Week 3 Analysis of Trajectory Data Lecture 5: Fundamental quantities; total energy, temperature, pressure, volume, density. Structural quantities; root.

Week 3

Analysis of Trajectory Data

•Lecture 5: Fundamental quantities; total energy, temperature,

pressure, volume, density. Structural quantities; root mean square

deviation (RMSD), distribution functions (pair or radial), conformational

analysis (Ramachandran plots, rotamers). Kinematic quantities; time

correlation functions and transport coefficients. Complex structure

determination via docking and MD.

•Lecture 6: Dynamical quantities; free energy of a system, Helmholtz

and Gibbs free energies, calculation of free energy using perturbation,

umbrella sampling, and steered MD methods.

Structural analysis of proteins and complexes

1. Fundamental quantities

Total energy, temperature, pressure, volume (or density)

2. Structural quantities

Root Mean Square Deviation (RMSD)

Distribution functions (e.g., pair and radial)

Conformational analysis (e.g., Ramachandran plots, rotamers)

3. Complex structure determination

Determination of protein-ligand and protein-protein complexes using

docking programs.

Refinement of complex structures using MD simulations.

Fundamental quantities

• Total energy: strictly conserved but due to numerical errors it may

drift.

• Temperature: would be constant in a macroscopic system

(fluctuations are proportional to 1/N, hence negligible). But in a small

system, they will fluctuate around a base line. Temperature coupling

ensures that the base line does not drift from the set temperature.

• Pressure: similar to temperature, though fluctuations are much larger

compared to the set value of 1 atm.

• Volume (or density): fixed in the NVT ensemble but varies in NPT.

Therefore, it needs to be monitored during the initial equilibration to

make sure that the system has converged to the correct volume

(density). Relatively quick for globular proteins in water but could take

a long time (~1 ns) for systems involving lipids (membrane proteins).

Root Mean Square Deviation

For an N-atom system, variance and RMSD at time t are defined as

Where ri(0) are the reference coordinates. Usually they are taken from the

first frame in an MD simulation, though they can also be taken from the

PDB or a target structure. Because side chains in proteins are different,

it is common to use a restricted set of atoms such as C or the backbone

(N, C, C,O) atoms.

RMSD is very useful in monitoring the approach to equilibrium (typically,

signalled by the appearance of a broad shoulder).

It is a good practice to keep monitoring RMSD during production runs

to ensure that the system stays near equilibrium.

RMSD,)0()(1

)(1

2N

iii t

Nt rr

Evolution of the RMSD for the backbone atoms of ubiquitin

rr

rNrg waterion

24

)()(

Distribution functions

Pair distribution function gij(r) gives the probability of finding a pair of

atoms (i, j) at a distance r. It is obtained by sampling the distance rij in

MD simulations and placing each distance in an appropriate bin, [r, r+r].

Pair distribution functions are used in characterizing correlations between

pair of atoms, e.g., hydrogen bonds, cation-carbonyl oxygen. The peak

gives the average distance and the width, the strength of the interaction.

When the distribution is isotropic (e.g., ion-water in an electrolyte), one

samples all the atoms in the spherical shell, [r, r+r]. Thus to obtain the

radial distribution function (RDF), the sampled number needs to be

normalized by the volume ~4r2r (as well as density)

Radial distribution functions exhibit oscillations, where each maxima is

identified with a coordination number. They are obtained by integrating

g(r) between two neighbouring minima, e.g. denoting the first minimum

by rmin1, the first coordination number is given by

Similarly, the second coordination number can be found from

where rmin2 denotes the second minimum in the RDF.

1min

0

21 )(4

r

drrgrN

2min

1min

)(4 22

r

r

drrgrN

RDF for a Lennard-Jones liquid

Example: Binding of charybdotoxin to KcsA potassium channel

Important charge int’s:

K27 – Y78 (ABCD)

R34 – D80 (D)

R25 – D64, D80 (C)

K11 – D64 (B)

R34K11

K27

Pair distribution functions for charge interactions

10

RMSD of charybdotoxin (ChTX)

11

Configurational changes in ChTx

during umbrella sampling simul.

The N-terminal moves away

and the helix is distorted

(L16-V20 H-bond is broken)

Changes are induced by the

tidal forces arising from the

competition between the

applied harmonic forces

pulling the toxin out and the

Coulomb forces pulling it in.

Such simulation artefacts has

to be avoided using restraints.

Conformational analysis

In proteins, the bond lengths and bond angles are more or less fixed.

Thus we are mainly interested in conformations of the torsional angles.

As the shape of a protein is determined by the backbone atoms, the

torsional angles, (N−C) and (C−C), are of particular interest.

These are conveniently analyzed using the Ramachandran plots.

Conformational changes in a protein during MD simulations can be most

easily revealed by plotting these torsional angles as a function of time.

