Protein Dynamics, Allostery and Function

Protein Dynamics, Allostery and Function Lecture 3. Protein Dynamics Xiaolin Cheng UT/ORNL Center for Molecular Biophysics SJTU Summer School 2017 1

Obtaining Dynamic Information Experimental Approaches X- ray Crystallography Nuclear MagneBc Resonance Neutron Spectroscopy H/D exchange mass spectrometry Single- Molecule Fluorescence Spectroscopy ComputaBonal Approaches Molecular Dynamics Normal Mode Analysis ElasBc Network Model Generalized Langevin EquaBon

X-ray Crystallography B- factors indicate the amount of thermal fluctuabon for a parbcular residue. Is anisotropic but usually only measurable as an isotropic mean value. MoBon extrapolated between crystal structures of two different conformabonal states. Weakness: ConformaBonal changes are not necessary linear. Maltose Binding Protein Engulfing Ligand

Nuclear Magnetic Resonance Solution NMR techniques cover the complete range of dynamic events in enzymes. Chem. Rev. 2006, 106, 3055-3079

Nuclear Magnetic Resonance Fast Backbone and Side-Chain Motions Occur on the ps to ns timescale, usually involving measuring relaxation times such as T1 and T2 to determine order parameters, correlation times, and chemical exchange rates. The spectral density function, J(ω), is directly related to the three relaxation rates, through the Lipari and Szabo model, where τ m is the correlation time for the overall rotational diffusion of the macromolecule, S 2 is the order parameter, and 1/τ ) 1/τ m + 1/τ e, where τ e is the time scale (ns) for the internal bond vector (e.g. N-H) motions.

Nuclear Magnetic Resonance Conformational Exchange The dipolar coupling between two nuclei depends on the distance between them, and the angle of bond relative to the external magnetic field. Residual dipolar coupling (RDC) provides spatially and temporally averaged information about an angle between the external magnetic field and a bond vector in a molecule RDCs also provide rich geometrical information about dynamics on a slow (>10 9 s) in proteins. A motion tensor B can be computed from RDCs: The RDC- derived mobon parameters are local measurements.

7 QENS / MD Neutron Scattering

8 QENS / MD Dynamic Structure Factor

Quasi-elastic Neutron Scattering Simple Translational Diffusion In the energy space, we typically fit with Lorentzians. For simple translational diffusion, particles follow Fick s Law: D s is diffusion coefficient Γ is the half width at half maximum (HWHM) which is this case = DQ 2, leading to a broadening of the elastic line following a given Q-dependence. 9 QENS / MD

Neutron Spin Echo Neutron Spin Echo (NSE) measures collective dynamics at the timescale of ns 100 ns. It provides the intermediate scattering function, I(Q, t), directly. 10 QENS / MD

Inelastic Neutron Scattering 11 QENS / MD Neutrons are unique for probing protein H dynamics both spatial and time resolution.

Hydrogen/Deuterium Exchange Mass Spectrometry Amide hydrogens exchange at rates that are characteristic of local backbone conformation and dynamics. In highly dynamic unstructured regions, the exchange reaction proceeds on the msec-sec timescale while amides that are hydrogen bonded will exchange more slowly (minutes to days).

Single-Molecule Fluorescence Spectroscopy Förster resonance energy transfer (FRET), fluorescence resonance energy transfer (FRET), resonance energy transfer (RET) or electronic energy transfer (EET) is a mechanism describing energy transfer between two light-sensitive molecules (chromophores). In FRET, the efficiency of the absorption of the photon emitted from the first probe in the second probe depends on the distance between these probes. Since the distance changes with time, this experiment probes the internal dynamics of the molecule.

In Silico Approaches Different representation of protein structures and dynamics Molecular Dynamic Simulation Accuracy Detail Computation Time Normal Mode Analysis Elastic Network Analysis Generalized Langevin Equation

m!! x V = Molecular Dynamics Simulation Newton s Equation of Motion = V(x,...,x i i 1 Energy Function (Force Field) + bond i K b ( b b ) q q 0 2 + angle K θ n ) ( θ θ ) σ σ i j ij ij + 4ε > > ij j i rij i j i rij rij 0 2 12 + dihe K φ (1 + cos( nφ φ )) 6 0 The ultimate detail modeling of protein fluctuation, every atom is accounted for. How the system evolves with time - computational microscope enzyme functional dynamics, receptor gating mechanism

MD in the 1970s and Now McCammon, Gelin, Karplus. Nature, 1977, 267, 585 1977 - Bovine Pancreatic Trypsin Inhibitor ~400 atoms in vacuum, for 9 ps 1979 - Simulations resulted in the recognition that B factors can be used to infer internal motions. Model of the cytoplasm of B. subtilis 60 million atoms in water, for >1µs ~10 15

Simulated Vibrational Spectra Amplitude Pitch POX Which Regions Contribute to the Observed Differences in the INS?

