Protein Dynamics, Allostery and Function

Lecture 1. Molecular Forces

Xiaolin Cheng UT/ORNL Center for Molecular Biophysics

SJTU Summer School 2017 1 Lecture 1, SJTU Summer School 2017 2 Lecture 1, SJTU Summer School 2017 3 Ribosome: an machine to produce proteins.

Mitochondrion: a factory to produce energy.

Lecture 1, SJTU Summer School 2017 4 Structures: protein

Lecture 1, SJTU Summer School 2017 5 Protein Structure

Lecture 1, SJTU Summer School 2017 6 Lecture 1, SJTU Summer School 2017 7 Protein Folding

Anfinsen’s dogma: The nave structure corresponds to the state with the lowest free energy of the protein-solvent system.

But how? Lecture 1, SJTU Summer School 2017 8 Levinthal Paradox

Suppose (i) each amino acid has 3 conformaons (ii) a protein consists of 100 amino acid residues, with a total conformaon number 3100 ≈ 5 x 1047 (iii) 100 psec (10-10 sec) are required for a conformaonal change

Then A random search of all conformaons would require around 1030 years. Nevertheless, folding of a real protein takes place in msec to sec order.

So Protein folding cannot be via a random search.

Lecture 1, SJTU Summer School 2017 9 Funnel Theory

For a muldimensional smooth landscape like (a), to find its minimum is simple. A real landscape is much more complex, with mulple local minima (folding traps) like (c).

Lecture 1, SJTU Summer School 2017 10 Funnel Theory

enthalpy-entropy compensaon Gain of binding energy pulls pepde down; reducon of entropy keeps pepde up.

Lecture 1, SJTU Summer School 2017 11 Model for Protein Folding

Lecture 1, SJTU Summer School 2017 12 Forces on Proteins

Lecture 1, SJTU Summer School 2017 13 Force Scales

room ATP covalent glucose temperature hydrolysis bond oxidation

secondary bond

vdw hydrophobic

HB

salt bridge covalent

Lecture 1, SJTU Summer School 2017 14 Bonded Forces

Lecture 1, SJTU Summer School 2017 15 Non-Bonded Forces

A repulsive component resulng from the Pauli exclusion principle that prevents the collapse of molecules. Aracve or repulsive electrostac interacons between permanent charges (in the case of molecular ions), dipoles (in the case of molecules without inversion center), quadrupoles (all molecules with symmetry lower than cubic), and in general between permanent mulpoles. Inducon (also known as polarizaon), which is the aracve interacon between a permanent mulpole on one molecule with an induced mulpole on another. This interacon is somemes called Debye force. Dispersion (usually named aer Fritz London), which is the aracve interacon between any pair of molecules, including non-polar atoms, arising from the interacons of instantaneous mulpoles.

Lecture 1, SJTU Summer School 2017 16 Force Field Functions

Lecture 1, SJTU Summer School 2017 17 Coulomb Interaction charge-charge interacons

N2 complexity – requires fast algorithms, e.g., PME, FMM, Mul-grid

Entropic effect is significant

The dielectric constant of water is strongly temperature dependent, and decreases by 0.46% per Kelvin near room temperature.

TS = -1.38G

Lecture 1, SJTU Summer School 2017 18 Coulomb Interaction

Charge-dipole interacons Induced dipoles

Dipole-dipole interacons Induced dipoles

Lecture 1, SJTU Summer School 2017 19 Cation–π Interactions

Lecture 1, SJTU Summer School 2017 20 Cation–π Interactions

+ + + + + + + M Li Na K NH4 Rb NMe4 -ΔG [kcal/ 38 27 19 19 16 9 mol] ri o n [Å] 0.76 1.02 1.38 1.43 1.52 2.45

Lecture 1, SJTU Summer School 2017 21 Cation–π Interactions

Lecture 1, SJTU Summer School 2017 22 Cation–π Interactions

Lecture 1, SJTU Summer School 2017 23 Cation–π Interactions

Lecture 1, SJTU Summer School 2017 24 π Stacking

The benzene dimer is experimentally bound by 8–12 kJ/mol (2–3 kcal/mol) in the gas phase with a separation of 4.96 Å between the centers of mass for the T-shaped dimer.

Lecture 1, SJTU Summer School 2017 25 π Stacking

intermolecular overlapping of p-orbitals in π-conjugated systems, so they become stronger as the number of π- electrons increases.