Also of interest are the torsional angles of the side chains, etc.

Each side chain has several energy minima (called rotamers), which are

separated by low-energy barriers (~ kT). Thus transitions between

different rotamers is readily achievable, and they could play functional

roles, especially for charged side chains.

-helix

~ 57o

~ 47o

(Sasisekharan)

2)0(

)()0()(

ttC

Bound atoms or groups of atoms fluctuate around a mean value.

Most of the high-frequency fluctuations (e.g. H atoms), do not directly

contribute to the protein function. Nevertheless they serve as “lubricant”

that enables large scale motions in proteins (e.g. domain motions) that

do play a significant role in their function. As mentioned earlier, large

scale motions occur in a ~ ms to s time scale, hence are beyond the

present MD simulations. (Normal mode analysis provides an alternative.)

But torsional fluctuations occur in the ns time domain, and can be studied

in MD simulations using time correlation functions, e.g.

These typically decay exponentially and such fluctuations can be

described using the Langevin equation.

Time correlation functions & transport coefficients

At room temperature, all the atoms in the simulation system are in a

constant motion characterized by their average kinetic energy: (3/2)kT.

Free atoms or molecules diffuse according to the relationship

Dtt2 6)( r

dttD

0

)()0(3

1 vv

While this relationship can be used to determine the diffusion coefficient,

more robust results can be obtained using correlation functions, e.g.,

D is related to the velocity autocorrelation function as

Similarly, conductance of charged particles is given by the Kubo formula

ii

iqdttVkT vJJJ

,)()0(3

1

0

(current)

Complex structure determination

Few proteins perform their function alone. In most cases, they form

complexes with other proteins to become functional, or binding of a

ligand to a protein triggers its function.

• Hemoglobin (oxygen carrying protein) is a complex of four proteins

• Ligand-gated ion channels are opened by binding of a ligand

• Most drugs act by binding to a receptor protein, moderating its

function

Structures of many biomolecules have been determined using x-ray

diffraction or nuclear magnetic resonance (NMR). However, only in a

handful of cases, structures of complexes have been determined

experimentally. Accurate determination of complex structures is

essential for understanding the function of proteins, and will have

many applications in pharmacology (rational drug design) and

biotechnology.

Basic methods for complex structure determination

1. Docking and scoring: The simplest and the fastest method. Typically

an analytical function is used to estimate the interaction energy of the

protein-ligand complex for various positions, orientations and

conformations of the ligand, and its minimum value is used to find the

binding pose of the ligand. Limited accuracy, especially in scoring.

2. Brownian dynamics (BD) simulations: Trajectory of the ligand is

followed in BD simulations (implicit water) until it binds to the protein.

Slower than docking and provides no apparent advantages.

3. MD simulations: Trajectory of the ligand is followed in MD simulations

until it binds to the protein. Accurate but too slow to be useful in

practice.

4. Docking combined with MD: Poses predicted by docking are refined in

MD simulations. Currently the most optimal method for this purpose.

AutoDock: The most popular docking method. Calculates the protein-

ligand interaction energies over a grid.

Scoring function: electrostatics + Lennard-Jones + hydrogen bond +

solvation + entropic term.

Protein is taken as rigid while ligand is allowed torsional flexibility.

Works well for small molecules (e.g. not good for > 30 AA peptide)

ZDOCK: Treats both protein and ligand as rigid, and hence can handle

much larger ligands. Searches for the binding position by optimizing

shape complementarity, electrostatics, and dehydration energies using a

Fast Fourier Transform (FFT) algorithm on a grid.

HADDOCK: Retains ligand flexibility and works for larger peptides. Much

more sophisticated (e.g. uses of exp. restraints, allows ensemble

docking, etc) but also much slower than the others.

Docking methods

Dynamical quantities – Free energy calculations

• Free energy of a simulation system

• Free energy difference between two states

• Calculation of free energy differences using free energy perturbation

(FEP) and thermodynamic integration (TI) methods; alchemical

transformation between two states

• Path dependent methods; potential of mean force along a reaction

coordinate; umbrella sampling with weighted histogram analysis

method (WHAM); steered MD with Jarzysnki’s equation.

• Calculation of binding free energies using path independent and path

dependent methods.

Free energy of a system

Free energy is the most important quantity that characterizes a dynamical

process. One can use either the Helmholtz or Gibbs free energy for this

purpose, which are given by

A = U – TS (Helmholtz)

G = H – TS = U + pV – TS (Gibbs)

In a typical biomolecular process, the volume and pressure do not

change, hence it is appropriate to use the Helmholtz free energy A.

Unfortunately, G has been used instead of A in the literature although the

calculated quantity is the Helmholtz free energy.

Calculation of the absolute free energies is difficult in MD simulations.

However, free energy differences can be estimated more easily and

several methods have been developed for this purpose.