Protein Conformational State PGK: 45 kd t = 100 ps 5, 10 ns 5, 500 ns 5 and 17 µs

Markov State Model

Markov State Model What can we know from an MSM? Thermodynamic information Equilibrium populations Kinetic information Mean first passage time Dominant Pathways (pathways with major flux)

Normal Mode Analysis A simple harmonic oscillator based system that analyzes dynamics near a local minimum. Specifically, if one expands the potential energy function U around a minimum on the energy surface, r o, the Hamiltonian of the system is given by:

Normal Mode Analysis Harmonic oscillator Newton law: ma = F ma = kx x = Acos(ωt + φ)

Normal Mode Analysis coordinate transformation

Normal Mode Analysis normal mode vector normal mode coordinate

Normal Mode Analysis

Normal Mode Analysis Atomic fluctuations

Normal Mode Analysis Cartesian displacement along normal mode

Normal Mode Analysis Overlap Gives information on how well a normal mode describe the conformational change. A value close to 1 indicates that the mode represents well the conformational change.

Normal Mode Analysis Correlation of atomic motion = 1 => atom i and j moves in the same direction with same amplitude: correlated motions = -1 => atom i and j moves in the opposite direction with same amplitude: anti-correlated motions = 0 => no correlation

Quasi-harmonic Analysis = Δ i i ij B j T k x 2 2 2 ω α ij B j i F T k x x ) ( 1 = Δ Δ ) / ( 2 j i ij x x V F Cartesian atomic fluctuations can be determined as a sum over all normal modes Under quasi-harmonic approximation, the sum can be simplified to yield x i x j H Δ Δ = = 2 2 1 0 0 0... 0 0 0 n B B T k T k L ω ω

Elastic Network Model Atoms connected via elastic springs Coarse grained models such as only Cα atoms can be used Tirion MM (1996) Phys Rev Lett. 77, 1905-1908

Elastic Network Model

Brownian Motion 1827: Robert Brown observed pollen particles moving erratically in water in a zigzag fashion 1905: Albert Einstein and Marian von Smoluchowski explained the phenomenon mathematically 1911: Jean Perrin validated the theory by the use of sedimentation equilibrium Small particles, moving with thermal motion, randomly collide with large particle à diffusive, random motion of large particle in a fluid http://www.deutsches-museum.de/ausstell/dauer/physik/e_brown.htm

Brownian Motion 1827: Robert Brown observed pollen particles moving erratically in water in a zigzag fashion 1905: Albert Einstein and Marian von Smoluchowski explained the phenomenon mathematically 1911: Jean Perrin validated the theory by the use of sedimentation equilibrium For n-d random walk: 2 x = 2nDt, D = k T / 6πη r k T / ζ B = B Average displacement of a particle is proportional to square root of time D = diffusion coefficient T = temperature r = hydrodynamic radius η = solvent viscosity ζ = friction coefficient Stokes-Einstein relation for Brownian motion

Brownian Motion in a Living Cell

Modelling Brownian Motions Discrete: stochastic motions of individual particles Monte Carlo Langevin Brownian dynamics Continuum: concentration distribution probability

! 2 d R( t) m 2 dt Langevin Equation Brownian motion of a particle in a fluid due to collisions with the molecules of the fluid can be described by the Langevin equation,! dr( t) = γ + dt! F! + S,!!! 2 6kBT R = 0, ( R( t) R0 ) = t = 6Dt mγ!!! γ S( t) = 0, S(0) S( t) = δ ( t) m Friction: force in opposite direction and proportional to velocity F! : systematic interaction forces S! : stochastic forces γ : friction coefficient = k B T/D Under over-dampened conditions, leading to Brownian dynamics! d R( t) 1! 1! = F + S dt γ γ

Ermak-McCammon Brownian Dynamics 1 ri ( t + Δt) = ri ( t) + DijFjΔt + Ri ( Δt), i = 1,..., k T B j r i : coordinates of particle i F j : systematic interaction forces D ij : diffusion tensor à hydrodynamic interactions R i : random stochastic displacement N R i = 0, R R = 2D i j ij Δt time step >> momentum relaxation time Δt >> m D i ii / Ermak and McCammon JCP(1978) 69,1352-60 k B T R R i j = 2D Δt Cholosky decomposition ij Hydrodynamic interaction (HI) between solute molecules results from induced fluid flow of solvent due to motion of solute molecules

Molecular Diffusion The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. When applied to particle position distributions, it becomes the Smoluchowski equation. When zero diffusion, it is known as the Liouville equation.

Smoluchowski Equation The Smoluchowski equation is the Fokker Planck equation for the probability density function of the particle positions of Brownian particles.

Smoluchowski Equation P t = D[ 2 P 1 k T B ( W P)] Continuity equation

Smoluchowski Equation P t = D[ 2 P 1 k T B ( W P)] Diffusive association of particles to a sphere c = D 2 c t 2 ( ) steady state: c / t = 0 2 = 1 d rc c 2 = 0 r dr Diffusion equation: (without friction) after integrating: c( r) = c 1 ( a ) r Dc a particle flux: ( ) D dc J r = = 2 dr r number of collisions per second at r = a: I( a) = J ( a) 4π a = 4π Dc 2 a association rate: k a = I( a) / c = 4π Da ~ 10 9 M -1 s -1

Fokker-Planck Equation a partial differential equation that describes the time evolution of the probability density function of the momentum of a particle undergoing a Brownian motion.

Fokker-Planck Equation a partial differential equation that describes the time evolution of the probability density function of the momentum of a particle undergoing a Brownian motion. The equation can be generalized to other observables: