set of [L]T1 and [L]T2 and determined i, and [L]F, using eqs. 10 11. Analogously, above the intersection point, where the function is monotonously decreasing

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with increasing values of [L]T, for each selected value of (Fobs)i, one obtains the set of [L]Tx and [L]Ty and determined i, and [L]F, using eqs. 10-11. Obviously, the accuracy of the analysis

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association in each binding phase.

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will strongly depend on the quality of the observed spectroscopic signal accompanying the

1. b. Fluorescence Anisotropy.

In the case of monitoring the multiple-ligand binding to a macromolecule, using the fluorescence anisotropy, the signal conservation equation is defined, by the relationship [39,40, preceding paper]

robs rFfF  firi



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(12a)

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where rF and ri, are the fluorescence anisotropies of the free macromolecule and a given

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same entities to the total fluorescence intensity of the sample.

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complex, Mi, respectively, while fF, and fi, are the corresponding fractional contributions of the

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As we discussed in the preceding paper, further analysis is greatly simplified if there is not

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fluorescence intensity change accompanying the binding process (eq. 52, preceding paper). The condition can be tested for, e.g., binding of two ligand molecules (see above). However, in a

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general case of the multiple-ligand binding, some complexes might have positive changes of the fluorescence intensities and some might have negative changes with similar affinities, which

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may compensate each other. In other words, the presence or absence of fluorescence intensity

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changes accompanying the reaction cannot be rigorous tested. However, what is required in any

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case is that robs is a monotonously increasing or decreasing function of [L]T (see above). In the absence of any fluorescence intensity changes, the fractional contributions of the complexes to the total fluorescence intensity of the sample are then the fractional contributions of corresponding complexes to the total concentration of the macromolecule, i. For instance, for the model of binding of two ligand molecules (preceding paper, see above), the observed fluorescence anisotropy of the sample is defined, as

robs rAfA  rB1fB1  rB2fB2  rCfC



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(12b)

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where rB1, rB2, and rC, are fluorescence anisotropies of the complexes B1, B2, and C, while fB1,

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fB2, and fC, are the corresponding fractional contributions of the same complexes to the total fluorescence intensity of the sample. No fluorescence intensity change means that the

1 1  0 2  Z 1(K1 +K 2 )[L]F K1K 2[L]

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fA 

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corresponding fractional contributions to the total fluorescence intensity of the sample are

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fB1 

K1[L]F



1(K1 +K 2 )[L]F K1K 2[L]2 F

K1[L]F  B1 Z

fB2 



fC 

K 2 [L]F

1(K1 +K 2 )[L]F  K1K 2[L]2 F

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

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

F

K1K 2 [L]2

F

1(K1 +K 2 )[L]F K1K 2[L]2 F





K 2 [L]F  B2 Z

K1K 2 [L]2

F

Z

(13)

 C



where, Z, is the partition function of the binding system. The fluorescence anisotropy of the sample for multiple ligand binding process, at any titration point, is then

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n

robs rF 0   iri

(14)

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i 1

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where the summation is overall possible complexes of the macromolecule with “i” ligand

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molecules bound.

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This is different from the fluorescence intensity because instead of the partial degrees of

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binding, i, one has the fractional contributions of every complex to the total macromolecule concentration, i. However, thermodynamically, this is a result that we need. The observed

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fluorescence anisotropy is the sum of all fractional contributions of each complex, i, to the total

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macromolecule concentration, weighted by the fluorescence anisotropies corresponding to a

by

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given complex. In the model of just two binding sites considered here, i and i are described

i =

and

(K1  K 2 )[L]F 2K1 K 2 [L]2

F 1(K1  K 2 )[L]F K1 K 2[L]2 F

(15)



i =

(K1  K 2 )[L]F K1 K 2[L]2

F

1(K1  K 2 )[L]F K1 K 2[L]2

F



19

(16)

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Both quantities are unique functions of the free ligand concentration, [L] F. The same is true for

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any multiple-ligand binding models. Ergo, for the observed fluorescence anisotropy monotonous increasing or decreasing with [L]T, a specific value of i always corresponds to a specific value

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of i, at a given, [L]F, independently of the macromolecular concentration. Therefore, the

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same value of robs at different [M]T’s indicates the same physical state of the macromolecule, i.e.,

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the same i, i, and [L]F [6,11,12,17-19,21-30,32-38].

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To convince ourselves further, we can perform a simple transformation of eq. 14 utilizing the fact that any partial degree of binding, i, is related to the fraction of the macromolecule in

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the complex containing “i” bound ligand molecules, i, by eq. 3, as

i 

i[M]i  ii [M]T

(17)

 The observed fluorescence anisotropy is then

n n   r robs rF 0   iri rF 0   i ii i i 1 i  1 

and

(18a)

 r  robs  rF 0   i i i i  1  n

20



(18b)

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Expression above is completely analogous to eq. 7 derived for the fluorescence intensity. The parameter, ri/i, is the average fluorescence anisotropy per bound ligand molecule in the complex

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containing "i" ligand molecules. Correspondingly to Fi/i (eq. 18b), the quantity, ri/i, in eq. 18b

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is an intensive molecular property of the macromolecule-ligand complex with “i” ligand

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molecules bound, i.e., independent of the ligand and macromolecule concentrations.

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Therefore, the method of transforming a titration curve into a quantitative thermodynamic binding isotherm, discussed above for the fluorescence intensity, as the applied signal, is also

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applicable here. For two titrations at suitable macromolecular concentrations (see above) and at

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two different total concentrations of the macromolecule, [M]T1 and [M]T2, the total ligand

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concentrations, ([L]T1)j and ([L]T2)j for the same value of robs are related to [L]F and i by eqs. 8 and 9. The total average degree of binding, i, and the free ligand concentration, [L]F, are then obtained using eqs. 10 and 11. Two such theoretical fluorescence titration curves, using the fluorescence anisotropy as the spectroscopic signal, for the hypothetical binding process where the ligand binds to two independent and different binding sites on the macromolecule are shown in Figure 3. The binding system is characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.2, rB2 = 0.1, and rC = 0.3. The quantity, robs, (eq. 12b) is plotted as a function of the logarithm of the total ligand concentration, [L]T, for two macromolecule concentrations, [M]T1 = 1x10-6 M and [M]T2 = 5x10-6 M respectively. The titration curves are well separated and would allow the experimenter to determine the thermodynamic binding isotherm, using the quantitative method described above. Notice, the system is characterized by 6 independent parameters, K1, K2, rA, rB1, 21

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rB2, and rC. However, once the binding isotherm is determined, K1 and K2, can be obtained,

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regardless of the values of rA, rB1, rB2, and rC. Moreover, rA and rC, can usually be measured as the initial anisotropy of the macromolecule fluorescence and at the plateau of the spectroscopic

NU

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determined from the fit of the original titration curves.

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titration curves. In this simple system, two remaining parameters, rB1 and rB2, are to be

We discussed above, that the quantitative method can be used for any parts of the titration

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curves where robs behaves like a monotonous increasing or decreasing function of [L]T. The dependence of the fluorescence anisotropy, robs, as a function of the total ligand concentration,

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[L]T, for the macromolecule with two different binding sites is shown in Figure 4a (solid line).

This is the monotonous increasing function of [L]T over the whole

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0.15, and rC = 0.2.

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The curve has been obtained using K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.2, rB2 =

spectroscopic titration curve and we can apply the general quantitative method to obtain the binding isotherm (see above). Analogous titration for the same system with the same binding constants, but different values of the fluorescence anisotropy characterizing the association with the high-affinity binding site, is also shown in the Figure 4a. It is obvious that for, rB1 = 0.3 and rC = 0.2, robs is not a monotonous function of [L]T over the entire titration curve, which contain two binding phases, corresponding to the association with the high- and low-affinity binding site, respectively.

