Thermodynamic cycles Powerful ways to exploit the properties of state functions
Energy Units: How much is a lot of energy? Boltzmann distribution: ( ) exp − E / k b T At any temperature, population of states decreases exponentially with increasing energy. “ kT ” provides a natural energy unit, typically @ T=300 K 1 kT = 0.6 kcal/mol = 2.5 kJ/mol From Δ G=-RTlnK: If the equilibrium constant changes by a factor of 10, Δ G changes by ~1.4 kcal/mol [If you remember nothing else, remember this]
Critical concept: Enthalpy, entropy, and free energy are all state functions path 1 H 2 H 1 Δ S = S 2 − S 1 Δ H = H 2 − H 1 G 1 G 2 S 1 S 2 Δ G = G 2 − G 1 path 2 State 1 State 2 (e.g., ligand and (e.g., ligand bound protein free) to protein) The key thermodynamic parameters H, S, and G depend only on the beginning and end state, and don’t depend on how you get between the two
Analogy: Traveling from San Francisco to NYC State 2 State 1 Elevation Elevation Latitude Latitude Longitude Longitude Path-dependent quantities include miles traveled, amount of gas guzzled, cups of coffee consumed, cost, etc.
Why does table salt dissolve in water? Think about the following thermodynamic cycle: Break apart crystal lattice NaCl(s) Na + (g) + Cl - (g) breaking Δ H » 0 Δ S > 0 lattice order Solution process Δ H = ? Δ S = ?? Na + (aq) + Cl - (aq) Strong interactions between ions and water dipole compensate for breaking ionic lattice (but for AgCl, which is insoluble, the unfavorable terms outweigh the favorable ones)
Thermodynamic cycles for computing pKa Gas phase reference state: relationship of proton affinity to pKa Model compound reference state: How does pKa change in protein?
Thermodynamic cycle for cooperative ligand binding
Thermodynamic cycle for coupled ligand binding Δ G oxygen(apo) Apo Hb (T state) Apo Hb (R state) Δ G bind(T) Δ G bind(R) Δ G oxygen(holo) Holo Hb (R state) Holo Hb (T state) ΔΔ G bind = Δ G ligand(T) – Δ G ligand(R) = Δ G oxygen(apo) – Δ G oxygen(holo)
Keep in mind that protons are ligands too Δ G oxygen(His0) Apo Hb (T state) Apo Hb (R state) pKa (T) pKa (R) Δ G oxygen(His+) Holo Hb (T state) Holo Hb (R state)
Binding thermodynamics in water Matt Jacobson
Goal of this lecture • The primary challenge to understanding the thermodynamics of molecular biology, such as binding events, is that it happens in water • We need to understand – How water modulates electrostatic forces – How binding events change water entropy • Molecular mechanics force fields allow us to quantify inter-molecular forces • Molecular dynamics applies Newtonian mechanics to predict macromolecular dynamics and ensembles
The 2 key properties of water are: 1) it’s polar 2) it’s small
Basic Electrostatics • Interactions of point-charges described by Coulomb ’ s law: q 1 q 2 /r 12 • A dipole is a pair of opposite charges separated by a small amount • Charge-dipole interactions ∝ 1/r 2 • Dipole-dipole interactions ∝ 1/r 3 • In molecular mechanics methods, assign partial charges to each atom (e.g., based on quantum mechanics calculations, or electronegativity concepts) • But there are limitations to the partial charge concept …
Dipole-dipole interactions depend on both distance and orientation
Electrostatics in Solution: Water has a strong dipole moment Strong interactions between ions and water dipole compensate for breaking ionic lattice • The ability of water to reorient around the ions gas phase: creates a “ dielectric screening ” between the ions. q q E + − • The “ first shell ” of waters moves with the ions. ∝ r But even waters further away tend to orient towards (or away from) the ions. • The dielectric constant of water is ~80 at room solution phase: temp, but decreases with increasing T. q q E + − ∝ r ε
Can we capture the effects of Implicit/Continuum Water Model water on ε =80 - electrostatics + - + + - ε =1 + – without simulating - + - + hundreds or - - + + thousands of individual waters? Basic idea: Treat solvent as a dielectric continuum.
