Viewpoint pubs.acs.org/JPCL
Relation between Natural and Practical Ligand Binding Constants: Strong Implications for Extraction of Molecular-Level Information from the Temperature Dependence of Ligand Binding Equilibria igand binding reactions of the form L + P → LP are ubiquitous in biological systems and therefore commonly studied in biochemical and biophysical laboratories. The species P might be a biopolymer like DNA or an actin filament, while the partner L for DNA might be a transcription or repair protein1,2 or a small drug molecule.3 For actin, the binding of myosin and tropomyosin is essential for muscle function.4 The ligand L could be an amyloid peptide or the protein antithrombin, and P could be the linear polysaccharide heparin.5,6 Antigen−antibody reactions or coenzyme−enzyme reactions fit into this framework, as does ATP−protein binding.7 Still, another possibility is that L is identical to P, and then, the reaction could represent a protein dimerization 2P → P2.8 In fact, the analysis given in this Viewpoint is applicable to any chemical reaction of the form A + B → AB or indeed to any reaction in which the number of reactants is not equal to the number of products. In biochemical literature, it is conventional to define a practical binding constant Kobs for L + P → LP in terms of the equilibrium molarities [L], [P], and [LP]
L
Kobs ≡
[LP] [L][P]
μ = μ′ + RT ln N
Aside from a pv contribution (p = pressure, v = molar volume), which typically gives rise to negligible effects for a condensed phase like a liquid (our primary focus), the term μ′ consists entirely of energy contributions from the kinetic energies of the mass centers of the molecules, rotational kinetic energy, vibrational kinetic and potential energies, the intramolecular potential energies of interactions among the various atoms comprising each molecule, and the intermolecular potential energy of interactions among the molecules (we omit mention of electronic and nuclear energies with an eye to applications not involving covalent bonding or changes in states of the nuclei), along with the entropies associated with the randomizing effect of the temperature dependence of the average energies. The RT ln N term, in contrast, is purely entropic. It accounts for the thermodynamic indistinguishability of configurations of the liquid differing only in the interchange of some subset of the N identical molecules; such configurations can give rise to only a single thermodynamic (i.e., macroscopically observable) state. The physical significance of this term becomes transparent if we take negative temperature derivatives of both sides of eq 3
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Thus defined, Kobs is not unit-free; it has units of inverse molarity M−1, or L mol−1. For the purposes of thermodynamic analysis, a quantity ΔG0 is conventionally introduced through the equation
ΔG 0 = −RT ln Kobs
s = s′ − R ln N
(4)
This equation states that the thermodynamic entropy per mole s of the liquid is overestimated by the term s′ associated with the temperature dependence of the various energy contributions. It must be reduced by the amount R ln N. On the other hand, μ′ = h′ − Ts′, an identity that highlights s′ as including the entropy per mole associated with the temperature dependence of the translational kinetic energy that is responsible for the actual physical interchange of positions of identical molecules, that is, mixing. Correct as it is, the appearance of the term RT ln N may seem at odds with the fact that the chemical potential is a thermodynamically intensive property. This problem vanishes with realization that μ′ depends on the total volume of the liquid in such a way that the two components of μ in eq 3, when combined, yield an overall intensive chemical potential. However, the volume dependence must be retained separately as part of μ′ if, as desired, μ′ is to have the significance of physical energy/entropy. The volume V of the liquid is an essential aspect of the entropy associated with the center-ofmass (translational) kinetic energy of the molecules because each molecule of the liquid can sample all locations within the volume containing it. If we now turn to the chemical potential μi representing Ni molecules of a solute species i in liquid solution, it also may be written as the sum of two contributions
(2)
This quantity ΔG0 is said to be the free-energy change when one mole of L combines with one mole of P to yield one mole of complex LP, with all three species in their “standard states”. However, ΔG0 cannot be a physically meaningful free-energy difference because if it were, it would have energy units, whereas −RT ln Kobs has units of energy multiplied by the logarithm of a quantity that is not unit-free. Sometimes, this difficulty is said to be overcome by the understanding that a molarity like [L] is actually the ratio [L]/(1 M), thus rendering Kobs unit-free. However, a sleight-of-hand can go nowhere if the goal is to interpret measured values of Kobs in terms of actual physical driving forces. The purpose of this Viewpoint is to establish an easily accessible, correct, physically meaningful basis for the extraction of molecular-level information from the equilibrium binding constant Kobs and its temperature dependence. To this end, we have to start at the beginning, which can be found in any introductory treatment of the molecular basis of thermodynamics (for example, Dill and Bromberg9 or BenNaim10) but with a different emphasis that only implicitly contains the main point that we make here. The chemical potential μ (Gibbs free energy per mole) of a pure liquid (or real or ideal gas) containing N identical molecules may be written as the sum of two components © 2016 American Chemical Society
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Published: November 17, 2016 4611
DOI: 10.1021/acs.jpclett.6b02452 J. Phys. Chem. Lett. 2016, 7, 4611−4613
Viewpoint
The Journal of Physical Chemistry Letters μi = μi′ + RT ln Ni
where NA is Avogadro’s number, and Vl is the volume in liters of the solution containing the reactants and products. The units of Vl cannot be changed at will; because of the definition of molarity, the numerical value of Vl is fixed at the volume in liters of the space in which the reaction takes place. Then
(5)
