If Binding Is a Favorable Process, What Is the Free Energy Relationship Between E+s and Es?

Free Energy of Binding

The free energy of binding of a ligand to a protein ΔGbinding is split as:Gbinding=ΔGvacuo+ΔGsComplex−ΔGsLigand−ΔGsProteinwhere ΔGvacuo is the free energy of binding in vacuo, and ΔGsComplex, ΔGsLigand, ΔGsProtein are the solvation energies of the protein/ligand complex, the free ligand and the free protein, respectively.

From: Frontiers in Computational Chemistry , 2015

Chemical and Synthetic Biology Approaches To Understand Cellular Functions – Part B

Andrew J. Schaub , ... Shiou-Chuan Tsai , in Methods in Enzymology, 2019

1.8 Free energy calculations

The accurate estimation of the free energy of binding for protein-protein and protein-substrate interactions in FAS, PKS and NRPS is crucial for the engineering of these megasynthases. Since there are often many modules, each with several domains, obtaining experimental free energy values can be difficult, if not impossible, using experimental techniques. The Helmholtz free energy, A, can be expressed in terms of a partition function (Eq. 2):

(2) $\mathrm{A}=-{\beta }^{-1}lnQ\left(N,V,T\right)\text{,}$

where N, V, and T is defined as the canonical ensemble with a fixed number of atoms, volume and temperature. Free energy calculation approaches, including thermodynamic integration (TI) and free energy perturbation (FEP) methods, involve calculating ΔA. TI, FEP and histogram methods can be computationally very expensive as adequate sampling is needed to ensure convergence (Leach, 2001).

Thermodynamic integration (TI) calculates the free energy change by transforming one configuration state of interest into another one, via a parameter λ (Michel & Essex, 2010). The formula for TI is (Eq. 3):

(3) $\mathrm{\Delta }G={\int }_0^1{〈\frac{\partial V\left(\lambda \right)}{\partial \lambda }〉}_{\lambda }\mathit{d\lambda }$

In this equation, V(λ) is the λ–coupled Hamiltonian that corresponds to initial state for λ  =   0 and end state for λ  =   1. Due to the difficulty of obtaining an analytical solution, this integral is typically calculated numerically by performing a series of simulations corresponding to different λ points from 0 to 1. Simonson et al. used TI to calculate the proton pK _a shifts of a number of ionizable residues in thioredoxin, where the protonated and deprotonated residues corresponded to initial and end states (Simonson, Carlsson, & Case, 2004).

An alternative approach of calculating free energy difference is Molecular Mechanics Poisson–Boltzmann/Generalized Born Surface Area (MM-PB/GBSA) (Homeyer & Gohlke, 2012). MM-PB/GBSA belongs to the category of end-state methods, which do not require the simulation of intermediate states as in TI (Miller et al., 2012). This method involves the calculation of (1) the solvation free energy contribution by solving the linearized Poisson Boltzmann or Generalized Born equation, (2) the gas phase energy contribution by performing a molecular mechanics calculation, and (3) the entropy contribution by performing normal mode analysis. In our group, Ellis et al. performed MM-PBSA to calculate the binding free energy between the enzyme DpsC and its native substrate or the recently developed oxetane-based substrate mimics in order to assess the validity of the mimics (Section 2.2.3) (Ellis et al., 2018). MM-PBSA has also been used to assess the contributions of interface residues in FAS (Section 2.1.3).

Although both methods are widely used during free energy calculation, they have certain limitations. TI is theoretically rigorous but computationally expensive. MM-PB/GBSA, on the other hand, has a lower computational cost but limited accuracy, especially for highly charged systems (Wang, Hou, & Xu, 2006). Homeyer et al. developed a user friendly Free Energy Workflow (FEW) allowing users to select different methods depending on the tradeoff between desired accuracy and available computational resources (Homeyer & Gohlke, 2013). There are three methods available in FEW, including MM-PB/GBSA, TI and Linear Interaction Energy (LIE), another end state free energy method (Aqvist, Luzhkov, & Brandsdal, 2002).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0076687919300825

G Protein Coupled Receptors

Adriano Martinelli , Gabriella Ortore , in Methods in Enzymology, 2013

3.3 MM/PBSA

The MM/PBSA approach is a method to evaluate the free energies of binding. This approach averages the contributions of gas-phase energy, solvation free energy, and solute entropy from snapshots (calculated through MD simulations) of the complex molecule, as well as the unbound components.