Lecture 1, SJTU Summer School 2017 26 Hydrogen Bond

aracve interacon of a hydrogen atom with an electronegave atom

occur between molecules (intermolecularly), or within different parts of a single molecule (intramolecularly)

5 to 30 kJ/mol - stronger than a van der Waals interacon, but weaker than covalent or ionic bonds. electrostac dipole-dipole interacon, but has some features of covalent bonding: direconal and strong, produces interatomic distances shorter than sum of van der Waals radii, and usually involves a limited number of interacon partners. Lecture 1, SJTU Summer School 2017 27 Hydrogen Bond

Lecture 1, SJTU Summer School 2017 28 Van der Waals Force

London dispersion forces - aracons between atoms, molecules, and surfaces.

caused by correlaons in the fluctuang polarizaons of nearby parcles (a consequence of quantum dynamics).

The strength of London dispersion forces is proporonal to the polarizability of the molecule, which in turn depends on the total number of electrons and the area over which they are spread.

Lecture 1, SJTU Summer School 2017 29 Van der Waals Force

Whereas the funconal form of the aracve term has a clear physical jusficaon, the repulsive term has no theorecal jusficaon. It is used because it approximates the Pauli repulsion well, and is more convenient due to the relave computaonal efficiency of calculang r12 as the square of r6. Lecture 1, SJTU Summer School 2017 30 Van der Waals Force

a relavely good approximaon and due to its simplicity is oen used to describe the properes of gases, and to model dispersion and overlap interacons in molecular models.

Lecture 1, SJTU Summer School 2017 31 Non-Bonded Forces

Other noncovalent interacons include hydrogen bonds, van der Waals forces, charge-transfer interacons, and dipole-dipole interacons. These forces exist between molecules when they are sufficiently close to each other. The forces consist of four types: Dipole–dipole forces, ion–dipole forces, dipole-induced dipole force or Debye forces, instantaneous dipole-induced dipole forces or London dispersion forces.

Bond type Dissociaon energy (kcal) Covalent 400 Hydrogen bonds 12-16 Dipole–dipole 0.5 - 2 London (van der Waals) Forces <1

Lecture 1, SJTU Summer School 2017 32 Hydrophobic Effect

a phenomenological (entropic) force resulng from the enre system's stascal tendency to increase its entropy, rather than from a parcular underlying microscopic force.

The hydrophobic effect represents the tendency of water to exclude non-polar molecules.

The effect originates from the disrupon of highly dynamic hydrogen bonds between molecules of liquid water by the nonpolar solute.

The hydrogen bonds are parally reconstructed by building a water "cage" around the hexane molecule. The water molecules that form the "cage" (or solvaon shell) have substanally restricted mobility.

Lecture 1, SJTU Summer School 2017 33 Hydrophobic Effect

The hydrophobic effect is responsible for effects related to biology, including: cell membranes and vesicles formaon, protein folding, inseron of membrane proteins into the nonpolar lipid environment and protein-small molecule associaons.

Lecture 1, SJTU Summer School 2017 34 Hydrophobic Effect

can be quanfied by measuring the paron coefficients between water and non-polar solvents: ΔG = ΔH - TΔS. These components are experimentally determined by calorimetry.

As a result of an entropy-enthalpy compensaon, the hydrophobic effect (as measured by the free energy of transfer) is only weakly temperature-dependent and became smaller at the lower temperature, which leads to "cold denaturaon" of proteins.

Lecture 1, SJTU Summer School 2017 35 Electrostatic Interaction

Electrostac or Coulomb potenal describes the interacons between pairs of paral charges.

This formula indicates that the electrostac interacon of two charges of 1e separated by the distance of 7 Å is about 50 kcal/mol.

Because VEL decays as 1/r, the electrostac interacons are considered as long-ranged. It is important to devise methods reducing their N2 scaling.

Lecture 1, SJTU Summer School 2017 36 Time Scale Challenge

V K (b b )2 K ( )2 = ∑ b − 0 + ∑ θ θ −θ0 bond angle K (1 cos(n )) + ∑ φ + φ −φ0 1 µs = 1 billion steps dihe 1 week of wall clock time q ⋅q + ∑∑ i j = 604, 800 seconds i j>i rij à 0.6048 millisecond per step 12 6 ⎡⎛σ ⎞ ⎛σ ⎞ ⎤ 4 ⎢⎜ ij ⎟ ⎜ ij ⎟ ⎥ + ∑∑ ε ij − i j i ⎢⎜ r ⎟ ⎜ r ⎟ ⎥ > ⎣⎝ ij ⎠ ⎝ ij ⎠ ⎦

1. N-body problem O(N2)àO(NlogN), O(N) e.g. PME, FMM Faster force evaluaon 2. Parallelization

Enhanced sampling techniques increase the barrier crossing rate

Lecture 1, SJTU Summer School 2017 37 Particle Mesh Ewald

1 N ' q q E(r ,...,r ) i j 1 N = ∑∑ 2 i, j=1 n | rj − ri + nL |

PME reduces the complexity in treating the Coulomb interaction to NlogN.

