Thermodynamic cycles Powerful ways to exploit the properties of - - PowerPoint PPT Presentation

Thermodynamic cycles

Powerful ways to exploit the properties of state functions

Energy Units: How much is a lot of energy?

Boltzmann distribution:

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 ΔG=-RTlnK: If the equilibrium constant changes by a factor of 10, ΔG changes by ~1.4 kcal/mol [If you remember nothing else, remember this]

exp −E / kbT

( )

Critical concept: Enthalpy, entropy, and free energy are all state functions

State 1 (e.g., ligand and protein free) State 2 (e.g., ligand bound to protein) S1 H1 G1 S2 H2 G2 ΔS = S2 − S1 ΔH = H2 − H1 ΔG = G2 −G1 path 2 path 1 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 1 Elevation Latitude Longitude State 2 Elevation Latitude 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: Na+(g) + Cl-(g) Na+(aq) + Cl-(aq) NaCl(s)

Break apart crystal lattice breaking lattice order ΔH » 0 ΔS > 0 Solution process ΔH = ? ΔS = ?? 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

Apo Hb (T state) Holo Hb (T state) Holo Hb (R state) Apo Hb (R state) ΔGoxygen(apo) ΔGoxygen(holo) ΔGbind(T) ΔGbind(R) ΔΔGbind = ΔGligand(T) – ΔGligand(R) = ΔGoxygen(apo) – ΔGoxygen(holo)

Keep in mind that protons are ligands too

Apo Hb (T state) Holo Hb (T state) Holo Hb (R state) Apo Hb (R state) ΔGoxygen(His0) ΔGoxygen(His+) pKa (T) pKa (R)

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: q1q2/r12

• A dipole is a pair of opposite charges

separated by a small amount

• Charge-dipole interactions ∝ 1/r2
• Dipole-dipole interactions ∝ 1/r3
• 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

creates a “dielectric screening” between the ions.

• The “first shell” of waters moves with the ions.

But even waters further away tend to orient towards (or away from) the ions.

• The dielectric constant of water is ~80 at room

temp, but decreases with increasing T.

gas phase: solution phase:

r q q E

− +

∝ r q q E ε

− +

∝

Implicit/Continuum Water Model

+

–

ε=80 ε=1

• -

+ + + + + + +

Can we capture the effects of water on electrostatics 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:

ΔGsolv ∝ q2/R (where R is atomic radius)

Poisson Equation Basically Generalizes This For Any Molecule

( ) ( ) ( )

r r r

• πρ

ϕ ε 4 − = ∇ ⋅ ∇

( )=

r

• ε

( )=

r

• ϕ

( )=

r

• ρ

electrostatic potential dielectric constant (small inside protein; 80 outside) 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

( )

( )

kT r q i i

i

e r

• ϕ

ρ ρ

−

=

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

( ) ( ) ( )

( ) ( )

( )⎟

⎠ ⎞ ⎜ ⎝ ⎛ = − = + =

+ − − +

kT r e e r r r

kT r kT r ionic

• ϕ

ρ ρ ρ ρ ρ ρ

ϕ ϕ

sinh 2

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)

Electrostatic forces speed the binding of the positively- charged substrate to acetylcholinesterase by a factor of more than 100. Role of Solvated Electrostatics in Molecular Recognition 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)

Alkane transfer (benzene→water) Ion transfer (benzene→water) H H O H H O H H O H H O Key point: The strongly favorable enthalpy of ions in waters frequently compensates for the entropic loss, but not for hydrophobic solutes. ΔH ~ 0 ΔS < 0 ΔH << 0 ΔS < 0 CH4(benzene) → CH4(water) ΔH° = -11.7 kJ/mol ΔS° = -76 J/K·mol ΔG°(298) = +11 kJ/mol

Salt Bridges

Questions:

• How much does a buried salt

bridge (i.e., interior of protein) stabilize a protein?

• How about a surface exposed
• ne?
• [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?

[Hydrogen bonds are similar, but the magnitude of the forces is smaller.]

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 streptavidin One of the strongest protein- ligand interactions known: Kd = 10-15

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/r6
• Magnitude of force: obviously

depends strongly on distance; generally small relative to kT. But it adds up (N2 interactions in protein).