A snapshot of every 50   ps is extracted from the last 4400   ps of MD for a total of 88 snapshots (at time intervals of 50 ps) for each species (complex, receptor, and ligand). The total MM-PBSA free energy of binding is then computed as:

$G=G_{\text{polar}}+G_{\text{nonpolar}}+E_{\text{MM}}-\text{TS}$

The E _MM term, which includes electrostatic, van der Waals, and internal energies, is calculated using AMBER 10. The polar energy term (G _polar) is calculated from the PBSA module of the AMBER 10 program through the Poisson–Boltzman method using dielectric constants of 1 and 80 to represent the gas and water phases, respectively. The nonpolar energy term (G _nonpolar) is calculated through the MOLSURF program. In order to compare the binding free energy of the various conformers with the receptor, only the first three terms are taken into account, while the entropic term TS is considered approximately constant.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780124078659000030

Insights into Enzyme Mechanisms and Functions from Experimental and Computational Methods

D. Kilburg , E. Gallicchio , in Advances in Protein Chemistry and Structural Biology, 2016

3.5 Protein–Peptide Binding Affinities

The ultimate goal of computational modeling of macromolecular binding is the estimation of binding free energies from physical principles. This is a very complex computational problem. After several decades of efforts, useful progress is being obtained in the area of protein–drug binding (Gallicchio et al., 2014; Wang et al., 2015). There are few examples of applications of binding free energy methods to protein–peptide bindings. They are reviewed here.

The binding of a series of four-residue phosphorylated peptides to an SH2 regulatory domain has been studied by the Roux group using a free energy PMF methodology (Gan & Roux, 2009; Woo & Roux, 2005) . The procedure involves several perturbation steps, each contributing to the overall free energy of binding. The general idea at the basis of this and later works by this group (Gumbart et al., 2012; Gumbart, Roux, & Chipot, 2013; Jo, Jiang, & Roux, 2015) is to reduce the amount of conformational sampling required by first restraining the peptide in solution, then carrying out the PMF calculation with the restrained peptide, and then finally releasing the restraints with the peptide bound to the receptor. The first step is the localization of the peptide in a fixed position in solution in front and at some distance from the receptor site to ensure an unobstructed binding path. In the second step, the conformation of the peptide is restrained so as to mimic its known or assumed conformation in the bound state. The PMF for binding has then been calculated along the peptide–receptor site distance with umbrella sampling using a series of harmonic restraining potentials distributed along the binding path.

Src homology 2 (SH2) domains are regulatory domains present in many enzymes. They recognize specific motifs containing a phosphorylated tyrosine. The free energy calculations by Gan and Roux (2009) recapitulated known trends in peptide recognition by five SH2 domains (Lck, Grb2, Cbl, p85αN, and Stat1) yielding binding free energies in good agreement with experimental measurements. The calculations elucidated the key electrostatic interactions responsible for the recognition of the phosphate group, and the more subtle interactions responsible for the modulation of binding by flanking residues. The free energy decomposition afforded by the method revealed the important role played by the conformational reorganization free energy as measured by the difference of the free energies for imposing and releasing the conformational restraints. The authors acknowledged that the success of the calculations were in part due to the rigidity of the receptor. Indeed, later applications of the PMF method to more flexible receptors encountered conformational sampling challenges (Gumbart et al., 2013).

Gumbart et al. (2012) compared the relative merits of PMF and alchemical routes to the calculation of the binding free energy between a proline-rich peptide and the SH3 domain of tyrosine kinase Abl. The double-decoupling alchemical approach requires the separate evaluations of the decoupling free energies of the peptide in solution and in the receptor site. Due to the size of ligand, these can be quite large and difficult to converge. On the other hand, the PMF approach models the direct transfer of the peptide from bulk water to the protein receptor site and is not as challenging. In either case the prior restraining of the peptide to limit the amount of conformational sampling necessary helped in reaching convergence of the calculations. As they should, the alchemical and PMF approaches yielded a similar estimate of the protein–ligand binding free energy.