Lecture 1, SJTU Summer School 2017 38 Particle Mesh Ewald

43,222 atom system using a 128 × 128 × 128

Jaguar XT5 benchmark, 2010 Fitch BG et al. SC2006

For biomolecular simulaons, the messages to be sent and their associated latency dominate the FFT computaon me.

Lecture 1, SJTU Summer School 2017 39 Particle Mesh Ewald

number of nodes 1 M Atom Virus on Titan GPU Lecture 1, SJTU Summer School 2017 40 Fast Multipole Method

FMM reduces the complexity in treating the Coulomb interaction to N.

FMM vs. PME (parcle-mesh Ewald) - Linear me complexity - Allows non-periodic or periodic boundaries - Avoids 3D FFTs for beer parallel scaling - Spaal separaon allows use of mulple me stepping - Can be extended to other types of pairwise interacons (e.g., vdW)

Lecture 1, SJTU Summer School 2017 41 Fast Multipole Method

Yokota et al., Computer physics communications, 2011,1272

Strong scaling performance of an FMM based electrostatic model.

Nagasaki Advanced Computing Center: 128 nodes, each with 4 GPUs

20 million atoms and one billion unknowns, required only one minute on 512 GPUs Lecture 1, SJTU Summer School 2017 42 Fast Multipole Method

“achieved 4.4 PFLOPS in a 520 million-atom simulation with FMM”

Lecture 1, SJTU Summer School 2017 43 Implicit Solvation Models

Explicit Solvent Average Density Implicit Solvent http://feig.bch.msu.edu/main-research-methodology.html

It is possible to construct an implicit solvent model by approximating the medium outside the water-excluded volume as a continuum with electrostatic, entropic, and viscous properties that match water.

Lecture 1, SJTU Summer School 2017 44 Implicit Solvation Models

The solvation free energy of the solute in conformation X:

ΔG(X ) = ΔGnp (X ) + ΔGelec (X ) where ΔGnp is the change of solvation free energy in going from nothing to the non- polar solute and ΔGelec is the change of free energy in going from the non-polar (uncharged) form of the solute to the polar (charged) form of the solute.

Lecture 1, SJTU Summer School 2017 45 Polar Solvation

Consider the free energy of placing a charge q at the center of a spherical cavity in a solvent.

1 ⎛ 1 ⎞ q2 Consider the solvent to be a uniform ΔGelec = − ⎜1− ⎟ 2 R dielectric medium with dielectric constant . -⎝ ε ⎠ ion ε - - The dielectric is polarized by the charge at the center of the spherical cavity and will - +q - produce a reaction field that opposes the - - electric field produced by the solute charge. -

1 ⎛ 1 ⎞ q2 ΔGelec = − ⎜1− ⎟ 2 ⎝ ε ⎠ Rion

This expression is known as Born model, it gives the charging free energy (hydraon energy) of a single ion in a spherical cavity.

Lecture 1, SJTU Summer School 2017 46 Polar Solvation For the general case of a solute of arbitrary shape with several paral charge sites, the electrostac free energy is given by, 1 G q rf Δ elec = − ∑ iφi 2 i rf aq vac φi = φi −φi

φ sasfies the Poisson equaon ∇⋅[ε(r)∇φ(r)] = −4πρ(r) - Analycal soluon available for spherical, - - ε0 cylindrical, or planar symmetry - a - - +q - -

Lecture 1, SJTU Summer School 2017 47 Poisson-Boltzmann Theory

The electrostatic potential related to charge density is given by Poisson’s law

∇⋅ε(r)∇φ(r) = −4πρ(r) Mobile ions and the Poisson-Boltzmann equation _ (r) z exp( z (r)) + ρm = ∑ iρi,bulk − iβφ Expand at low-salt concentration + ∇⋅ε(r)∇φ(r) −κ 2φ(r) = −4πρ(r) _ where, κ 2 ~ βI + + in: ε1 _ The Tanford and Kirkwood model for protein _ N q + 2 (r!) i (r! r!) ∇ φ1 = −∑ δ − i i=1 ε1 2 ! 2 ! + ∇ φ2 (r) −κ φ2 (r) = 0 out: ε2