Simplest example: monoatomic ion in water ε =80 - - ε =1 - - + - - - Born equation: Δ Gsolv ∝ q 2 /R (where R is atomic radius)
Poisson Equation Basically Generalizes This For Any Molecule � � � ( ) ( ) ( ) r r 4 r ∇ ⋅ ε ∇ ϕ = − πρ � ( ) = r electrostatic potential ϕ � ( ) = r ε dielectric constant (small inside protein; 80 outside) � ( ) = r ρ charge density (partial charges inside; ions outside) This is one of the fundamental equations of classical electrostatics. In fact, Coulomb’s law can be derived as a special case where the dielectric is constant.
If there are high concentrations of salt, that creates another screening effect Debye-Huckel theory gives the density of ions as � � q ( ) r − ϕ i 0 ( ) r e kT ρ = ρ i i Ionic density in bulk solution This just gives a model for the enrichment of, e.g., negative ions in places where the potential is positive. So, for a 1:1 salt solution, we have � � � ( ) ⎟ � � � r ( ) r ( ) r ϕ − ϕ + ϕ ⎛ ⎞ 0 0 0 ( ) ( ) ( ) r r r e e 2 sinh kT kT ρ = ρ + ρ = ρ − ρ = ρ ⎜ ionic + − kT ⎝ ⎠ Typical physiological ionic strength is 200 mM, which leads to a “Debye length” of ~8 Ang
Visualizing Electrostatic Fields Red = negative Blue = positive Grasp (Honig and Nicholls)
Role of Solvated Electrostatics in Electrostatic forces Molecular Recognition speed the binding of the positively- charged substrate to acetylcholinesterase by a factor of more than 100. AChE and Fasciculin 2 bind with electrostatically-steered, diffusion- controlled kinetics. Honig group, Columbia U.
Water entropy: Probably the most important force driving binding in solution, but it’s easy to forget about it … and come to incorrect conclusions
Both Hydrophobic and Polar/Charged Solutes Reduce Water Entropy (in different ways) Ion transfer Alkane transfer (benzene → water) (benzene → water) H H H O O Δ H << 0 Δ S < 0 H Δ H ~ 0 Δ S < 0 H O O H H H CH 4 (benzene) → CH 4 (water) Δ H° = -11.7 kJ/mol Δ S° = -76 J/K · mol Δ G°(298) = +11 kJ/mol Key point: The strongly favorable enthalpy of ions in waters frequently compensates for the entropic loss, but not for hydrophobic solutes.
Salt Bridges [Hydrogen bonds are similar, but the magnitude of the forces is smaller.] Questions: • How much does a buried salt bridge (i.e., interior of protein) stabilize a protein? • How about a surface exposed one? • [Hint: What is the role of solvent? What is the salt bridge energy in the gas phase? What is the solvation free energy of the ions?] • How is a salt bridge different from sodium chloride in solution?
Hydrophobicity at small (volume) and large (SA) length scales Chandler and Weeks, J. Phys. Chem. B, 103 (22), 4570 -4577, 1999
Hydrophobic cavities and drug binding • Hydrophobic effect is generally the most important favorable interaction for potent binders. • Entropy gained by displacing waters overcomes entropy losses of ligand and protein. biotin One of the strongest protein- ligand interactions known: K d = 10 -15 streptavidin
And not so exotic, but still arising from quantum mechanics
VDW Part 1: Dispersion Forces • Consider 2 He atoms – the least chemically reactive, most “ ideal ” gas. They still interact with each other! • Quantum mechanical effect • Long-range, weak attraction • Can be described classically as a spontaneously induced dipole-induced dipole interaction • As r → ∞ , the interaction scales as 1/r 6 • Magnitude of force: obviously depends strongly on distance; generally small relative to kT. But it adds up (N 2 interactions in protein).
VDW Part 2: Close-Range Repulsion • Direct consequence of Pauli exclusion principle: 2 electrons (which necessarily have same spin) cannot simultaneously occupy same space • Formally increases exponentially with decreasing internuclear separation • However, frequently modeled as 1/r 12 . • Magnitude of force: gets extremely large very quickly ( “ steric clash ” )
VDW Part 3: Complete Potential • Dispersion and short- range repulsion are then combined in the Lennard-Jones formula: A/r 12 – C/r 6 • Narrow, rather shallow minimum at the sum of the “ VDW ” radii (when the atoms are just touching).
Computational Methods Matt Jacobson matt.jacobson@ucsf.edu Some slides borrowed from Jed Pitera (IBM, Adjunct Faculty UCSF)
Recommend
More recommend