With implicit solvation the distinction between the PMF and alchemical approaches becomes more blurred. The BEDAM method (Gallicchio et al., 2010), for example, is based on the effective binding energy function u(x) which can be interpreted as either the adiabatic free energy change for turning on ligand–receptor interactions or the adiabatic free energy change for transferring the ligand from the solvent phase to the receptor. An early application of the BEDAM single-decoupling approach has been to study protein–peptide interactions in an mRNA silencing complex involved in fragile X syndrome (Di Marino, D'Annessa, Tancredi, Bagni, & Gallicchio, 2015), one of the primary causes of autism in children (Bagni & Oostra, 2013). As discussed previously, control of mRNA translation is a crucial step in many cellular processes as dysregulation leads to cancer and other diseases (Lama et al., 2013). In addition to the eIF4E–eIF4G binding discussed above, eIF4E is known to share the same binding site with several other proteins and peptides called 4E-BPs that are implicated in many processes including the development of synaptic plasticity (Banko et al., 2005). CYFIP1 (cytoplasmic FMRP interacting protein 1) is a member of the 4E-BP family that, together with FMRP (fragile X mental retardation protein), binds to eIF4E acting as an inhibition complex (Napoli et al., 2008). Mutations in FMRP inhibit complex formation with CYFIP1 and eIF4E, leading to fragile X syndrome (Bagni & Oostra, 2013; Rubeis et al., 2014). Di Marino et al. (2015) set out to computationally model the key amino acid sequence of CYFIP1 that binds to eIF4E with the hope that the information gained from this study will lead to the engineering of a peptide inhibitor that can serve to reregulate eIF4E if FMRP is defective.

As no crystal structures of the eIF4E–CYFIP1 complex exist, the authors built homology models of the complex of eIF4E with a CYFIP1-derived peptide based on the crystal structures of 4E-BP- and eIF4G-derived peptides bound to eIF4E. Interestingly, molecular dynamics simulations with explicit solvation started from these homology models, resulting in unbinding and unfolding of the CYFIP1-derived peptide. It was therefore concluded that CYFIP1 must bind eIF4E in a unique binding mode that differed from the other 4E-BPs. The authors set out to predict the structure of eIF4E bound to a CYFIP1-derived peptide starting from a docking model of the eIF4E–CYFIP1 complex obtained using the HADDOCK program (Dominguez et al., 2003; Marino et al., 2015). In this model the CYFIP1 peptide was bound to a different site than that shared by 4E-BPs. To ensure exhaustive sampling during the conformational search, the authors used multidimensional replica exchange molecular dynamics with the BEDAM protocol (Gallicchio et al., 2010), normally used to predict binding free energies (Gallicchio et al., 2014). BEDAM simulations showed a distinct binding mode for CYFIP1 that was near the canonical binding site but altered by significant rotation. MD simulations on this new binding mode showed stability in the peptide secondary structure and stable binding to eIF4E in contrast to earlier MD where the complex lacked stability. The authors were also able to ascertain unique binding interactions with the canonical binding site that are not present in the other peptides. Overall the work of Di Marino et al. (2015) shows that high-level modeling using different techniques can help gain invaluable insights in protein–peptide complexation. The work also validated the BEDAM binding free energy method as a conformational search tool in difficult cases such as this where peptide folding is coupled to binding.

The convergence properties of single-decoupling protocols such as BEDAM, which are already well established for protein–drug binding (Gallicchio, 2012; Gallicchio et al., 2014; Lapelosa, Gallicchio, & Levy, 2012), are currently being studied in the context of protein–peptide binding. The benchmark system being evaluated (Fig. 1) consists of a series of cyclic peptide inhibitors of the interaction of HIV integrase with the lens epithelium-derived growth factor host protein, which has been shown to be critical for the integration of the viral genome into the host chromosome (Rhodes et al., 2011; Tsiang et al., 2009). As shown in Fig. 1 the BEDAM calculation for the wild-type sequence converges to a value in reasonable agreement with the experimental measurements (Rhodes et al., 2011); however, a double-mutant sequence fails to converge within a similar simulation time. The simulations were started from the crystal structure of the complex with the wild-type peptide (Rhodes et al., 2011), so it is likely that the difference in convergence times is related to the fact that the double-mutant peptide is undergoing slow conformational reorganization to assume a binding pose different from the one assumed by the wild-type peptide.