VDW Part 2: Close-Range Repulsion

• Direct consequence of Pauli exclusion

principle: 2 electrons (which necessarily have same spin) cannot simultaneously

• ccupy same space
• Formally increases exponentially with

decreasing internuclear separation

• However, frequently modeled as 1/r12.
• 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/r12 – C/r6

• 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)

Putting It All Together: Molecular Mechanics Models of Macromolecules

All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

2 0)

( r r kr −

Bonds

2 0)

( θ θ

θ

− k ∑

n n

n k ) (cos ϕ Angles Torsions Nonbonded: Lennard-Jones

ij j i

r q q ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

6 12 ij ij ij ij ij

r r σ σ ε Electrostatic Sources of parameters:

• Gas-phase QM
• Macroscopic

properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS)

• Spectroscopic and

crystallographic data (small molecules) θ r φ H N C O

Putting It All Together: Molecular Mechanics Models of Macromolecules

All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

2 0)

( r r kr −

Bonds

2 0)

( θ θ

θ

− k ∑

n n

n k ) (cos ϕ Angles Torsions Nonbonded: Lennard-Jones

ij j i

r q q ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

6 12 ij ij ij ij ij

r r σ σ ε Electrostatic Sources of parameters:

• Gas-phase QM
• Macroscopic

properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS)

• Spectroscopic and

crystallographic data (small molecules) θ r φ H N C O

Covalent forces are very strong

Bond Bond energy, kJ/mol C–C 347 C=C 615 C≡C 812 C–O 360 C=O 728 F–F 158 Cl–Cl 244 C–H 414 H–H 436 H–O 464 O=O 498

1 σ, 2 π bonds 1 σ, 1 π bonds

Torsion potentials

Putting It All Together: Molecular Mechanics Models of Macromolecules

All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

2 0)

( r r kr −

Bonds

2 0)

( θ θ

θ

− k ∑

n n

n k ) (cos ϕ Angles Torsions Nonbonded: Lennard-Jones

ij j i

r q q ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

6 12 ij ij ij ij ij

r r σ σ ε Electrostatic Sources of parameters:

• Gas-phase QM
• Macroscopic

properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS)

• Spectroscopic and

crystallographic data (small molecules) θ r φ H N C O

Putting It All Together: Molecular Mechanics Models of Macromolecules

All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

2 0)

( r r kr −

Bonds

2 0)

( θ θ

θ

− k ∑

n n

n k ) (cos ϕ Angles Torsions Nonbonded: Lennard-Jones

ij j i

r q q ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

6 12 ij ij ij ij ij

r r σ σ ε Electrostatic Sources of parameters:

• Gas-phase QM
• Macroscopic

properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS)

• Spectroscopic and

crystallographic data (small molecules) θ r φ H N C O

Putting It All Together: Molecular Mechanics Models of Macromolecules

All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

2 0)

( r r kr −

Bonds

2 0)

( θ θ

θ

− k ∑

n n

n k ) (cos ϕ Angles Torsions Nonbonded: Lennard-Jones

ij j i

r q q ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

6 12 ij ij ij ij ij

r r σ σ ε Electrostatic Sources of parameters:

• Gas-phase QM
• Macroscopic

properties via liquid state simulation, e.g., density, heat capacity, compressibility

• Spectroscopic and

crystallographic data (small molecules) This is sufficient to describe a macromolecule by itself; but what about solvent? θ r φ H N C O

As Briefly Discussed Earlier: Models of Solvation

Explicit Pro: water models fairly mature Con: ensemble averaging extremely expensive for large system SPC, TIP4P, etc. Adjustable parameters: Partial charges, bond lengths, etc.

+

–

O H H O H H O H H O H H O H H

Semi-analytical approximation

Implicit/Continuum Pro: solvation free energy estimates cheap and generally accurate Con: dynamics, first shell effects ??? Poisson-Boltzmann Generalized Born Adjustable parameters: radii

+

–

ε=80 ε=1

• -

+ + + + + + +

Heuristic Distance-dependent dielectric

( )

r r ∝ ε

∑

= Δ

i i i solv

A G σ

Surface-area based methods

Molecular Dynamics

• Very simple idea: Just use the simple molecular mechanics models of

forces, then feed them into Newton’s equations of motion, basically F=ma. Then watch the molecules move!

• In the realm of biology, Martin Karplus deserves a lot of credit for early

work that convinced people to think about macromolecules as dynamic, not static, structures. His program, Charmm, is still widely used.