Free energy perturbation (FEP)

The starting point for most approaches is Zwanzig’s perturbation formula for

the free energy difference between two states A and B, which are described

by the Hamiltonians HA and HB . Assuming that A and B differ by a small

perturbation, the free energy difference is given by:

The equality should hold if there is sufficient sampling.

However, if the two states are not similar enough, this is difficult to achieve

in finite simulations, and there will be a large hysteresis effect.

(i.e. the forward and backward results will be very different)

)()(

expln)(

expln)(

/)(

/)(

ABGBAG

kTGGABG

kTGGBAG

B

kTHHBA

A

kTHHAB

BA

AB

Derivation of the perturbation formula

From statistical mechanics, the Helmholtz free energy is given by

Free energy difference between two states A and B can be written as

Inserting in the in top integral and using the probability

A

kTHH

AkTHH

kTH

kTH

A

AB

AB

A

A

kT

dqdppqPkTBAG

dqdpe

epqP

/)(

/)(

/

/

expln

),(expln)(

),(

dqdpeNh

ZZkTG kTHN

/3 !

1,ln (Z: partition function)

dqdpe

dqdpekT

ZZ

kTGGBAGkTH

kTH

A

BAB

A

B

/

/

lnln)(

kTHkTH AA ee //

FEP with alchemical transformation

To obtain accurate results with the perturbation formula in finite time, the

energy difference between the states should be a few kT, which is not

satisfied for most biomolecular processes. To deal with this problem, one

introduces a hybrid Hamiltonian

and performs the transformation from A to B gradually by changing the

parameter from 0 to 1 in small steps. That is, one divides [0,1] into n

subintervals with {i, i = 0, n}, and for each i value, calculates the free

energy difference from the ensemble average

BA HHH )1()(

ikTHHkTG iiii /))]()((exp[ln)( 11

24

1

01)()10(

n

iiiGG

The total free energy change is then obtained by summing the contributions

from each subinterval

The number of subintervals is chosen such that the free energy change at

each step is < a few kT, otherwise the method may lose its validity.

Points to be aware of:

1.Most codes use equal subintervals for i. But the changes in Gi

are usually highly non-linear. One should try to choose i such

that Gi remains around a few kT for all values, e.g., for charge

interactions, using exponential spacing gives better results.

2.The simulation times (equilibration + production) have to be chosen

carefully. It is not possible to extend them in case of non-convergence

(have to start over).

25

1

0

)(

d

HG

Thermodynamic integration (TI)

Another way to obtain the free energy difference is to integrate the

derivative of the hybrid Hamiltonian H():

This integral is evaluated most efficiently using a Gaussian quadrature.

In typical calculations for ions, 7-point quadrature is found to be sufficient.

(But one should check that other quadratures give the same result)

The advantage of TI over FEP is that the production run can be extended

as long as necessary and the convergence of the free energy can be

monitored (when the cumulative G flattens, it has converged). 26

H

dqdpe

dqdpeH

ddG

kTH

kTH

/

/

Example : Binding of a K+ ion to a protein

Initial state (A): K+ ion in bulk, a water molecule at the binding site.

Final state (B): K+ ion at the binding site, water in place of the ion.

The free energy difference between A and B gives free energy gain in

translocating a K+ ion from bulk to the binding site of the protein. Note that

this is not the binding free energy – that includes entropic changes as well.

W K+

protein + K+ (bulk) protein + W (bulk)

A B

)()()( AGBGBAG

Example: Free energy change due to mutation of a ligand

A very common question is how a mutation in a ligand (or protein) changes

the free energy of the protein-ligand complex.

+

GA

Gbulk(AB) Gbs(AB)

GB

Thermodynamic cycle

)()( bulkbs BAGBAGGG AB

+

+ mutant

wild type

Standard binding free energy from FEP/TI methods

The total binding free energy of a ligand can be expressed as

The various sigma’s are the translational and rotational rmsd’s of the ligand

in the binding site. The last term is the interaction (or translocation) energy,

which is calculated using FEP or TI.29

siteres

bulkresres

rot

zyxtr

intresrottrb

GGG

ekTG

VV

ekTG

GGGGG

2321

2/3

30

0

2/3

8

)2(ln

1660,)2(

ln

Path dependent methods

Methods such as FEP and TI are independent of the physical path

connecting the two states. As seen in the previous example, calculation

of free energy difference for binding of a ligand involves annihilation of

the ligand in bulk and its creation in the binding site. Such calculations

can be done fairly accurately for small, uncharged ligands, but FEP/TI

methods fail for large or charged ligands. The reason is that the hydration

energies of such ligands are quite large and the errors committed in each

(bulk and binding site) calculation is usually larger than the free energy

difference between the two states.

To calculate the binding free energy of such ligands, one has to use path

dependent methods, where the ligand is physically moved from the

binding site to bulk along a reaction coordinate, and its free energy profile

along this path is calculated using various methods.