Fig. 1. (A) Representation of the complex catalytic core domain of HIV integrase (blue (dark gray in the print version), red (gray in the print version), and gray) with two cyclic peptides (green (light gray in the print version)). BEDAM convergence plots (computed binding free energy as a function of simulation time) for two cyclic peptides: SLKIDNLD (B, wild-type sequence) and the ALKIDNMD double mutant (C).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S1876162316300293

Anti-Tubercular Drug Designing Using Structural Descriptors

Manish C. Bagchi , Payel Ghosh , in Advances in Mathematical Chemistry and Applications, 2015

Molecular Docking & Combinatorial Library

Docking involves the placement of a ligand within a binding site and the prediction of the free energy of binding for such poses. The goal is to find the global energy minimum of the complex, and numerous programs have been developed to solve this non-trivial problem. One of the first docking programs was DOCK, developed by Kuntz et al. [53], which treated both the ligand and target as rigid bodies. Today, most algorithms deal with a flexible ligand and a rigid target, although significant effort has been devoted to also include protein flexibility, and the modeling of structural waters.

In molecular docking, the most important aspect is the calculation of binding energy so as to fit a ligand in a binding site. The binding softwares like Gold [54] and AutoDock [55] are frequently used to compute the binding affinities and scoring functions of ligands. The descriptor based QSAR models are often very useful in predicting biological activities of molecules and thus enriches the concept of virtual screening. The fast expanding protein bank information coupled with molecular descriptor based virtual screening methods supplement the whole drug design process by identifying lead molecules.

In the treatment of tuberculosis, fluoroquinolones play a major role as these are considered to be second line anti-TB agents and are very effective when the disease becomes drug resistant [56]. DNA gyrase protein is the main target of such fluoroquinolone derivatives [57, 58] and interaction pattern of fluoroquinolones and DNA gyrase is the key feature in molecular docking. Combinatorial library generation program is employed with fluoroquinolone template for obtaining a virtual library which consists of permissible substituents at particular positions. The virtual library is again screened by applying Lipinski's rule of five criteria and the remaining molecules are then tested with QSAR models for activity prediction. Molecules with high predicted activities are subjected to docking studies to scrutinize the dock score and interaction patterns. A selected number of molecules with high activity profiles, minimum dock scores and desired interaction patterns are recommended for further chemical synthesis and testing for lead identification.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781681080536500089

Pore-Forming Toxins

Rajat Desikan , ... K. Ganapathy Ayappa , in Methods in Enzymology, 2021

3.2.1 Protomer-membrane binding energy via umbrella sampling

We employed MARTINI simulations, with a polarizable water model and PME electrostatics, to assess the binding free energy of a single ClyA protomer with a DMPC membrane (Fig. 7A ) using umbrella sampling simulations (Desikan, Maiti, & Ayappa, 2017). The reaction coordinate, "ζ," was defined as the distance between the centers of mass of the protomer and the membrane along the membrane normal. The ClyA protomer was equilibrated in the DMPC membrane for 100 ns, and subsequently, short steered MD simulations were utilized to create 24 starting configurations for umbrella sampling along ζ, with Δζ = 0.2 nm. Each of these configurations were restrained around their respective initial values of ζ with weak harmonic potentials. From the variance of ζ in independent unrestrained simulations (ζ*), the spring constant for these weak harmonic restraints was calculated as $\frac{RT}{{\zeta }^{*}}=10$ kJ mol⁻¹ nm⁻² at 310 K (see Desikan, Maiti, & Ayappa, 2017 for more details). Movement of the membrane along its normal was restrained by applying a strong harmonic potential on the DMPC phosphate atoms (force constant of 100,000 kJ mol⁻¹ nm⁻²). Each umbrella sampling window was then simulated for 100 ns. From examining the composite histogram plot along ζ from all umbrella sampling trajectories, it was observed that umbrella sampling windows 11 and 12, corresponding to the protomer at the water–membrane interface, exhibited large fluctuations and poor sampling (overlap). Thus, only these windows were rerun with a higher force constant of 100 kJ mol⁻¹ nm⁻². Note that the N-terminus of the protomer, which is a part of ClyA's transmembrane domain, undergoes significant conformational changes in the fully solvated state (umbrella sampling simulations at higher ζ). The potential of mean force (PMF) was obtained by using the weighted histogram analysis method (WHAM) along ζ, and error bars on the PMF were computed by bootstrapping. From the PMF (Fig. 7A), the membrane-inserted protomer state is observed to be a thermodynamically favorable state, with a free energy difference of ~−13 kcal/mol compared to the fully solvated protomer.