• Now there are many thousands of papers using molecular dynamics,

and lots of widely used programs.

• Some of the areas of current interest where MD continues to play an

important role:

• Mechanisms of action of membrane proteins
• Mechanisms of allostery
• Protein folding
• Quantitative prediction of binding affinities
• A nice review article: Karplus and McCammon, “Molecular dynamics

simulations of biomolecules”, Nat Struct Biol. 2002 Sep;9(9):646-52.

Molecular dynamics integrators

( ) ( ) ( ) ( ) …

+ ⋅ Δ + ⋅ Δ + = Δ + t a t t v t t x t t x

2 2 1 Variable definitions x: position v: velocity a: acceleration (All of these are obviously vectors of size 3N) Basic idea: If we know (x,v,a) at time t, estimate their values at time t+Δt There are many integrators, and they basically all start from Taylor expansions of position and/or velocity:

Velocity Verlet integrator

This is not quite how it works in practice, but this is good enough for our purposes.

( ) ( ) ( ) ( ) …

+ ⋅ Δ + ⋅ Δ + = Δ + t a t t v t t x t t x

2 2 1

Position is updated first, based on current x, v, a

( ) ( ) ( ) ( )

[ ]

t t a t a t t v t t v Δ + + ⋅ Δ + = Δ +

1 2

Velocity updates are more accurate if you use both the current/future acceleration. Q: How to get a(t + Δt)?? A: Update positions, calculate new forces, use F=ma

Possible to play some tricks to get beyond 1 fs, e.g., freezing bonds, multiple timescale methods. Currently limited to ~microsecond simulations now, soon getting up to milliseconds, probably, although these will be huge calculations, not something everyone can do. Typical simulations: nanoseconds.

This leads to some challenges in computing electrostatic interactions

Can in principle extract two types of information from MD simulations

• Thermodynamic properties: e.g.,

– ΔH, ΔS, ΔG, etc. – experimental observables, e.g., NOEs An important point about these is that they are properties of the ensemble, not a single snapshot.

• Kinetic properties: How long does a process take? This

can be very tricky to predict accurately, for several

• reasons. Kinetic properties are also ensemble

properties, in general.

Computing free energies

Having discussed binding thermodynamics qualitatively, and introduced quantitative models of molecular energies, we now return to free energy from a quantitative standpoint.

The Configuration Integral

The partition function in classical statistical mechanics (from which free energy and many other ensemble properties can be derived) is directly proportional to the configuration integral:

Z = exp −U ! x1, ! x2,…, ! xN

( ) / kbT

" # $ %

∫

d! x1d! x2…d! xN

This integral will of course have high dimensionality, and the integrand is extremely complicated.

• MD methods implicitly give you information about this integral, in the long-

time limit.

• There are also other approaches to estimating this integral (by generating

ensembles), such as Monte Carlo methods, which I won’t discuss.

• Some simple 1D examples help with intuition [Matt will draw on board]
• Usually we are not interested in ‘absolute’ free energies, but how they

change when something happens, such as a ligand binding. [and the free energy can be computed as ΔG = - kbT ln Z]

Free Energy Perturbation

This nice simple version of the derivation taken from Severance, Essex, and Jorgensen, J. Comp.

• Chem. 16 (1995) 311.

“Alchemical" free energy calculations

§ State function

• path independence

§ Introduce new coordinate to connect states of interest by an unphysical path

• lambda (λ), aka coupling coordinate
• all paths are OK, some are more efficient. . .

§ Simple linear interpolation

• U(r,λ) = λU'(r) + (1 - λ) U(r)
• end states U(r) (λ=0), U'(r) (λ=1)
• example: U(r)=ethane, U'(r)=ethanol

§ Integrate free energy along λ

• ΔGsolv(ethane->ethanol)

Using thermodynamic cycles to compute free energies

§ ΔΔG

• relative affinities of two drugs for one receptor
• relative stabilities of two protein mutants

§ Often of more experimental interest than ΔG § Use two small nonphysical changes to compare two large physical changes

Protein + LigA Protein + LigB ProteinLigB ProteinLigA ΔGbind,A ΔGbind,B ΔGsolv,AB ΔGbound,AB ΔΔGbind,AB= ΔGbind,B - ΔGbind,A = ΔGbound,AB - ΔGsolv,AB