22

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The dependence of the fluorescence anisotropy, robs, as a function of the total ligand

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concentration, [L]T, for the same binding system as in Figure 4a, with K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.3, rB2 = 0.15, and rC = 0.2, obtained for two different macromolecule

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concentrations, is shown in Figure 4b. In the high-affinity binding phase, robs is a monotonously

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increasing function of [L]T. Above the intersection point of two titration curves, robs is a

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monotonously decreasing function of [L]T. The general method (see above) applies to the first binding phase, up to [L]T corresponding to the intersection point, and at the values of [L]T above

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the intersection point, where robs is monotonously decreasing with the increasing values of [L]T

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(see above).

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The situation may seem more complex when the changes of the fluorescence intensity of

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different complexes accompany the binding reaction, which is monitored by the fluorescence anisotropy. For the considered model of two different binding sites, the values of fA, fB1, fB2, and fC, (eq. 13) are described by the relationships (preceding paper)

fA 

1 1(FB1K1 +FB2 K 2 )[L]F FC K1K 2[L]2

F

(19)

fB1 

FB1K1[L]F

1(FB1K1 +FB2 K 2 )[L]F FC K1K 2[L]2

F



23

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fB2 

FB2K 2 [L]F

1(FB1K1 +FB2 K 2 )[L]F FC K1K 2[L]2 FC K1K 2 [L]2

F

1(FB1K1 +FB2 K 2 )[L]F FC K1K 2[L]2

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fC 

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F

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F

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 fluorescence intensities, FB1, FB2, and FC of corresponding complexes are where all molar

expressed as the multiplicities of the fluorescence intensity of the free macromolecule, taken as

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FF = 1. Expression 12b together with eq. 19 contain 8 independent parameters, which make the quantitative analysis of even this simple binding system practically intractable in the case of a

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single titration curve. The solid line in Figure 5a is the dependence of the fluorescence

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anisotropy, robs, as a function of the total ligand concentration, [L]T, for the same binding system

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as in Figure 4a, with K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.3, rB2 = 0.15, rC = 0.2, FA =1, FB1 = 1, FB2 = 1, and FC = 1. The dashed line in Figure 5a is the dependence of the fluorescence anisotropy for the system characterized by the same binding constants and fluorescence anisotropies of all complexes of the reaction, but with but with the fluorescence intensities as, FA =1, FB1 =1.5, FB2 = 1, and FC = 1.3. Thus, in this case, the binding reaction is accompanied by the fluorescence changes of formed complexes. The titration curves do not superimpose each other and would provide different binding parameters, if the computational analysis were limited to a single titration curve.

24

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However, notice, that the fractional fluorescence intensities in eq. 19 are all unique

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functions of the free ligand concentration, [L] F, with FB1, FB2, FC K1, and K2 being constants characterizing the system. Ergo, the observed fluorescence anisotropy, defined by eq. 12b, is also

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a unique function of [L]F, in any part of the titration curve where robs is a monotonously

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increasing or decreasing function of [L]T (see above). The general quantitative method to

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construct the binding isotherm and obtain the maximum stoichiometry, as well as extract the binding parameters, K1 and K2, is clearly applicable here. An example of the approach is

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depicted in Figure 5b, which contains two titration curves obtained at two different macromolecule concentrations and obtained for K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 =

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0.3, rB2 = 0.15, rC = 0.2, FA =1, FB1 =1.5, FB2 = 1, and FC = 1.3. On the other hand, even if the

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experimenter somehow knew the maximum stoichiometry of 2:1 of the examined system, a

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single titration curve, like the one in Figure 5a, would be prohibitively complex to analyze and would require determination of all 8 parameters.

1. c. Fluorescence Polarization.

Analysis of the titration curved obtained using the fluorescence polarization is analogous to the analysis of the experiments performed using the fluorescence anisotropy. Recall, in the binding studies, instead of the total emission, IVV + 2GIVH, the expression, IVV + GIVH, enters the final formulas into molar fluorescence intensity factors describing the titration curves [39,40, preceding paper]. In other words, the difference between the emissions enters into the molar fluorescence intensities proportional to the concentrations of the fluorescing species. For the 25

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multiple-ligand binding reaction, the signal conservation equation for the fluorescence

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(20)

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pobs pFfF'  fi'pi

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polarization becomes

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 where pF and pi, are the polarizations of the free macromolecule and the complex, Mi, respectively, while fF' and fi' , are the corresponding fractional contributions of the emission



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intensity of same complexes to the observed fluorescence intensity of the sample, IVV + GIVH,.



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If there is not fluorescence intensity change accompanying the binding process, a condition that cannot be rigorous tested for the multiple-ligand binding systems (see above), the fractional

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contributions of the complexes to the observed fluorescence intensity of the sample are then the fractional contributions of corresponding complexes to the total concentration of the macromolecule, i. The fluorescence polarization of the sample, at any titration point, is then

n

p obs p F 0   ip i i 1

(21)

where the summation  is over all possible complexes of the macromolecule with “i” ligand molecules bound.

26

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Using eq. 21 for the relationship between the partial degrees of binding and the fractional

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contributions of a given complex with ‘i” ligand molecules bound, provides (see above) the

(22)

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n   p p obs  p F 0   i i i i  1 

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observed polarization of the sample as

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Expression above is completely analogous to eq. 7 derived for the fluorescence intensity or eq. 18b for the fluorescence anisotropy. The parameter, pi/i, is an intensive property, i.e., the

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average fluorescence polarization per bound ligand molecule in the complex containing "i"

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ligand molecules independent of the ligand and macromolecule concentrations. The discussion

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about the monotonous character of pobs as a function of [L]T applied also here (see above). As in the case of the fluorescence anisotropy (see above), for two titrations at two different total concentrations of the macromolecule, [M]T1 and [M]T2, the total ligand concentrations, ([L] T1)j and ([L]T2)j for the same value of (pobs)j are related to [L]F and i by eqs. 10 and 11, wherever pobs is the monotonous increasing or decreasing function of [L]T. Having I and [L]F, obtained at different total ligand concentrations, one can construct the thermodynamic binding isotherm and obtain the maximum stoichiometry, as well as extract the binding parameters.

27

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1. d. Cumulative Heat of Reaction.

Analysis of the multiple-ligand binding process using the heat of reaction as the physico-

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chemical signal to monitor the binding is completely analogous to the analysis using the

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fluorescence anisotropy or polarization (see above). The signal conservation equation is defined,

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by the relationship (preceding paper), as

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Q obs VS[M]T i H o i

(23)

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 enthalpy the formation of the complex with ‘i” number of ligand where Hoi is the molar

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reaction in the sample as

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molecules bound [41-43]. Using the relationship, i = ii, one obtains the observed heat of the

H o  Q obs VS[M]T  i i   i  1 i  n

(24)

 Once again, expression above is completely analogous to eqs. 7 and 18b derived for the fluorescence intensity and anisotropy. The parameter, Hoi/i, is an intensive property, i.e., the average molar enthalpy per bound ligand molecule in the complex containing "i" ligand molecules, independent of the ligand and macromolecule concentrations. The observed heat of reaction is the total average degree of binding weighted by molecular parameters, Hoi/i. For two

28

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titrations at two different total concentrations of the macromolecule, [M] T1 and [M]T2, one can

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obtain two total ligand concentrations, ([L]T1)j and ([L]T2)j at the titration point “j”, for the same value of Qobs. [M]T1 and [M]T2, and ([L]T1)j and ([L]T2)j are then related to [L]F and i by eqs.