Fig. 7. Martini model simulations for (A) potential of mean force (PMF) computations of a ClyA protomer with a phospholipid (DMPC) membrane (Desikan, Maiti, & Ayappa, 2017), (B) lipid evacuation in membrane-inserted ClyA arcs (Desikan, Patra, et al., 2017) (similar to Fig. 4B), and distortions in membrane-inserted pore complexes of (C) AHL and (D) ClyA pores (Desikan, Patra, et al., 2017). Only protein atoms shown for clarity in (C) and (D).

Panel (A): Reprinted with permission from Desikan, R., Maiti, P. K., & Ayappa, K. G. (2017). Assessing the structure and stability of transmembrane oligomeric intermediates of an α-helical toxin. Langmuir, 33 (42), 11496–11510. Copyright (2017) American Chemical Society; Panels (B)–(D): Reprinted with Permission from Desikan, R., Patra, S. M., Sarthak, K., Maiti, P. K., & Ayappa, K. G. (2017). Comparison of coarse-grained (martini) and atomistic molecular dynamics simulations of α and β toxin nanopores in lipid membranes. Journal of Chemical Sciences, 129 (7), 1017–1030. Copyright (2017) Springer Nature.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0076687921000434

Advances in structure-based drug design

Divya Jhinjharia , ... Shakti Sahi , in Chemoinformatics and Bioinformatics in the Pharmaceutical Sciences, 2021

3.2.5.1 Forcefield-based scoring functions

Forcefield is a mathematical function defining the conformations based on energy terms. These scoring schemes approximate the free binding energy of protein–ligand complexes using forcefields. In other words, forcefields are sums of terms that correspond to bonded and nonbonded interaction, namely bond, angle, torsion, van der Waals, and electrostatic interaction energies as functions of conformation.

The forcefield parameters from AMBER (Weiner and Kollman, 1981), CHARMM (Brooks et al., 1983), OPLS (Jorgensen et al., 1996), OPLS3 (Harder et al., 2016), and MMF forcefields are used. The solvent effect is considered using (1) distance-dependent dielectric constant or (2) explicit solvents such as free energy pertubation and thermodynamic integration (Wang et al., 2001) or (3) implicit solvents such as Poisson–Boltzmann/surface area models (Rocchia et al., 2002; Grant et al., 2001) and the generalized-born/surface area models (Zou et al., 1999; Liu et al., 2004). The limitations of these methods are in the calculation of entropic effects and free energy calculations. Examples include AutoDock, G-Score, GOLD, DockScore, GoldScore, and HADDOCK scoring functions.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128217481000099

IMMUNE FUNCTION AND ANTIBODY STRUCTURE

CAROLYN S. FELDKAMP , JOHN L. CAREY , in Immunoassay, 1996

3.1. Antigen-Antibody Interaction

Most antigen-antibody complexes have the characteristics of other well-studied protein-protein interactions. The free energy of binding is a function of the amount of surface of each protein (antigen and antibody) which is hidden within the complex from exposure to solvent( 13, 20). It has been estimated that at least 600 Ų of buried surface is associated with a stable complex. The antigen-antibody bond occurs through multiple noncovalent bonds—electrostatic, hydrogen, hydrophobic, and Van der Waals. Long-range forces such as electrostatic and hydrogen bonds are important in the rate of formation of antigen-antibody complexes at the points of contact. The short-range forces contribute significantly to bond strength by reducing the rate of complex dissociation. Cross-reacting antigens show a wide range of affinity depending on the balance of attractive and repulsive forces within the binding site (21). Typical dissociation constants for antigen–antibody reactions are >10⁻⁹ M.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780122147302500030