Potential of mean force (PMF)

Interaction of two molecules in solution is described by their PMF, W(r),

whose gradient gives the average force acting between them.

The PMF can be determined from MD simulations by sampling the relative

positions of the two molecules. Assuming one is at the origin, the density

of the second will be given by (r). From Boltzmann equation, the density

is given by

Here is a reference point in bulk where W is assumed to vanish.

Inverting the Boltzmann equation yields

Thus by sampling the density along a reaction coordinate, one can

determine the PMF between two molecules.

kTWWe /)]()([0

0)()( rrrr

)(

)(ln)()(

00 r

rrr

kTWW

0r

Umbrella sampling simulations

In general, position of a molecule cannot be adequately sampled at

high-energy points in finite simulations. To counter that, one introduces

harmonic potentials, which restrain the particle at desired points, and then

unbias its effect (this can be done easily for a harmonic biasing potential).

For convenience, one introduces umbrella potential at regular intervals

along a reaction coordinate (e.g. ~0.5 Å). After adequate sampling of each

window, the raw density data are unbiased, and the PMFs obtained from

all the windows are optimally combined using the Weighted Histogram

Analysis Method (WHAM).

For a simple introduction to the WHAM method and how it is implemented

in practice, see Alan Grossfield’s web page (search for “wham method” in

google and download the pdf file).

Points to consider in umbrella sampling

Two main parameters in umbrella sampling are the force constant, k

and the distance between windows, d. In bulk, the position of the

ligand will have a Gaussian distribution given by

The overlap between two Gaussian distributions separated by d

The parameters should be chosen such that the percentage overlap

satisfies, 10-20% > % overlap > 5%

If the overlap is too small, PMF will have discontinuities.

If it is too large, simulations are not very efficient.

)8/(1% derfoverlap

kTkzzezP Bzz /,,

21

)( 02)( 22

0

Standard binding free energy from PMF

Experimentally, the standard binding free energy is determined by

measuring the dissociation constant Kd (or IC50), which is defined as the

ligand concentration at which half the ligands are bound. The standard

binding free energy is related to the binding constant Keq = 1/Kd via

Where C0 is the standard concentration of 1 M (i.e., 1/ C0 = 1660 Å3)

The binding constant is determined by integrating the 3-D PMF over the

binding site

Because it virtually impossible to determine the PMF in a 3-D grid, it is

necessary to invoke a 1-D approximation to the PMF.

34

site

3)( rdeK kTWeq

r

eqdb KCkTKCkTG 00 lnln

1-D approximation of PMF

In most cases, there is a well-defined path from the binding site to bulk,

along which the PMF is determined. Thus the simplest approximation is to

perform the integral along this path assuming a flat-bottomed potential in the

transverse directions which will make a constant contribution. Taking the

path along the z-axis and assuming a cross sectional area of R2 for the

flat-bottomed potential, we obtain

The most sensible choice for R is to determine it from the transverse

fluctuations of the COM of the ligand in the binding pocket. In fact, this

choice can be rigorously justified using the simulation data without assuming

a flat-bottomed potential. 35

bulk

bs

)(2 dzeRK kTzWeq

Justification of the 1-D approximation of PMF

Distribution of the x and y components of the peptide COM follows a

Gaussian distribution in all of the PMF windows (Kv1.3-HsTx1 complex)

Justification of the 1-D approximation of PMF

Gaussian distribution in x-y directions:

Comparing with the Boltzmann distribution due to W(x,y)

shows that the transverse PMF can be represented by a harmonic pot.

Assuming W(x,y,z) = W(x,y) + W(z), the x-y integrals separate from z

1-D formula for Keq follows upon using 37

222 2/)(0),( yxeyx

TkyxW Beyx /),(0),(

222 with)(

21

),(Tk

kyxkyxW B

22)(),( 2222

kTk

dydxedydxe BTkyxkTkyxW BB

2222 2 yxR

For the validity of the 1-D

approximation, the average

transverse position of the

ligand and its fluctuations

should not deviate much from

the binding site.

B) The average transverse position of the ligand in each umbrella window.

C) The R values (fluctuations) in each umbrella window.

38

Justification of the 1-D approximation of PMF

Steered MD (SMD) simulations and Jarzynski’s equation

Steered MD is a more recent method where a harmonic force is applied

to the COM of a peptide and the reference point of this force is

pulled with a constant velocity v. It has been used to study binding of

ligands and unfolding of proteins. The discovery of Jarzynski’s equation

in 1997, enabled determination of PMF from SMD, which boosted its

applications.

kTWkTG

f

i

ee

tkW

//

0 )]([,.

vrrFdsFWork done by the harmonic force

Jarzynski’s equation:

This method seems to work well in simple systems and when G is large

but beware of its applications in complex systems!

0rr