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10 and 11. The discussion about the monotonous character of Qobs as a function of [L]T applied

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also here (see above). Once i and [L]F are determined from the titrations curves at different

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total ligand concentrations, one can construct the thermodynamic binding isotherm.

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Theoretical titration curves, using the heat of reaction as the physico-chemical signal, for the hypothetical binding process where the ligand binds to two independent and identical binding

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sites on the macromolecule are shown in Figure 6. The plots have been normalized to the

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maximum observed heat of the reaction at the saturation, Qmax = VS[M]THoC. The binding

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system is characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, HoB1 = 20.92 kJ/mol, HoB2 = 4.184 kJ/mol, and HoC = 25.10 kJ/mol. The quantity, Qobs, (eq. 61b, preceding paper) is plotted as a function of the logarithm of the total ligand concentration, [L] T, for two macromolecule concentrations, [M]T1 = 1x10-6 M and [M]T2 = 5x10-6 M respectively. The separation of the titration curves on [L]T scale would allow the experimenter to determine the thermodynamic binding isotherm, using the quantitative method described above (eqs. 10 and 11).

29

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1. e. The Sedimentation Coefficient.

Finally, the expression for the average sedimentation coefficient used as the physico-

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chemical signal to monitor the binding, in the presence of the ligand is [44]

(25)

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n   s sav  sF 0   i i i i  1 

 been obtained in the same way as the analogous expression for the The expression above has

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fluorescence anisotropy (eq. 18b) and is completely analogous to eq. 7 derived for the

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fluorescence intensity. The parameter, si/i, is an intensive property, i.e., the average

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sedimentation coefficient per bound ligand molecule in the complex containing "i" ligand molecules, independent of the ligand and macromolecule concentrations. In other words, the observed average sedimentation coefficient is the total average degree of binding weighted by molecular parameters, si/i and, similar to the fluorescence anisotropy and polarization, sav([L]T) does not require any normalization. Construction the thermodynamic binding isotherm proceeds using the quantitative method (Figure 1a), as described above [44]. The discussion about the monotonous character of sav as a function of [L]T applied also here (see above).

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2. Practical Application of the Quantitative Method of Analysis of Physico-chemical

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Titrations.

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2. a. Fluorescence Intensity Change Accompanying the Ligand Binding Originates From

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the Macromolecule.

The polymerase X (pol X) of the African Swine Fever Virus (ASFV) is involved in repair

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processes of the viral DNA protecting the nucleic acid from to the host defense mechanisms. The enzyme has a molecular weight of ~20000 and is currently the smallest known DNA polymerase

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[45,46]. The problem of the DNA-repair polymerase mechanism, which is still not well

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understood, is how the enzyme recognizes the damaged DNA containing a small ssDNA gap,

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surrounded by the large excess of the dsDNA. We addressed the energetics of pol X interactions with the gapped DNA using the DNA substrate depicted in insert in Figure 7a [45]. One of bases in the gap has been replaced by the fluorescent marker (fluorescein), which provided the signal to monitor the association reaction [45].

Fluorescence intensity of the of DNA substrate, having the ssDNA gap with five nucleotide residues, titrated with the ASFV pol X at two different DNA concentrations, are shown in Figure 7a. The maximum relative quenching of the nucleic acid fluorescence, Fobs, reaches the value of Fmax = 0.58 ± 0.03 at saturation. As discussed above, the fluorescence titration curves alone do not

provide

any

indication

as

to

the

maximum

stoichiometry.

They

are

just

spectroscopic/physico-chemical titration curves. Moreover, the character of the binding process, 31

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e.g., whether or not it is a cooperative process, is not obvious either. Furthermore, the structure

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of the DNA substrate or the enzyme does not provide any possibility to make an educated guess about the maximum stoichiometry. To obtain the total average degree of binding, Σi,

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independent of any assumption about the relationship between the observed signal and Σi, we

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used the quantitative method discussed above [45]. The dependence of ΔFobs, upon Σi is shown

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in Figure 7b. The values of Σi could be determined up to ~1.6. Also, the analysis has been limited to the region of the titration curves, where the bound enzyme constitutes a significant

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(~10 – 15%) fraction of the total polymerase concentration (see above). The plot is, within experimental accuracy, linear. However, binding process continues above the stoichiometry of

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~1.6, reached at Fobs ≈ 0.47. The maximum value of the observed relative fluorescence

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quenching, Fmax, is obtained from the original titration curves (Figure 7a) and a short

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extrapolation of Σi to ΔFmax = 0.58 ± 0.03, provides ΣΘi = 2 ± 0.2. Thus, the data show that, at saturation, two ASFV pol X molecules bind to the examined gapped DNA substrate [45].

Determination of the maximum stoichiometry is the first step in the analysis of a new binding system. To extract binding parameter, one has to apply a statistical thermodynamic model, which is based on know molecular characteristics of the participating components of the reaction [6,11,12,17-19,21-30,32-38]. This is an extra-thermodynamic knowledge and always it is subjected to interpretation. The gapped DNA contains the ssDNA gap and two dsDNA parts (Figure 7a). The enzyme specifically binds to the gap and can form a high affinity gap complex. Nevertheless, the polymerase molecule can also associate with two dsDNA parts replacing the gap complex [45]. The simplest partition function, ZG, for the considered binding system, which 32

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describes the observed behavior and takes into account the determined maximum stoichiometry,

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is described by

G

D

F

D F

(26)

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G

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Z 1 (K  K )L K 2 L2

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 where KG and KD are intrinsic binding constants characterizing the formation of the gap complex

and the complex of the enzyme with the dsDNA part of the DNA substrate, respectively. The

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quantity, , is the cooperative interactions parameter, which characterizes possible cooperative interactions between two polymerase molecules associated with dsDNA parts. The total average

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degree of binding, i, is defined as

i 

(K  K )L +2K 2 L2 G

D

F

D F

(27)

ZG

 model, the observed relative fluorescence quenching, F , is then For the considered binding obs

Fobs =

F K L  F K L 2F K 2 L2 G

G

F

D

D F

Z

D

D F

(28)

G

where, FG, andFD are the relative molar fluorescence quenchings of the DNA fluorescence intensity (see above) [45]. 33

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Even such a simple system with only two enzyme molecules binding to the DNA is

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characterized by a large number of independent parameters, FG, FD, KG, KD, and  (eq. 28). However, in applying the quantitative analysis, we have two plots available, the parental titration

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curves (Figure 7a) and the plot of Fobs as a function of i (Figure 7b). The values of FG can

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be obtained from the plot in Figures 7b, with FG = ∂F/∂(i). Moreover, FG = FD,

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otherwise the plot in Figure 7b would not be linear. Finally, FD can be obtained from the fact that, Fmax = 2FD. The solid lines in Figure 7a are nonlinear least squares fits of the

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experimental titration curves to eq. 28 with three fitting parameters, which provide, KG = (6.5 ± 1.6) x 108 M-1, KD = (6.5 ± 1.6) x 108 M-1, and  = 1. The data indicate that affinities the two

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ASFV pol X molecules in the complex with the DNA substrate containing five nucleotide residues in the ssDNA gap are virtually the same, independent of the presumed nature of the

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formed complexes. This is a profound result indicating that with the substrate containing five nucleotide residues in the gap, the ASFV pol X does not form the high-affinity gap complex to any detectable extent. The interested reader is referred to the original work to consult the full discussion of the biological importance of these results [45,46].