Applications of Isothermal Titration Calorimetry to Lectin–Carbohydrate Interactions

Tarun K. Dam , C. Fred Brewer , in Lectins, 2007

3.1.4 Range of microscopic affinity constants for multivalent carbohydrates binding to ConA and DGL

Based on Fig. 5, Eq. (5) was derived to describe the relationship between the observed macroscopic free energy of binding and the microscopic free energies of binding of the various epitopes of a multivalent carbohydrate binding to a lectin [41].

(5) $\Delta G(\text{obs})=\frac{\Delta G_1+\cdots +\Delta G_n}{n}$

Eq. (5) states that the observed macroscopic ΔG value (ΔG(obs)), determined by ITC, of a multivalent carbohydrate is the average of the microscopic ΔG values of the individual epitopes, n the number of epitopes of the multivalent ligand [41]. This equation correctly estimates the difference in microscopic ΔG values of the two epitopes of bivalent analog 19 binding to ConA [44]. In this case, Eq. (6) for binding of 19 is

(6) $\Delta G(\text{obs})=\frac{\Delta G_1+\Delta G_2}{2}$

Eq. (6) shows that ΔG(obs) obtained from an ITC experiment allows calculation of ΔG ₁, the first epitope of the divalent carbohydrate, assuming that ΔG ₂ for the second epitope is the same as that of a monovalent ligand analog. This latter assumption was shown to be true from a reverse ITC experiment that allows direct determination of ΔG ₁ and ΔG ₂ [44]. The difference between ΔG ₁ and ΔG ₂ calculated from Eq. (6) using ΔG(obs) from an ITC experiment agreed well with that determined from the reverse ITC [44]. Eq. (5) can also be used to estimate the spread in microscopic ΔG values for the tetraantennary analog 21 that binds to ConA and DGL. Eq. (7) describes the relationship between the macroscopic ΔG(obs) and four microscopic ΔG values for binding of 18 to DGL.

(7) $\Delta G(\text{obs})=\frac{\Delta G_1+\Delta G_2+\Delta G_3+\Delta G_4}{4}$

ΔG ₁ in Eq. (7) is associated with the binding of the first carbohydrate epitope of tetraantennary analog 21, ΔG ₂ with the second, ΔG ₃ with the third, and ΔG ₄ with the fourth. The ITC determined macroscopic ΔG(obs) for binding of 21 to DGL is −10.6 kcal/mol [40], while ΔG ₄ in Eq. (7) can be taken as the ΔG(obs) for binding monovalent trimannoside 1 to DGL which is −8.3 kcal/mol [40]. Since ΔG(obs) is the average of the four microscopic ΔG values, then

(8) $\Delta G_1-\Delta G(\text{obs})=\Delta G(\text{obs})-\Delta G_4$

assuming that ΔGobs – ΔG ₂ ~ ΔG ₃ – ΔGobs (i.e., there is a symmetrical distribution of microscopic ΔG values on either side of ΔG(obs)). The numerical value of ΔG ₁ calculated from Eq. (8) is −12.9 kcal/mol, which is 4.6 kcal/mol greater than ΔG ₄. This difference between ΔG ₁ and ΔG ₄ translates to a difference in microscopic K _a values, K _a1 and K _a4, of approximately 2,800 fold. In absolute terms, K _a1 is approximately 0.3 nM, while K _a4 is approximately 0.8 mM. Thus, the microscopic K _a1 of the first unbound epitope of tetraantennary analog 21 binding to DGL is 2,800-fold greater than K _a4 for binding of the fourth. For 21 binding to ConA, this difference between K _a1 and K _a4 is nearly 1,200-fold [40]. This indicates a decreasing gradient of microscopic binding constants of the four epitopes of 21 binding to ConA and DGL. These differences have been postulated to be due to kinetic effects on the off rates of the various fractionally bound complexes of the multivalent carbohydrates [40]. The microscopic off rate (k ₋₁) for K _a1 in Fig. 5 (K _a1 = k ₁/k ₋₁) would be expected to be slower than the microscopic off rate for K _a2, etc., due to binding and recapture of the first bound lectin molecule by the remaining unbound trimannoside residues of the tetravalent analog before complete dissociation of the complex.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444530776500053

