Molecular Biophysics III - DYNAMICS
References
Molecular Biophysics – Structures in Motion, Michel Daune, Oxford U Press, 1999 (reprinted 2003, 2004).
Computational Molecular Dynamics: Challenges, Methods, Ideas LNCSE (Lecture Notes In Comp Sci & Eng) Series, Eds: Griebel, Keyes, Neimien, Roose, Schlick, Springer-Verlag, 1999.
Protein Physics A.V. Finkelstein and O. G. Ptitsyn, Academic Press, 2002.
A Theoretical Perspective of Dynamics, Structure, and Thermodynamics. C. L. Brooks, III, M. Karplus, and B.M. Pettitt. Wiley Interscience, New York, 1988.
Biophysical Chemistry. C.R. Cantor and P.R. Schimmel. Vol.1,2,3. W.H. Freeman and Company, San Fransisco, 1980.
Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding. A. Fersht. W. H. Freeman and Company, New York, 1999.
Molecular Modelling. Principles and Applications. A. R. Leach. Addison Wesley Longman, Essex, England, 1996.
Dynamics of Proteins and Nucleic Acids. J.A. McCammon and S.C. Harvey. Cambridge University Press, Cambridge, 1987.
Physics With Illustrative Examples from Medicine and Biology – Statistical Physics Benedek & Villars, Springer-Verlag, 2000 ‘Protein folding problem’ Sequence ------> Structure
B io inf or s m n o at ti ics la . u Se im quenc s d n a e g al ig llin n e m d e o nt M s Function
Fundamental paradigm: Sequence encodes structure; structure encodes function Structure Æ Function
ToTo whatwhat degreedegree isis foldfold associatedassociated withwith function?function?
Structure Æ Dynamics Æ Function IsIs therethere aa uniqueunique relationrelation betweenbetween foldfold andand function?function? Frequency in database of 229 folds Frequency in Hegyi & Gerstein, JMB 288: 147
Number of functions associated with a fold Knowledge of sequence or structure does not permit us to
• Understand the mechanism of function • Device methods of controlling/inhibiting function • Predict the behavior in different forms, different environments • Answer the questions ‘how’ or ‘why’!
Biological function is a dynamic process
Structure Æ Dynamics Æ Function To understand the principles that underlie the passage
to function... from structure,
We need to examine the conformational dynamics.
Dynamics ÅÆ Function State-of-the-art in computational/mathematical biology
?
Å ------25 Å ------Æ Molecular computations Subcellular/cellular computations
● Limited to small systems (one ● Simple mass-action kinetics macromolecule) or short times (~ ns) ● No spatial-structural realism ● Dependent on force field ● Lack of data for model parameters ● Solvent effect – a problem SupramolecularSupramolecular dynamicsdynamics
Wikoff, Hendrix and coworkers
Multiscale modeling – from full atomic to multimeric structures Subcellular and cellular simulations
Monte Carlos simulations using MCell and DReaMM - Stiles and coworkers (PSC) Dynamics systems - Mathematical modeling using ODEs Progresses in molecular approaches: Coarse-grained approaches for large complexes/assemblies
Å ------25 Å ------Æ Å ------250 Å ------Æ
Example: EN models for modeling ribosomal machinery (Frank et al, 2003; Rader et al., 2004) Stuctures suggest mechanisms of function
A. Comparison of static structures available in the PDB for the same protein in different form has been widely used as an indirect method of inferring dynamics. Bahar et al. J. Mol. Biol. 285, 1023, 1999.
B. NMR structures provide information on fluctuation dynamics ProteinProtein dynamicsdynamics
Folding/unfoldingFolding/unfolding dynamicsdynamics
Passage over one or more energy barriers B. Ozkan, K.A. Dill & I. Bahar, Protein Sci. 11, 1958-1970, 2002 Transitions between infinitely many conformations
FluctuationsFluctuations nearnear thethe foldedfolded statestate
Local conformational changes Fluctuations near a global minimum SeveralSeveral modesmodes ofof motionsmotions inin nativenative statestate
Hinge site MacromolecularMacromolecular ConformationsConformations
(i-4) (i+1) li+2 (i-2) ϕ i-2 θi li-1 ϕ i (i-3) (i) (i-1)
Schematic representation of a chain of n backbone units. Bonds are labeled from 2 to n, and structural units from Schematic representation of a portion of the main chain of a 1 to n. The location of the ith unit with respect to the macromolecule. li is the bond vector extending from unit i- laboratory-fixed frame OXYZ is indicated by the position 1 to i, as shown. ϕi denotes the torsional angle about vector Ri. bond i. How/whyHow/why doesdoes aa moleculemolecule move?move?