Discussion above includes analysis of the thermodynamic and spectroscopic parameters characterizing the binding system. This is usually performed, as one is interested in both sets of data. The corresponding thermodynamic binding isotherm, i.e., the dependence of i upon the logarithm of [L]F, for the binding of the polymerase to the considered DNA substrate, is shown in Figure 7c. The values of i could be reliably determined from ~0.14 to ~1.6 for the applied concentrations of the nucleic acid (see above). Notice, the available range of i comprises from 34

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~10% to ~80% of the maximum value of i = 2. The solid line is a nonlinear least squares fit of

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the statistical thermodynamic model, defined by eqs. 26 - 27. The obtained binding parameters, KG ≈ 6.5 x 108 M-1, KD ≈ 6.5 x 108 M-1, and  ≈ 1, and within experimental accuracy, are the

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same as those independently determined, in the analysis of both spectroscopic and

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thermodynamic parameters described above.

Accompanying the Ligand Binding.

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2. b. The Changes of the Average Sedimentation Coefficient of the Macromolecule

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Interactions between the E.coli primary replicative helicase DnaB protein and the essential

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replication factor DnaC protein are absolutely necessary for the initiation of the DNA replication

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[44]. Interactions between the DnaB helicase and the DnaC protein also constitute a major step in the assembly of the primosome, a multi-protein-DNA complex, involved in priming the DNA synthesis. Because the DnaB protein is a hexamer of ~300000 Daltons and the DnaC protein has a molecular mass of ~30000 Daltons, they significantly differ in the values of the their sedimentation coefficients [44]. The DnaB protein has been labeled with the fluorescein marker, which allowed us to specifically monitor the hydrodynamics of the DnaB hexamer complex with the DnaC protein using absorption band (495 nm) of the fluorescein residues and eliminating any interference from the absorption of the protein and the nucleotide cofactors [44].

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The dependence of the average sedimentation coefficient, sav, of the DnaB hexamer,

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corrected to standard conditions, as a function of the total DnaC concentration, for two different concentrations of the helicase, in the buffer, 10 mM Tris/HCl pH 8.1, 100 mM NaCl, 5 mM

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MgCl2, 10 °C, 2mM AMP-PNP, is shown in Figure 8a. At higher concentration of the helicase, a

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given value of sav is reached at higher DnaC protein concentration, as a higher DnaC protein

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concentration is necessary to saturate the enzyme. Using the titration curves, shown in Figure 8a, the total average degree of binding, i, of the DnaC protein on the DnaB hexamer, has been

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obtained using the general approach described above (eqs. 10, 11, and 25). Figure 8b shows the dependence of the observed sav of the DnaB hexamer as a function of i. The plot is linear,

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indicating that the increase of sav is a linear function of the average degree of binding of the

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DnaC protein. Short extrapolation of the plot to the maximum value of sav = 16.5 S ± 0.3, shows

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that at saturation 6 ± 0.5 molecules of DnaC protein bind to the DnaB hexamer. Thus, the data show that, at equilibrium, the DnaB hexamer forms a complex with the DnaC protein where a maximum of six DnaC molecules associate with the hexamer. The solid lines are computer fits of the experimental binding isotherms to the hexagon model using single set of binding parameters, the intrinsic binding constant, K = 1 x 105 M-1 and  = 30 [44].

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3. Quantitative Method of Analysis of Physico-Chemical Titrations Using the Spectroscopic

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Signal Originating From the Ligand. The LBDF Function Method.

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The “normal” titrations discussed above are intuitively easier to analyze than the situation

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where the binding reaction is monitored by the signal originating from the ligand because, in a

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natural way, the saturation of the macromolecule increases during the titration. In a “reverse” titration, where the ligand is titrated with the macromolecule, the saturation of the ligand with the

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increase of the macromolecule concentration increases while the saturation of the macromolecule with the ligand decreases. In the course of the titration, some physico-chemical property of the

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ligand (predominantly spectroscopic property) changes upon binding to the macromolecule, as a

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result of the association reaction. These changes of the observed signal monitor the apparent

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degree of saturation of the ligand with the macromolecule. In the case of protein - nucleic acid interactions, formation of complexes have been routinely examined by changes of the protein fluorescence in the presence of the nucleic acid [6,11-14,20,21,31,47-56]. However, the studies were mainly performed using the assumption, more or less reasonable for specific cases, about the linearity between the observed signal and the total average degree of binding. For the first time, the quantitative approach of transforming the physico-chemical/spectroscopic titration curves into thermodynamic binding isotherms has been developed for the “reverse” titration method in fluorescence titration studies of E. coli SSB protein - nucleic acid interactions [12].

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The approach is based on the same signal and mass conservation equations as described

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above for normal titrations. We will illustrate the method using the fluorescence intensity as the measured signal. In the complex with a macromolecule, the ligand can be in "i" bound states, in

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each state the ligand possessing a different molar fluorescence intensity, Fi. The signal

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conservation equation for the observed fluorescence, Fobs, at the total concentrations, [L]T and

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[M]T, respectively, contains terms describing the free ligand fluorescence and the fluorescence of

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the ligand bound in any of its "i" possible bound states, as

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

(29)

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Fobs FF [L]F Fi [L]i

In the expression above, FF and [L]F are the molar fluorescence and concentration of the free ligand, respectively, and Fi and [L]i are the molar fluorescence and concentration of the ligand bound in a state "i", respectively. The mass conservation equation for the total ligand concentration is

where

[L]T  [L]F  [L]i

(30a)

[L]i  i [M]T

(30b)



38 

ACCEPTED MANUSCRIPT

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and i, is the partial degree of binding of the ligand in the “i” state. Introducing eqs. 30a, and

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30b to eq. 29, provides

(31)

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Fobs  FF [L]T  [M]T (Fi FF ) i

The product, FF[L]T, is  the initial fluorescence of the ligand in the absence of the macromolecule.

D

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[L]  Dividing both sides of eq. 31 by FF[L]T and multiplying by the ratio,  T , gives [M ]T 

and

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 F )  (Fobs  FF [L]T )  [L]  (F T     i F i     FF [L]T  [M]T   FF 

(32)



(Fobs  FF [L]T ) FF [L]T



39

[L]   T   Fii [M]T 

(33)

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The quantity,

(Fobs FF [L]T )  Fobs , is the fractional change in the fluorescence intensity of FF [L]T

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the ligand in the sample in the presence of the macromolecule, at any given titration point.

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 (F F ) Whereas, (F) i  i F , is the relative fluorescence intensity change, characterizing the ligand FF

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bound to the macromolecule in a state "i", i.e., an intensive quantity independent of the ligand

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 concentration in a given “i” state. Expression 33 can be written is a more concise form as

(34)

D

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[L]  Fobs  T    Fii [M]T 

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  [L] The quantity, Fobs  T , is equal to Fi i , the sum of all partial degrees of binding for "i" [M]T  states of the bound ligand, weighted by the molar fluorescence change of the ligand, i.e.,   molecular intensive quantities independent of the ligand and macromolecule concentrations, for the ligand in each bound state. Thus, the weighting factors ∆Fi are constant for a particular binding state "i". For given specific values of i and [L]F, Fii is constant, reflecting the distribution of the ligand among different possible bound states. Expression 34 is general and  independent of the spectroscopic signal used to monitor the interactions [6,11,12]. In other

[L]  words, the values of [L]F and i are constant for a given specific value of Fobs  T , [M]T  [L]  independent of [L]T and [M]T, in any region of Fobs  T , where the function is [M]T   40



ACCEPTED MANUSCRIPT

monotonously increasing or decreasing with [M]T (see above). Therefore, quantitative estimates

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[L]  of i and [L]F can be obtained from plots of Fobs  T  vs. [M]T for titrations performed at [M]T 

SC

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[L]  different total ligand concentrations, [L]T. For a given value “j” of Fobs  T , the plots  [M]T  provide a set of concentrations [L]T and [M]T. These total ligand and macromolecule



MA

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concentrations form linear functions, as

([L]Tx ) ([L]F ) (i ) ([M]Tx ) j j

j

(35)