Biochemistry of Glycoconjugate Glycans; Carbohydrate-Mediated Interactions

T.K. Dam , C.F. Brewer , in Comprehensive Glycoscience, 2007

3.21.3.2.1.1.6 Range of microscopic affinity constants for multivalent carbohydrates binding to ConA and DGL

Based on Figure 50, eqn [1] was derived to describe the relationship between the observed macroscopic free energy of binding and the microscopic free energies of binding of the various epitopes of a multivalent carbohydrate binding to a lectin. ¹⁶⁴

[1] $\text{Δ}G(\text{obs})=\{\text{Δ}{G}_1+\cdots +\text{Δ}{G}_n\}/n$

Equation [1] states that the observed macroscopic ΔG value (ΔG(obs)), determined by ITC, of a multivalent carbohydrate is the average of the microscopic ΔG values of the individual epitopes, where n is the number of epitopes of the multivalent ligand. ¹⁶⁴ This equation correctly estimates the difference in microscopic ΔG values of the two epitopes of bivalent analog 17 binding to ConA. ¹⁶³ In this case, [2] for the bivalent carbohydrate,

[2] $\text{Δ}G(\text{obs})=\{\text{Δ}{G}_1+\text{Δ}{G}_2\}/2$

shows that ΔG(obs) from a normal ITC experiment allows calculation of ΔG ₁, the first epitope of the divalent carbohydrate, assuming that ΔG ₂ for the second epitope is the same as that of a monovalent ligand. This latter assumption was shown to be true from a reverse ITC experiment that allows direct determination of ΔG ₁ and ΔG ₂. ¹⁶³ The difference between ΔG ₁ and ΔG ₂ calculated from eqn [2] using ΔG(obs) from a normal ITC experiment agreed well with that determined from the reverse ITC. ¹⁶³

Equation [1] can also be used to estimate the spread in microscopic ΔG values for the tetraantennary carbohydrates that bound to ConA and DGL. Equation [3] describes the relationship between the macroscopic ΔG(obs) and four microscopic ΔG values for binding of the tetraantennary analog 19 to DGL.

[3] $\text{Δ}G(\text{obs})=\{\text{Δ}{G}_1+\text{Δ}{G}_2+\text{Δ}{G}_3+\text{Δ}{G}_4\}/4$

ΔG ₁ in eqn [3] is associated with the binding of the first carbohydrate epitope of tetraantennary analog 19, ΔG ₂ with the second carbohydrate epitope, ΔG ₃ with the third carbohydrate epitope, and ΔG ₄ with the fourth epitope. The ITC-measured macroscopic ΔG(obs) for binding of the 19 to DGL is –10.6   kcal   mol⁻¹, ¹⁶¹ while ΔG ₄ in eqn [3] can be taken as the ΔG(obs) for binding monovalent trimannoside to DGL which is –8.3   kcal   mol⁻¹. ¹⁶¹ Since ΔG(obs) is the average of the four microscopic ΔG values, then