Here we emphasize that in this context μ′i is not an ill-defined quantity pertaining to some fictitious hypothetical “standard state”. Instead, μi′ = hi′ − Tsi′ and has a sharp physical meaning as a free-energy combination of the energy of a mole of solute molecules i (their kinetic energies of translation, rotation, and vibration, their potential energy of vibration, their internal interaction energy, and their energy of interaction with their environments of solvent and solute molecules, including small ions), along with the entropies associated with the temperature dependence of these energies (this would include the entropic flexibility of i and the probabilities of multiple conformational states). Next, we consider the ligand binding reaction L + P → LP. The fundamental equation expressing the equilibrium of this reaction is
Δμ = 0
NLP NLNP
(8)
(9) 9
at equilibrium. The three parallel lines indicate that K is defined by the equilibrium combination of molecule numbers Ni on the right-hand side. This K is not the conventional one. In particular, K is unit-free. From the previous two equations, we have
Δμ′ = −RT ln K
(10)
Δs′ = Δsobs − R ln NAVl
relating the natural binding constant K to the energy and associated entropy change Δμ′ = Δh′ − TΔs′ when a mole of L combines with a mole of P to give a mole of LP. Notice that both sides of this equation have proper energy units; the argument of the logarithm is unit-free. The only caveat is that the entropy changes inherent in Δμ′ do not include the entropy due to the indistinguishabilities of L molecules, P molecules, and LP molecules. Included, however, is the entropy change associated with the disorder, such as thermal mixing, created by the temperature dependence of the various energy contributions. The natural binding constant K as defined is not directly measurable; therefore, the next step is to convert it to a practical, observable, binding constant Kobs. To do this, we convert the number of molecules Ni of each solute i to its corresponding molarity [i], which is the number of moles per liter
Ni = [i]NAVl
(14)
When the two logarithmic terms are combined, it is seen that the right-hand side has the proper energy units of RT; the argument of the single combined logarithmic term is unit-free. The left-hand side of this equation, Δμ′, is the physical freeenergy change when one mole of L and one mole of P combine to form one mole of LP. As Δh′ − TΔs′, it contains all of the molecular-level energies and entropies of concern to the biochemist/biophysicist. It contains nothing else. The first term on the right-hand side does not suffice to yield the physically meaningful free-energy change. It must be supplemented by the second term, which is very large. The interpretation of the free energy of reaction in terms of driving forces for binding would proceed through the enthalpy and entropy components of Δμ′. From the usual temperature derivatives of both sides of eq 14, we find that Δhobs = Δh′ to within a term involving the thermal expansion coefficient of water (assumed as the solvent). We have evaluated this term as a typically negligible 0.19 kJ mol−1. Thus, the enthalpy of reaction Δhobs calculated from the temperature dependence of Kobs is in fact equal to the actual physical enthalpy of reaction for practical purposes. A corollary is that heat capacity is also correctly calculated from Kobs. For the entropy, however, the situation is different. We find
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Note well that both terms of the left-hand side of this equilibrium equation have pure energy units. The argument of the logarithm in the second term is unit-free, not because of sleight-of-hand but because the statistical thermodynamic analysis works at the outset with the physical reality of molecules and their interactions. It is now natural to introduce a short-hand notation, namely, the natural binding constant K K≡
(13)
Δμ′ = −RT ln Kobs + RT ln NAVl
Written out, with substitution of eq 5 for the chemical potentials of the three solute species L, P, and LP, we get NLP =0 NLNP
K = Kobs/NAVl
The practical binding constant Kobs is a quantity directly available through measurements of the equilibrium concentrations. However, it is not unit-free. It has units of reciprocal molarity M−1. The units of NAVl are also M−1. The final step in our analysis is to substitute eq 13 into eq 10
where
Δμ′ + RT ln
(12)
and
(6)
Δμ = μLP − μL − μP
[LP] [L][P]
Kobs ≡
(15)
to within a negligible term involving the thermal expansion coefficient. This equation states that the actual physical binding entropy Δs′ (when one molecule of L combines with one molecule of P to produce one molecule of LP, multiplied by Avogadro’s number) is overestimated by the value of Δsobs calculated from the measured temperature dependence of Kobs. To get the physically meaningful entropy Δs′, the very large positive term R ln NAVl must be subtracted from the pseudoentropy Δsobs. Thus, if the pseudo-entropy Δsobs is found to be positive but not especially large, it is not unlikely that the actual physical entropy Δs′ will be strongly negative, reflecting the large amount of translational and rotational entropy lost when two molecules L and P freely moving throughout the entire volume of the solution while also freely rotating combine to form one molecule of LP. The estimation from Kobs of potentially favorable entropic effects such as solvent reorganization, counterion release, and possibly increased flexibility would require the overall negative entropy to be corrected at least for
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DOI: 10.1021/acs.jpclett.6b02452 J. Phys. Chem. Lett. 2016, 7, 4611−4613
Viewpoint
The Journal of Physical Chemistry Letters the decreased translational and rotational degrees of freedom using standard formulas from statistical mechanics.11 Finally, we apologize for and correct an error in a previous paper12 describing a consistent procedure for the separation of the electrostatic component of the reaction free energy (counterion release) from the overall free energy of reaction. The analysis there of ion effects is correct, but the unnumbered equation appearing above eq 8 of that paper should be replaced by eq 13 here, while the term log V appearing in eq 8 itself should be replaced by log NAVl.
Gerald S. Manning*
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Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, New Jersey 08854-8087, United States
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The author declares no competing financial interest.
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REFERENCES
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DOI: 10.1021/acs.jpclett.6b02452 J. Phys. Chem. Lett. 2016, 7, 4611−4613