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j

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where the subscript x  refers to the set of [L]Tx and [M]Tx obtained at a given value of the

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[L]  quantity, ( Fobs  T )j [4,6,31,33]. Therefore, the plot of eq. 35 has a slope i and intersect [M]T 

L  the [L]TX)j axis at [L]F)j, corresponding to a given value of ( Fobs  T )j. The quantity,  M T  L  Fobs  T , is referred to as the Ligand Binding Density Function (LBDF) [6,11,12,53]. M T  



41

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A series of theoretical fluorescence titration curves of ligand with the macromolecule,

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using the fluorescence intensity of the ligand as the physico-chemical signal, at different total ligand concentrations is shown in Figure 9. The relative fluorescence change of the ligand, Fobs,

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is plotted as a function of the logarithm of the total ligand concentration, [L] T. The plots are

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generated for the hypothetical binding process where the ligand binds to two independent and

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identical binding sites on the macromolecule. The binding system is characterized by K1 = 5x106 M-1, K2 = 1x106 M-1, FB1 = 2, FB2 = 1, and FC = 3. For the higher total ligand concentration,

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[L]T, the titration curves are shifted toward the higher macromolecule concentration range, as

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more macromolecule is required to saturate the ligand.

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[L]  The dependence of the ligand binding density function, Fobs  T , upon the logarithm [M]T  of [M]T, for the spectroscopic titrations shown in Figure 9, is shown in Figure 10. There is a fundamental difference between the situation wherethe macromolecule is titrated with ligand and the situation where the ligand is titrated with the macromolecule. Titration with the ligand (normal titration) always starts with the zero value of the total average degree of binding. In the case where the ligand is titrated with the macromolecule, each titration starts from a different value of i, at different total ligand concentration. Plots in Figures 9 and 10 span different values of the total average degree of binding and sets of different curves are necessary in determination of the i over the largest possible range of the binding isotherm.

42

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L  For the selected “j” value of ( Fobs  T )j, a horizontal line intersects all titration curves M T 

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at a given value of i. The points of intersections of the horizontal line and the LBDF curves

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 ligand and macromolecule concentrations, ([L] ) and ([M] ) , identify values of the total Tx j Tx j

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related by expression 35. Identical calculations are performed for the entire set of LBDF curves

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to provide the model-independent values of (LF)j and (Σi)j, i.e., the thermodynamic binding

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isotherm (Figure 10) [6,11,12,14,53].

4. Practical Application of the LBDF Function method When the Physico-Chemical Signal

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Change Accompanying the Ligand Binding Originates From the Ligand.

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The practical application of the LBDF function method will be illustrated using the DnaB hexamer - etheno-ADP (ADP, etheno-adenosine 5’-diphosphate) system [23,53]. The DnaB protein is the primary replicative helicase of E. coli [54,32-37]. The enzyme is a homo-hexamer, which binds six nucleotide cofactors [33,50]. The titrations have been performed in the presence of acrylamide to increase the resolution of the experiments [53]. The induced dynamic quenching process, does not affect the energetics of the nucleotide - protein interactions [53]. We refer the reader to the original work where the application of the differential dynamic quenching to increase the resolution of the binding experiments is discussed [53].

43

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Fluorescence titrations of ADP sample with the DnaB protein, performed at different

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concentrations of the nucleotide cofactor are shown in Figure 11. The shift of the titration curves for higher nucleotide concentrations toward higher concentrations of the helicase reflects the fact

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that more of the enzyme is needed to saturate the increased amount of ADP. It is obvious that

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the spectroscopic titration curves alone do not provide any information as to the maximum

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stoichiometry of the formed complex, not to mention the nature of the binding process (see

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above).

D

L  The plots of the LBDF function, Fobs  T , upon the logarithm of the DnaB protein M T 

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concentration, for different total concentrations of ADP and corresponding to the fluorescence

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 shown in Figure 12. As discussed above, neither titrations at titration curves in Figure 11, are different total ligand concentrations nor the LBDF functions span the same range of the total average degree of binding (Figures 11 and 12). Thus, the situation that only two titration curves can be used for the quantitative analysis, as in the case of the normal titrations, is not applicable here. Multiple titrations at different [L] T, are necessary to span as large as possible range of i (Figures 11 and 12) [12,14,52,53]. Usually, 6 to 8 reverse titrations, applying successive total ligand concentrations that differ by a factor of 1.5 - 2, provide an accurate set of i and [L]F. In other words, the lowest and the highest total macromolecule concentrations should be different by at least one order of magnitude. Such a difference is enough to obtain the binding data spanning ~70-80% of the complete binding isotherm [6,11,12,14,53,55]. If necessary, one can

44

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add additional titrations, a process which is only limited by the ligand and macromolecule

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solubility/aggregation.

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The LBDF plots are smooth functions and can be interpolated using any suitable

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commercial computer software [6,11,12,14]. As discussed above, the points of intersection of the

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L  horizontal line, defining a given “j” constant value of Fobs  T  (Figure 12) with each LBDF M T  curve, provide the set of values ( (M Tx ) , (L Tx ) ), for which (L F ) and (i ) are constant. This j

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j

j

j

 is a linear function of (L Tx ) vs. (M Tx ) (eq. 35) with the slope equal to (i ) and the j



j



j





TE

D

L  intercept to (L F ) , respectively. The analysis is performed for different values of Fobs  T , j M T    



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resulting in a function of i with respect to [L]F, i.e., the thermodynamic binding isotherm.



The thermodynamic binding isotherm for the DnaB helicase - ADP system is shown in Figure 13. The values of i and [L]F could be reliably determined from ~0.5 up to the value of ~4.5 and their span were limited by the range of the examined concentration of [ADP]T. Nevertheless, the obtained isotherm comprises from ~8% to ~78% of the maximum value of i = 6 of the DnaB protein - nucleotide complex [6,11,12,53]. The solid line is a theoretical plot of the statistical thermodynamic model for the binding of six ligands to a short circular lattice with six identical discrete sites, characterized by the intrinsic binding constant, K, and cooperativity parameter,  [53]. Within experimental accuracy, the values of both parameters are the same as

45

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those independently determined, using the quantitative method, in which the quenching of the

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DnaB fluorescence has been used to monitor the complex formation (see above) [23,32].

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5. The Empirical Function Approach to Physico-Chemical Titrations.

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Once the thermodynamic binding isotherm for multiple-ligand binding process is constructed, the final step in the data analysis is extraction of interaction parameters using a

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suitable statistical thermodynamic model [1-5]. This can be done without any considerations as to the specific values of molar spectroscopic signals originating from various complexes

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[6,11,12,14,22]. The situation is different in the case of spectroscopic titration curves, because

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spectroscopic parameters of all possible complexes must enter the relationship between the

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observed change of the measured signal, Sobs, and the ligand concentration. The advantage of fitting the parental spectroscopic titration curves is that it spans a larger range of i than the experimentally accessible binding isotherm, resulting in more accurate estimate of the binding parameters (e.g., Figure 13). Nevertheless, determination of all spectroscopic parameters characterizing the complex interacting system, exclusively from physico-chemical titrations, is simply impossible (see below). Thus a method that allows the experimenter to fit titration curves for a given statistical thermodynamic binding model, is an invaluable tool [6,11,12,22,53].