[4] $\text{Δ}{G}_1-\text{Δ}G(\text{obs})=\text{Δ}G(\text{obs})-\text{Δ}{G}_4$

assuming that ΔG(obs) – ΔG ₂ ∼ ΔG ₃ – ΔG(obs) (i.e., there is a symmetrical distribution of microscopic ΔG values on either side of ΔG(obs)). The numerical value of ΔG ₁ calculated from eqn [4] is –12.9   kcal   mol⁻¹, which is 4.6   kcal mol⁻¹ greater than ΔG ₄. This difference between ΔG ₁ and ΔG ₄ translates to a difference in microscopic K _a values, K _a1 and K _a4, of approximately 2800 times. In absolute terms, K _a1 is approximately 0.3   nM, while K _a4 is approximately 0.8   μM. Thus, the microscopic K _a1 of the first unbound epitope of tetrantennary analog 19 binding to DGL is 2800-fold greater than K _a4 for binding of the fourth epitope. For 19 binding to ConA, this difference between K _a1 and K _a4 is nearly 1200-fold. ¹⁶¹ This indicates a decreasing gradient of microscopic binding constants of the four epitopes of 19 binding to ConA and DGL. These differences have been postulated to be due to kinetic effects on the off-rates of the various fractionally bound complexes of the multivalent carbohydrates. ¹⁶¹ The microscopic off-rate (k _–1) for K _a1 in Figure 50 (K _a1  = k ₁/k _–1) would be expected to be slower than the microscopic off-rate for K _a2, etc., due to binding and recapture of the first bound lectin molecule by the remaining unbound trimannoside residues of the tetravalent analog before full dissociation of the complex.

The literature often presents 'valency-corrected' binding data in studies involving multivalency. Instead of using the concentration on the basis of whole multivalent molecule (molar concentration), these 'valency corrections' are made on the basis of per site or per branch (equivalent concentrations) of the multivalent molecules. Two groups ^155,164 have independently demonstrated the validity and necessity of using the concentration on the basis of whole multivalent molecule (molar concentration). Valid analysis of thermodynamic binding data can only be achieved by expressing concentrations on a molar basis, since the units in the thermodynamic binding equations are molar. ^155,164,165

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444519672000581

In Silico Drug Discovery Tools

Robin Taylor , in Comprehensive Medicinal Chemistry III, 2017

3.05.2.4 Enthalpy–Entropy Compensation

The phenomenon of EEC is often observed in ITC studies of protein–ligand binding. Within a series of ligands with similar free energies of binding (ΔG), the enthalpic (ΔH) and entropic (TΔS) contributions can vary significantly but in an opposing manner. As an example, a series of trypsin inhibitors with constant structure except for a variable side chain, R, were found to have ΔG values in a narrow range but greatly differing enthalpic and entropic terms. ⁵⁸ For instance, at 20°C the inhibitors with R   =   methyl, propyl, and n-hexyl had, respectively, ΔG  =   −   27.5, −   25.7, −   28.9; ΔH  =   −   16.7, −   9.7, −   6.6; TΔS  =   10.8, 16.0, 22.2   kcal   mol^{−   1}. The most common explanation for EEC focuses on ligand and protein flexibility: stronger binding (greater enthalpic contribution) is tighter binding, meaning less mobility, hence less entropy. In the case of the trypsin inhibitors, the entropic contribution becomes more favorable as the R group increases in length, suggesting that the more flexible alkyl chains retain greater mobility when bound, but in consequence form weaker interactions with the protein pocket. However, EEC can also be connected to differences in the structure of water networks around bound ligands. ^{59 , 60}

It should be noted that there is a question mark over the evidence for EEC. Results from ITC experiments can have large unreported errors. In a test, identical aliquots of protein–ligand preparations were sent to 14 laboratories; the resulting ΔH values had a root-mean-square error of 24%! ⁶¹ Further, the propagation of errors in ITC is such that errors in ΔG tend to be smaller than those in ΔH, which can give the appearance of EEC even when none is present. ⁶² In a careful study, Olsson et al. modeled the effects of these experimental constraints on 32 ITC data sets relating to diverse proteins. ⁶³ They concluded that the tendency for EEC is strong, but by no means universal. It can be weak, and it is occasionally observed that enthalpic and entropic binding contributions move in the same rather than opposing directions. Overall, theoretical studies have focused on explaining a stronger tendency for EEC than appears to exist. It remains to be seen why some changes to ligands result in compensation and others do not.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780124095472123406

If Binding Is a Favorable Process, What Is the Free Energy Relationship Between E+s and Es?

Source: https://www.sciencedirect.com/topics/chemistry/free-energy-of-binding

Share this post