Lecture 1, SJTU Summer School 2017 48 Numerical Solution of PB

Numerical solution (FD, BE, FEM) The finite difference formulation: spatial derivatives are approximated using neighboring points. A successive overrelaxation method used to get rapid convergence in solving the linear systems obtained from the finite difference discretization; The boundary element method: utilizes analytical solutions obtained in terms of Green’s functions and discretization on the domain surface (molecular surface); The finite element method: an adaptive multilevel approach based on tetrahedral elements to create a dense mesh to capture the dielectric discontinuity across the molecular surface.

Lecture 1, SJTU Summer School 2017 49 Numerical Solution of PB

Advantages Disadvantages

Finite difference and • Fast solvers • Non-adaptive uniform mesh methods • Low memory overhead • Poor solution resolution • Cartesian mesh • Previous parallel methods complicated and inefficient

Boundary element methods • Smaller numerical systems • Less efficient solvers • Easier interaction • Only applicable to linear evaluation problem

Finite element methods • Highly adaptive • Previous solver and • Relatively fast solvers adaptive methods inadequate • Previous parallel methods complicated and inefficient adapted from Nathan A. Baker’s slides, North Dakota State University, 2003

Lecture 1, SJTU Summer School 2017 50 AMFPB

300

250 direct calculation 6-digit accuracy 3-digit accuracy 200

150

100

50 Time (seconds) Time

0 0 5000 10000 15000 20000 25000 30000 30S ribosome Number of Nodes 500 subunit 3-digit accuracy 400 6-digit accuracy 88,431 atoms 300 343,028 elements and 171,288 nodes

200 21 min on our desktop machine (Intel(R) 100 Xeon(TM) CPU 3.00 GHz, 4GB memory), with memory (megabytes) memory 2.6GB memory 0 0 5000 10000 15000 20000 25000 30000 Number of Nodes <1 min on our 64 core Dell workstation

Lecture 1, SJTU Summer School 2017 51 Reaction Field Method

An approximate local algorithm for electrostatic calculation.

ε0

ε1 qi

rc

Tironi et al. J. Chem. Phys. 1995, 102, 1

Lecture 1, SJTU Summer School 2017 52 Reaction Field Method

Schulz et al., J. Chem. Theory Comput., 2009, 2798

28 ns/day for a 5.4 million atom system running on 30k cores; >300TF for pure water systems on 150K cores

Lecture 1, SJTU Summer School 2017 53 Tree-code GB Generalized Born electrostatics 1 1 q q G GB (1 ) i j Δ pol = − − ∑ 2 ε ij fGB 2 2 rij 1/ 2 fGB = [rij + αiα j exp(− )] 4αiα j

The particle-cluster interaction energy

Xu et al. J Chem Phys, 2011, 064107

Lecture 1, SJTU Summer School 2017 54 Image Charge Method

The reaction potential can be approximated by M +1 image point charges plus a few correction terms,

Xu et al, SIAM J. Appl. Math. 2013

Lecture 1, SJTU Summer School 2017 55 56 Model for protein folding: (i) Framework model describes a step-wise mechanism to greatly narrow the conformational search. This involves a hierarchical assembly whereby local elements of secondary structure are formed according to the primary sequence, but independent from tertiary structure. These elements then diffuse until they collide, whereupon they coalesce to form the tertiary structure. (ii) Nucleation model suggests that tertiary structure forms as an immediate consequence of the formation of secondary structure. Nucleation occurs through the formation of native secondary structure by only a few residues (e.g. a beta-turn, or the first turn of an alpha-helix), and structure propagates out from this nucleus. (iii) Hydrophobic collapse model hypothesises that the native protein conformation forms by rearrangement of a compact collapsed structure. Hydrophobic collapse to form a molten globule therefore constitutes an early step in the folding pathway. (iv) Nucleation-condensation model suggests that a diffuse folding nucleus is formed in (ii) and consolidated through the transition state, concomitant with tertiary structure formation. (i), (ii), and (iv) suggest the formation of kinetic intermediates, whereas (iii) does not. (A. R. Fersht. Curr. Opin. Struc. Biol., 1997, 7, 1, 3-9.)

Lecture 1, SJTU Summer School 2017 57 Dynamics of Protein Structures

Lecture 1, SJTU Summer School 2017 58