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We will illustrate the method using as example the binding of the ADP analog, ADP, to

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the E. coli primary replicative helicase, the DnaB protein [22]. The DnaB helicase is a homohexamer, which possesses six nucleotide-binding sites (see above). Binding of ADP to the

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protein induces a significant quenching of the protein fluorescence, indicating that the cofactor

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binds near the tryptophan residues of the enzyme [22]. The dependence of the relative

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fluorescence quenching of the DnaB helicase, Fobs, upon the ADP concentration in the sample, at two different protein concentrations, is shown in Figure 14a. At saturation with the cofactor,

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the maximum quenching of the helicase fluorescence reaches the value of Fmax = 0.53 ± 0.03. At the applied concentrations of the enzyme, the separation of the titration curves on the total

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ligand concentration scale, allows us to apply the quantitative thermodynamic approach, as

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described above, and obtain i and [L]F over a large range of the observed change of the

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fluorescence intensity (eqs. 7, 10 and 11).

To address the direct fitting of the physico-chemical/spectroscopic titration curves, we obviously need the original titration curves but also the plot of the observed relative change of the spectroscopic signal, as a function of i. The observed fluorescence quenching, Fobs, as a function of i, of ADP on the DnaB hexamer is shown in Figure 14b. The values of i could be determined up to ~5.1 of the cofactor molecules per DnaB hexamer. The plot of in Figure 14b is clearly nonlinear. Binding of the first three nucleotide cofactors induces larger changes of Fobs than the association of the remaining, three cofactor molecules. The dashed line in Figure 14b represents the hypothetical case when a strict proportionality between i and Fobs, would 47

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exist. Clearly, any analysis of the discussed spectroscopic titration curves (Figure 14a), assuming

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the strict proportionality between Fobs and i, would result in a significant error in the

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determined binding parameters.

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The statistical thermodynamic description of the nucleotide cofactor binding requires

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introduction of non-thermodynamic/structural characteristics of the system. We know that the DnaB protein is a homo-hexamer, i.e., a ring-like structure, which can be described as a short

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circular lattice of six identical binding sites [22]. The simplest physical picture of such a system is the hexagon model with only two interaction parameters, the intrinsic binding constant, K, and

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function for the hexagon model is

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the nearest-neighbor cooperativity parameter, for the nucleotide binding [22]. The partition



2

3



3

4

4 5

6 6

ZH =1+6x +3(3 +2)x +2(1+6 + 3 )x + 3(3 +2 )x +6 x + x

(36)

The total average degree of binding, i, is described by

6x +6(3 +2)x2 +6(1 +6 +3 )x 3 +12(3  +2 3 )x 4 + 30 4 x 5 +6 6 x6 i  ZH

48

(37)

ACCEPTED MANUSCRIPT

where x = K[L]F, stands for the product of the intrinsic binding constant, K, and the free cofactor

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concentration, [L]F. The thermodynamic binding isotherm, i.e., the dependence of i upon the logarithm of [ADP]F, is shown in Figure 15 [22]. The solid line is the nonlinear least squares fit

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to the hexagon model (eq. 36) of the determined isotherm, with the intrinsic binding constant K =

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(2.0 ± 1.5) x 105 M-1 and  = 0.35 ± 0.05.

As stated above, with the thermodynamic isotherm available, determination of K and 

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does not require any considerations of the spectroscopic parameters of different DnaB - ADP complexes. However, this is not the possible for the original fluorescence titration curves. Here,

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one must consider all possible different spectroscopic properties of various DnaB - cofactor

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complexes with a given number of nucleotide molecules bound. For instance, a hexamer with

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three cofactor molecules bound has sixty possible configurations (eq. 36) and they may differ by the values of the individual molecular spectroscopic parameters. Even an intuitively minimum, analytical relationship between Fobs and the molar spectroscopic parameters and the interaction parameters, based on only considering differences in the presence of cooperative interactions, is

Fobs = (6(F11x+6(3F21+2F22)x2+6(F31+6F32+32F33)x3+12(32F41+23F42)x4+ 304F51x5+66F61x6)/ZH

(38)

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Expression 38 contains ten spectroscopic parameters, Fij, corresponding to each complex with

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“i” cofactor molecules bound in a given “j” state differing in the density of the cooperative interactions. To obtain interaction parameters, K and , and all ten optical constants, Fij, from a

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single, or multiple spectroscopic titration curves is a hopeless task.

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The empirical function approach circumvents this problem by using the numerical representation of the observed change in the spectroscopic signal from the macromolecule, Fobs,

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as a function of i (Figure 14b), via an empirical function [22,23]. The simplest function is a polynomial, which relates, Fobs, to the experimentally determined, total average degree of

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binding, i, as

n

Fobs   a j (i ) j

(39)

j0

 where aj are the fitting constants.

The procedure of applying the empirical function is the following. First, one calculates i for a given free ligand concentration and initial estimates of the binding parameters of the statistical thermodynamic model. Second, the calculated value of i is introduced into eq. 39 and Fobs is obtained. The analysis is then performed for the entire titration curve. For the ADP - DnaB protein system, the plot of Fobs(i) in Figure 14a is described by a third-degree polynomial with the coefficients a1 = 0.286 and a2 = , and a3 = 3.194x10-3, i.e., 50

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(40)

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Fobs  0.141(i )  0.01302(i )2  6.9x104 (i )3

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 lines in Figure 14a are the nonlinear least squares fits of the fluorescence titration The solid

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curves for ADP binding to the DnaB helicase, using eqs. 36, 37, and eq. 40, which provide the

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intrinsic binding constant K = (2.0 ± 1.5) x 105 M-1 and  = 0.35 ± 0.05, i.e., within experimental accuracy, identical to those obtained using the thermodynamic binding isotherm in Figure 15

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D

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[22,23].

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FIGURE LEGENDS

Figure 1. a. Theoretical fluorescence titrations of a macromolecule with a ligand,

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obtained at two different macromolecule concentrations, [M]1 = 1 x 10-6 M (solid line);

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and [M]2 = 5 x 10-6 M (dashed line), respectively, for the binding of two ligand molecules

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to two independent and different binding sites on the macromolecule (eq. 42, preceding paper). The binding system is characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, FmaxB1

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= 2.5, and FmaxB2 = 0.5, FmaxC = 3 (eq. 50a, preceding paper). The vertical arrows

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indicates the total ligand concentrations, ([L]T1)j and ([L]T2)j, at the selected value of the

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observed relative fluorescence change, (Fobs)j, marked by the horizontal dashed arrow,

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at which the total average degree of binding, i, and the free ligand concentration, [L]F, are the same for both titration curves (details in text). b. Theoretical dependence of the relative fluorescence change, Fobs, for the binding of two ligand molecules to two independent and different binding sites on the macromolecule (eq. 42, preceding paper) upon the total average degree of binding, i. The binding system is characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, FmaxB1 = 2.5, FmaxB2 = 0.5, and FmaxC = 3 (eq. 50a, preceding paper). The dashed line is the theoretical dependence of Fobs upon i for the same binding system, assuming the linear character of the function Fobs (i). The dot lines indicate the slopes of the high- and low-affinity binding phases. c. The large solid line is the same plot as in panel b but ending at the value of i ≈ 1.4. The dashed line is 52

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the extrapolation of the low-affinity phase to the value of Fmax = 3, which provides the

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maximum stoichiometry of the binding process ~1.8 (details in text).

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Figure 2. a. The dependence of the relative fluorescence intensity, Fobs, as a function of the total ligand concentration, [L]T, for the macromolecule with two independent and

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different binding sites generated using K1 = 5x106 M-1, K2 = 5x105 M-1, and FmaxC = 3,

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and different values of the relative fluorescence changes characterizing the association with individual sites (eq. 50a, preceding paper); (______) FmaxB1 = 1.5 and FmaxB2 = 1.5;

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(_ _ _) FmaxB1 = 2.0 and FmaxB2 = 1.0; (- - -) FmaxB1 = 3.0 and FmaxB2 = 0.0; (_ · _ · _ )

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FmaxB1 = 4.0 and FmaxB2 = -1.0; (_ · · _ · · _) FmaxB1 = 5.0 and FmaxB2 = -2.0 (details in

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text). b. Theoretical fluorescence titrations of a macromolecule with a ligand, obtained at two different macromolecule concentrations, [M]1 = 1 x 10-6 M (solid line); and [M]2 = 5 x 10-6 M (dashed line), respectively, for the binding of two ligand molecules to two independent and different binding sites on the macromolecule (eq. 50a, preceding paper). The binding system is characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, FmaxB1 = 5.0, FmaxB2 = -2.0, and FmaxC = 3. There are two phases of the plots where the relative fluorescence change, Fobs, is a monotonously increasing or decreasing function of [L]T. The vertical arrows in the high-affinity phase, where Fobs is the monotonously increasing function of [L]T, indicates the total ligand concentrations, ([L]T1)j and ([L]T2)j, at the selected value of the observed relative fluorescence change, (Fobs)j, marked by the 53

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horizontal dashed arrow, at which the total average degree of binding, i, and the free

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ligand concentration, [L]F, are the same for both titration curves. Analogously, the

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vertical arrows in the low-affinity phase, where Fobs is the monotonously decreasing

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function of [L]T, indicates the total ligand concentrations, ([L]Tx)i and ([L]Ty)i, at the selected value of the observed relative fluorescence change, (Fobs)i, marked by the

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horizontal dashed arrow, at which the total average degree of binding, i, and the free

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ligand concentration, [L]F, are the same for both titration curves (details in text).

D

Figure 3. Theoretical dependence of the fluorescence anisotropy for the binding process

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where the ligand binds to two independent and different binding sites on the

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macromolecule characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.2, rB2 = 0.1, and rC = 0.3 (eq. 12b), and obtained for two macromolecule concentrations, [M]T1 = 1x10-6 M (solid line) and [M]T2 = 5x10-6 M (dashed line), respectively. The vertical arrows indicate the total ligand concentrations, ([L]T1)j and ([L]T2)j, at the selected “j” value of the observed fluorescence change, (robs)j, marked by the horizontal dashed arrow, at which the total average degree of binding, i, and the free ligand concentration, [L]F, are the same for both titration curves, allowing the experimenter to construct the binding isotherm at any value of “j”, exemplified by the horizontal dashed lines in the panel (details in text).

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Figure 4. a. The solid line is the dependence of the fluorescence anisotropy, robs, as a

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function of the total ligand concentration, [L]T, for the macromolecule with two independent and different binding sites generated using K1 = 5x106 M-1, K2 = 5x105 M-1,

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rA = 0.1, rB1 = 0.2, rB2 = 0.15, and rC = 0.2 (eq. 12b). The function, robs([L]T]), is

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monotonously increasing in the entire range of the total ligand concentration. The dashed

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line is the dependence of the fluorescence anisotropy, robs, as a function of the total ligand concentration, [L]T, for the same binding system, generated using rA = 0.1, rB1 = 0.3, rB2 =

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0.15, and rC = 0.2 (eq. 12b) The function, robs([L]T]), is not a monotonous function of [L]T

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over the entire titration curve, which contain two binding phases, corresponding to the

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association with the high- and low-affinity binding site, respectively (details in text). b.

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Theoretical fluorescence anisotropy titration of a macromolecule with a ligand, obtained at two different macromolecule concentrations, [M]1 = 3 x 10-7 M (solid line); and [M]2 = 2x10-6 M (dashed line), respectively, for the binding of two ligand molecules to two independent and different binding sites on the macromolecule. The binding system is characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.3, rB2 = 0.15, and rC = 0.2. There are two phases of the plots where the fluorescence anisotropy, robs, is a monotonously increasing or decreasing function of [L]T. The vertical arrows in the highaffinity phase, where robs is the monotonously increasing function of [L]T, indicates the total ligand concentrations, ([L]T1)j and ([L]T2)j, at the selected value of the observed relative fluorescence change, (robs)j, marked by the horizontal dashed arrow, at which the 55

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total average degree of binding, i, and the free ligand concentration, [L]F, are the same

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for both titration curves. Analogously, the vertical arrows in the low-affinity phase,

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where robs is the monotonously decreasing function of [L]T, indicates the total ligand concentrations, ([L]Tx)i and ([L]Ty)i, at the selected value of the observed relative

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fluorescence change, (robs)i, marked by the horizontal dashed arrow, at which the total

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average degree of binding, i, and the free ligand concentration, [L]F, are the same for

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both titration curves (details in text).

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Figure 5. a. The solid line is the dependence of the fluorescence anisotropy, robs, as a

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function of the total ligand concentration, [L]T, for the macromolecule with two

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independent and different binding sites, generated using, with K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.2, rB2 = 0.1, rC = 0.3, FA =1, FB1 =1, FB2 = 1, and FC = 1, i.e., binding process is not accompanied by any fluorescence intensity change (eqs. 12b and 19). The dashed line is the dependence of the fluorescence anisotropy for the system characterized by the same binding constants and fluorescence anisotropies of all complexes of the reaction, but with the fluorescence intensities, FA =1, FB1 =1.5, FB2 = 1, and FC = 1.3. Thus, in this case, the binding reaction is accompanied by the fluorescence changes of formed complexes (details in text). b. Theoretical fluorescence anisotropy titration of a macromolecule with a ligand, obtained at two different macromolecule concentrations, [M]1 = 3 x 10-7 M (solid line); and [M]2 = 2x10-6 M (dashed line), 56

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respectively, for the binding of two ligand molecules to two independent and different

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binding sites on the macromolecule. The binding system is characterized by K1 = 5x106 M-1, K2 = 5x105 M-1, rA = 0.1, rB1 = 0.2, rB2 = 0.12, and rC = 0.3, FA =1.0, FB1 =1.5, FB2 =

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1, and FC = 1.3. Thus the binding process is accompanied by the fluorescence changes in

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the formation of different complexes. The function, robs, is monotonously increasing in

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the entire range of [L]T. The vertical solid arrows indicate the total ligand concentrations, ([L]T1)j and ([L]T2)j, at the selected value of the observed relative fluorescence change,

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(robs)j, marked by the horizontal dashed arrow, at which the total average degree of

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binding, i, and the free ligand concentration, [L]F, are the same for both titration

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curves (details in text).

Figure 6. Theoretical dependence of the observed heat of reaction, Qobs, as a function of the logarithm of the total ligand concentration, [L]T, obtained at two different macromolecule concentrations, [M]1 = 1 x 10-6 M (solid line); and [M]2 = 5 x 10-6 M (dashed line), respectively, for the binding of two ligand molecules to two independent and different binding sites on the macromolecule (eq. 42, preceding paper). The binding system is characterized by binding constants, K1 = 5x106 M-1 and K2 = 5x105 M-1, and molar enthalpies, HoB1 = 20.920 kJ/mol, HoB2 = 4.184 kJ/mol, and HoC = 25.104 kJ/mol (eq. 61b, preceding paper). The plots have been normalized to the corresponding maximum values of the observed cumulative heat of reaction. The vertical arrows 57

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indicate the total ligand concentrations, ([L]T1)j and ([L]T2)j, at the selected “j” value of

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the observed fluorescence change, (Qobs)j, marked by the horizontal dashed arrow, at

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which the total average degree of binding, i, and the free ligand concentration, [L]F, are the same for both titration curves, allowing the experimenter to construct the binding

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isotherm at any value of “j”, exemplified by the horizontal solid lines in the panel (details

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in text)

Figure 7. a. Fluorescence titrations (ex = 495 nm, em = 520 nm) of the gapped DNA

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substrate with five nucleotides in the ssDNA gap (insert) with the ASFV pol X at two

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different concentrations of the nucleic acid, 5 x 10-9 M () and 1.2 x 10-8 M (),

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respectively [45]. The solid lines are the nonlinear least squares fits of the experimental titration curves to the model of two distinct cooperative binding sites (eq. 28), using a single set of binding and spectroscopic parameters, KG = 6.5 x 108 M1, KD = 6.5 x 108 M1,  = 1, FG = 0.29, and FD = 0.29 (details in text). b. The dependence of the observed relative fluorescence quenching, Fobs, upon the total average degree of binding, i, of the ASFV pol X on the DNA substrate with five nucleotide residues in the ssDNA gap. The solid line follows the experimental points and does not have a theoretical basis. The dashed line is an extrapolation of the plot to the maximum observed fluorescence quenching, Fmax, marked by the solid horizontal line. Reprinted with permission from ref. 45 (details in text) [45]. c. The dependence of the total average 58

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degree of the binding, i, upon the logarithm of the free ASFV pol X, i.e., the

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thermodynamic binding isotherm of the ASFV- gapped DNA system. Reprinted with

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permission from ref. 45 (details in text).