AmongAmong thethe 3N3N--66 internalinternal degreesdegrees ofof freedom,freedom, bondbond rotationsrotations (i.e.(i.e. changeschanges inin dihedraldihedral angles)angles) areare thethe softest,softest, andand mainlymainly responsibleresponsible forfor thethe functionalfunctional motionsmotions TwoTwo typestypes ofof bondbond rotationalrotational motionsmotions
FluctuationsFluctuations aroundaround isomericisomeric statesstates JumpsJumps betweenbetween isomericisomeric statesstates
Most likely near native state Definition of dihedral angles
Ci+2' C i+2 4
ϕ 3 i+1 2
(i+1) ) (kcal/mol)
Ci+1 ϕ C 1 i " E( Ci+2
0 i 0 60 120 180 240 300 360 (i) π−θ ϕ (°)
Rotational energy as a function of dihedral angle for a threefold symmetric torsional potential (dashed curve) and a three-state Ci-1 potential with a preference for the trans isomer (j = 180°) over the gauche isomers (60° and 300°) (solid curve), and the cis (0°) state Spatial representation of the torsional mobility around the bond i+1. being most unfavorable. The torsional angle ϕi+1 of bond i+1 determines the position of the atom Ci+2 relative to Ci-1. C'i+2 and C"i+2 represent the positions of atom i+2, when ϕi+1 assumes the respective values 180° and 0°. Rotational Isomeric States (Flory – Nobel 1974) BondBond--basedbased coordinatecoordinate systemssystems
Transformation matrix between frames i+1 and i
Virtual bond representation of protein backbone HomeworkHomework 1:1: PassagePassage betweenbetween CartesianCartesian coordinatescoordinates andand generalizedgeneralized coordinatescoordinates
Take a PDB file. Read the position vectors (X-, Y- and Z-coordinates – Cartesion coordinates) of the first five alpha-carbons
Evaluate the corresponding generalized coordinates, i.e. the bond lengths li (i=2-5), bond angles θi (i=2-4), and dihedral angles φ3 and φ4 using the Flory convention for defining these variables.
Using the PDB position vectors for alpha-carbons 1, 2 and 3, generate the alpha carbons 4 and 5, using the above generalized coordinates and bond-based transformation matrices. Verify that the original coordinates are reproduced. Side chains enjoy additional degrees of freedom HarmonicHarmonic OscillatorOscillator ModelModel
RapidRapid movementsmovements ofof atomsatoms aboutabout aa valencevalence bondbond OscillationsOscillations isis bondbond anglesangles FluctuationsFluctuations aroundaround aa rotationalrotational isomericisomeric statestate DomainDomain motionsmotions –– fluctuationsfluctuations betweenbetween openopen andand closedclosed formsforms ofof enzymesenzymes HarmonicHarmonic OscillatorOscillator ModelModel
A linear motion: Force scales F = - k x linearly with displacement
The corresponding equation of motion is of the form m d2x/dt2 + k x = 0
The solution is the sinusoidal function x = x0sin(ωt+φ) 1/2 where ω is the frequency equal to (k/m) , x0 and φ are the original position and velocity. EnergyEnergy ofof aa harmonicharmonic oscillatoroscillator
2 KineticKinetic energy:energy: EEK == ½½ mvmv
where v = dx/dt = d [x0sin(ωt + φ)]/dt = x0ω cos(ωt +φ) 2 2 2 2 2 2 ÆEK = ½ mx0 ω cos (ωt+φ) = ½ mω (x0 -x )
2 2 2 2 2 2 2 (because x = x0 sin(ωt+ φ) or x = x0 [1- cos (ωt+φ)] Æ x0 cos (ωt+φ) = x0 -x )
2 PotentialPotential energy:energy: EEP == ½½ kxkx
2 TotalTotal energy:energy: EEP ++ EEK== ½½ kxkx0 Always fixed RouseRouse chainchain modelmodel forfor macromoleculesmacromolecules
Connectivity matrix R Connectivity matrix 3 1 -1 R R -1 2 -1 1 4 -1 2 -1 Γ = R .. 2 ... -1 2 -1 R n -1 1
2 2 2 Vtot = (γ/2) [ (∆R12) + (∆R23) + ...... (∆RN-1,N) ]
2 2 = (γ/2) [ (∆R1 - ∆R2) + (∆R2 - ∆R3) + ...... (1) HomeworkHomework 2:2: PotentialPotential energyenergy forfor aa systemsystem ofof harmonicharmonic oscillatorsoscillators
(a) Using the components ∆Xi, ∆Yi and ∆Zi of ∆Ri, show that Eq 1 (Rouse potential) can be decomposed into three contributions, corresponding to the fluctuations along x-, y- and z-directions:
Vtot = VX + VY + VZ. where 2 2 VX = (γ/2) [ (∆X1 - ∆X2) + (∆X2 - ∆X3) + ...... (2) and similar expressions hold for Vy and Vz.
(b) Show that eq 2 can alternatively be written as
V = ½ ∆XT Γ ∆X (3)
T where ∆X = [∆X1 ∆X2 ∆X3.....∆XN], and ∆X is the corresponding column vector. Hint: start from eq 3, obtain eq 2. III. Understanding the physics
Harmonic oscillators Æ Gaussian distribution of fluctuations
Consider a network formed of beads/nodes (residues or groups of residues) and springs (native contacts)
Residues/nodes undergo Gaussian fluctuations about their mean positions – similar to the elastic network (EN) model of polymer gels (Flory)
2 2 W(∆Ri) = exp{ -3 (∆Ri) /2 <(∆Ri) >} Proteins can be modeled as an ensemble of harmonic oscillators
Gaussian Network Model - GNM MolecularMolecular MovementsMovements
Physical properties of gases – a short review (Benedek & Villars, Chapter 2) PV = NkT Ideal gas law: PV = RT M PV = nRT
where VM is the molar volume, T is the absolute temperature, R is the gas constant (1.987 x 10-3 kcal/mol or 8.314 J/K), k is the Boltzmann constant, N
is the number of molecules, n is the number of moles = N/N0 , N0 is the Avogadro’s number.
Mean kinetic energy of a molecule of mass m and its mean-square es as f g velocity: o ry 2 2 eo <½ mv >= (3/2) kT Æ
2 ½ ½ vrms =
Molecule M (g/mol) vrms (m/s) H 2 1880 2 Brownian motion 32 (Brown, 1827) O2 32 474 Macromolecules 104 - 106 2.6 - 26
Viruses 108 - 1010 0.026 – 0.26 (e.g. tobacco (5 x 107 g/mol) (35 cm/s) mosaic virus) (35 cm/s)
These numbers provide estimates on the time/length scales of fluctuations or Brownian motions Equipartition law
An energy of ½ kT associated with each degree of freedom
For a diatomic molecule, there are three translational (absolute), two rotational degrees of freedom, and the mean translational energies are
2 2 2 < ½ mvx >= < ½ mvY >= < ½ mvZ >= ½ kT
And the mean rotational energy is kT. For non interacting single atom molecules (ideal gases), there are only translational degrees of freedom such that the total internal energy is
U = 3/2 kT and specific heat is Cv = ∂U/∂T = 3/2 k RandomRandom WalkWalk
Probability of R steps to the right and L N steps to the left in a random walk of N steps PN(R, L) = (1/2 ) N! / R! L!
R + L = N Æ N R – L = m PN(m) = (1/2 ) N! / [(N + m/2)! (N - m/2)!]
Probability of ending up at m steps away from the origin, at the end of N steps
http://mathworld.wolfram.com/BinomialDistribution.html Binomial (or Bernoulli) Distribution P(n|N) N = 15
Properties of Binomial Distribution Mean Np n Variance Npq = Standard deviation (Npq)1/2 GaussianGaussian formform ofof BernoulliBernoulli distributiondistribution
N PN(m) = (1/2 ) N! / [(N + m/2)! (N - m/2)!]
As m increases, the above distribution may be approximated by a continuous function
½ 2 Gaussian approximation PN(m) = (2/πN) exp {-m /2N}
Length of Each step Examples of Gaussianly distributed variables: •Displacement (by random walk) along x-direction Æ W(x) ≈ exp {-x2/2Nl2} where m=x/l 2 2 •Fluctuations near an equilibrium position Æ W(r) ≈ exp {-3(∆r) /2<(∆r) >0} ½ 2 •Maxwell-Boltzmann distribution of velocities Æ P(vx) = (m/2πkt) exp (-½mvx /kT} •Time-dependent diffusion of a particle Æ P(x,t) = √[4πDt] exp(-x2/4Dt} Examples of Gaussianly distributed variables:
• Displacement (by random walk) along x-direction Æ W(x) ≈ exp {-x2/2Nl2} where m=x/l
2 2 • Fluctuations near an equilibrium position Æ W(r) ≈ exp {-3(∆r) /2<(∆r) >0}
½ 2 • Maxwell-Boltzmann distribution of velocities Æ P(vx) = (m/2πkt) exp (-½mvx /kT}
• Time-dependent diffusion of a particle Æ P(x,t) = √[4πDt] exp(-x2/4Dt}