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Figure 8. a. The dependence of the average sedimentation coefficient of the DnaC -

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DnaB complex, sav, corrected to standard values (s20,w), at two different fluorescein-

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labeled DnaB helicase concentrations, upon the total DnaC protein concentration, in buffer, 10 mM Tris/HCl pH 8.1, 100 mM NaCl, 5 mM MgCl2, 2 mM AMP-PNP, 10 °C. 6

M (

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The concentrations of the DnaB helicase are 4.2 x 107 M (

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respectively. The solid lines are computer fits of the experimental binding isotherms to

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the hexagon model using single set of binding parameters, the intrinsic binding constant, KM = 1 x 105 M1 and  = 30 (eqs. 25, 36-37). b. The average sedimentation coefficient of the DnaC - DnaB complex as a function of the total average degree of binding, i, of the DnaC protein on the fluorescein-labeled DnaB hexamer. The values of i have been determined using the quantitative approach described in the text (eqs. 10 - 11). The solid line follows the experimental points and has no theoretical basis. Reprinted with permission from ref. 44.

59

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Figure 9.

Theoretical fluorescence titrations of a fluorescing ligand with a

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macromolecule at different total concentrations of the ligand, [L]T. From the left, [L]T1, to

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the right, [L]T7, the total concentrations of the ligand are: 1x10-7 M; 2x10-7 M; 3x10-7

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M; 4x10-7 M; 6x10-7; 9x10-7 M; 1.5x10-6 M. The plots are for the binding model of two ligand molecules to two independent and different binding sites on the

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macromolecule (eq. 42, preceding paper). The binding system is characterized by K1 =

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5x106 M-1, K2 = 1x106 M-1, FB1=2, FB2 = 1, and FC = 3 (eq. 50a, preceding paper).

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Figure 10. Theoretical dependence of the binding density function, Fobs([L]T/[M]T),

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upon the logarithm of the total macromolecular concentration, [M]T, at different total

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concentrations of the ligand, [L]T. From the bottom to the top, the values of [L]T are: 1x10-7 M; 2x10-7 M; 3x10-7 M; 4x10-7 M; 6x10-7; 9x10-7 M; 1.5x10-6 M. The plots are for the binding model of two ligand molecules to two independent and different binding sites on the macromolecule (eq. 42). The binding system is characterized by K 1 = 5x106 M-1, K2 = 1x106 M-1, FB1 = 2, FB2 = 1, FC = 3 (eq. 50a, preceding paper). Horizontal dashed lines connect the points at the same value of the binding density function, Fobs([L]T/[M]T), at different [L]T, at which [L]Free and the total average degree of binding, i, of the ligand on the macromolecule are the same (details in text).

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Figure 11. Fluorescence titrations of ADP at different concentrations of the nucleotide

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with the DnaB protein (ex = 325 nm, em = 410 nm), in the presence of 50 mM

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Acrylamide [53]. The concentrations of the nucleotide are: () 3 x 10-6 M; () 5 x 10-6

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M; (◊) 1 x 10-5 M; () 2 x 10-5 M; () 3 x 10-5; ( ) 4 x 10-5 M [53]. Solid lines are

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nonlinear least squares fits of the experimental titration curves, according to the hexagon model, using a single set of binding parameters, K = 4.3 x 105 M-1, = 0.5, and Fmax

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Dependence of the binding density function, Fobs(LT/MT), on the

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Figure 12.

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= 1.23 (details in text).

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logarithm of the total DnaB protein concentration at different concentrations of ADP: () 3 x 10-6 M; () 5 x 10-6 M; ( ) 1 x 10-5 M; () 2 x 10-5 M; ( ) 3 x 10-5; () 4 x 10-5 M [53]. Solid lines separate different data sets and do not have theoretical basis. Horizontal dashed lines connect points at the same value of the binding density function, at different ADP concentrations, at which [ADP]Free and the total average degree of binding, i, of the nucleotide on the DnaB hexamer are the same (details in text).

61

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Figure 13. The dependence of the total average degree of binding of ADP, i, on the

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DnaB hexamer, upon the logarithm of the free nucleotide concentration, log[ADP]Free,

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i.e., the thermodynamic binding isotherm of the system, obtained using the LBDF

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method [53]. The solid line is the theoretical binding isotherm, according to the hexagon model, using intrinsic binding constant K = 4.3 x 105 M-1 and cooperativity parameter 

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= 0.35.

Figure 14. a. Fluorescence titration of the DnaB hexamer (ex = 300 nm, em = 345

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nm) with ADP, monitored by the quenching of the intrinsic protein fluorescence at two different concentrations of the DnaB hexamer: () 3.61x10-7 M; () 9.51x10-6 M [22].

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Solid lines are the nonlinear least squares fits of the experimental titration curves to the hexagon model, using the empirical function method (details in text). b. The dependence of the relative fluorescence quenching of the DnaB protein fluorescence, Fobs, upon the average number of ADP molecules bound per DnaB hexamer, i. The values of i have been determined using two titrations shown in panel a. The dashed line represents the theoretical situation when the strict proportionality between i and the quenching of the DnaB protein fluorescence existed. Solid line is generated using the empirical polynomial function as defined by eq. 40.

62

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Figure 15. The dependence of the total average degree of the binding, i, upon the

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logarithm of the free ADP, i.e., the thermodynamic binding isotherm of the ADP -

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DnaB hexamer system (details in text) [22]. The solid line is the theoretical binding

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isotherm, according to the hexagon model, using intrinsic binding constant K = 2 x 105

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M-1 and cooperativity parameter  = 0.33.

63

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T.L. Hill, Cooperativity Theory in Biochemistry. Steady State and Equilibrium

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Systems (Springer-Verlag, New York, 1985) chapter 4.

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1.

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Binding Interactions, J. Mol. Biol. 145 (1981) 75-104.

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of Bacteriophage T4-coded Gene 32 Protein with Nucleic Acids I. Characterization of the

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Research Highlights

In general, for the binding systems with maximum stoichiometry higher than 1, there is

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no linear relationship between the observed physico-chemical signal, originating from the

Signal and mass conservation relationships applied at different macromolecular

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macromolecule or the ligand and the total average degree of binding of the ligand.

concentrations allows the experimenter to construct the thermodynamic binding isotherm

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from physico-chemical titration curves.

Ligand Binding Density Function (LBDF) Method allows the experimenter to construct

The physico-chemical titration curves for the multiple ligand binding systems can be

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the ligand.

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the thermodynamic binding isotherm using the physico-chemical signal originating from

directly analyze using the Empirical Function (EF) Method, which relates the observed signal to the total average degree of binding.

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