Module-3

Ab Initio Molecular Dynamics

April 08

How MD Simulations are Useful?

• Computing Time Dependent Propertiesas MD generates conformation as a function of time

Correlation Functions:C

xy

=1

M

MX

i

xiyi ⌘ hxiyii

Normalised correlation function:

c

xy

=1

M

PMi xiyi⇣

1

M

PMi x

2

i

⌘⇣1

M

PMi y

2

i

⌘ ⌘ hxiyiihx2

i i hy2i i

(this varies between -1 and +1)

• time correlation functionsC

xy

(t) = hx(t)y(0)i

This is called as cross correlation function if x and y are different quantities and auto correlation function if x

and y are the same quantities

• velocity auto correlation:

Cvv(t) =1

N

NX

I

DRI(t) · RI(0)

E

0

average over time

origins

cvv(t) =1

N

NX

I

DRI(t) · RI(0)

E

0DRI(0) · RI(0)

E

0

Normalized velocity auto correlation function

highly correlated:

I(⌫) =

Z 1

�1d⌧ exp(�2⇡i⌫⌧)cvv(⌧)

2840 2880in

tens

ity, I

freq. (cm-1)

less correlated (short memory!)

• vibrational spectrum can thus be obtained

• dipole moment time correlation data can be used for simulating IR spectrum

• Transport Properties

Self Diffusion constant:

D = limt!1

1N

PNI

D|RI(t)�RI(0)|2

E

2t

=

Z 1

0dt

1

N

NX

I

DRI(t) · RI(0)

EGreen-Kubo

Einstein3

• Constant temperature MD take a molecular system to equilibrium configurations (with highest Boltzman weight, )

PE

t

PE

r

• Computing Structural Properties

exp(��H)

R

P

PE

t trxn?

PE

rxn coordinate

P

R

Depends on the Boltzman weight near TS!

Timescale

F

ΔF‡

ΔF

ΔF‡

A → B B → A

1/⌧ = k ⇠ kBT

hexp

✓��F ‡

kBT

◆A B

A→B A→B

A→B

Barrier (kJ/mol) Timescale MD steps (1fs timestep)

2.5 0.1 ps 100

10 1 ps 1000

20 0.1 ns 106

60 1 ms 1012

100 8 hrs 1020

at T=300K

F (s) = �kBT lnP (s)

state 1

and unfolding rates are measured for a series ofmutants. The results are typically presented asf values (30). A f value of ~1 suggests that theinteractions formed by a residue in the nativestate are also present in the TSE, whereas a valueclose to zero indicates that the native-state inter-actions are not present in the TSE. Commonly,f values are calculated from simulation by ap-proximating the mutational free-energy changesfrom the fraction of native side-chain contactslost upon mutation (31). We used this approachto calculate f values for side chains and a free-energy perturbation approach (25) for the back-bone, and compared the results to experimentalmeasurements in a related WW domain (Fig.2E) (17). The values obtained confirm the ob-servation that the first hairpin is essentially fullyformed in the TSE, whereas the second hairpinonly makes a fraction of the contacts found in thenative state.

Although the agreement between simulationand experiment (Fig. 2E) is encouraging, it mayalso be somewhat fortuitous, as the number ofatomic contacts is only a rough approximationfor the free energy (32). We thus also calcu-lated f values directly from the folding andunfolding rates obtained from reversible fold-ing simulations of mutant proteins (33, 34) andcompared these results to f values calculatedby applying the contact approximation to theTSE for FiP35.

We chose to study the effects of six mutationsthat we expected to have different effects on thefolding and unfolding rates (Fig. 2E). Ser13 →Ala, located in the tip of the first hairpin, andArg11 → Ala, located in the central b strand, areexpected to have high f values; Tyr19→ Leu andPhe21→ Leu, both located in the central b strand,are expected to have intermediate f values; andLeu4 → Ala and Trp8 → Phe, located in thehydrophobic core, are expected to have low fvalues.

All six mutants folded reversibly to the na-tive state, albeit with different rates and stabil-ities (Table 1). For most mutants, the changesin stability upon mutation were in good agree-ment with experimental data. Most of the fvalues calculated from the folding and unfold-ing rates were in reasonable agreement with themagnitude calculated from the TSE of FiP35(Table 1), although our results support the no-tion that individual f values are best interpretedqualitatively (35, 36).

A notable exception is the Arg11 → Ala mu-tation, whose low f value calculated from thefolding kinetics (0.2) is substantially smaller thanthe high value expected from the contact ap-proximation (0.8). Although the reasons for thisdiscrepancy remain unclear, our results overallsupport the use of experimentally derived f valuesto infer the structural properties of the TSE, withthe caveat that the contact approximation may failin individual cases (35); all the experimental fvalues should thus be considered simultaneouslywhen inferring the overall structural properties for

a TSE (31). Finally, we note that all mutant pro-teins fold via the same overall pathway as FiP35,although some of the mutations appear to cause anoticeable Hammond-like shift in the structure ofthe TSE (fig. S4).

It is perhaps worth noting that these simu-lations may also provide a computational goldstandard for future studies exploring the accuracyand efficiency of methods for the prediction ofmutational free-energy differences and foldingrates.

FiP35 folds across a small free-energy bar-rier. We determined the free-energy profileand position-dependent diffusion constant alongthe optimized reaction coordinate (25). We foundthat the free-energy barrier for folding is small(1.6 kcal mol−1 or ~2kBT), consistent with thesuggestion that FiP35 is an incipient downhillfolder (12). The transition-state region is broadand flat (Fig. 3A), helping to explain the longcommitment time observed in the Pfold analysis.Langevin simulations on the free-energy profile(Fig. 3B) approximate well the folding dynamicsobserved in the MD simulations, and we arethus able to use this kinetic model to simulate atemperature-jump experiment (25). In addition tothe slow phase associated with folding, we ob-served a fast “molecular” phase whose amplitudeand time constant depend on both the size of thetemperature jump and the spectroscopic probeused (25). Such features are spectroscopic indi-cations of protein folding across a low free-

energy barrier, and they support the notion thatexperimental studies of fast-folding proteins mightbe used to probe directly the spectroscopic prop-erties of the TSE (37).

It has been argued that the fast molecularphase provides an estimate of the time scale fortransition paths in folding of FiP35 (37). Thevalue obtained in these experiments (≤0.7 ms) isin agreement with theoretical estimates (0.3 ms)as well as with the upper bound (200 ms) obtaineddirectly through single-molecule experiments (6).These values also agree with the average tran-sition path time observed in our equilibrium sim-ulations (0.4 T 0.1 ms). Thus, a range of differenttechniques (simulation, theory, ensemble, andsingle-molecule experiments) provide indepen-dent evidence for transition path times for proteinfolding on the order of 1 ms.

Native-state dynamics of BPTI. Dynamicchanges in protein structure typically occur notonly during but also after the folding process.The 58-residue protein BPTI was the subject ofthe first nuclear magnetic resonance (NMR)experiments of the internal motions of proteins(38). NMR studies showed that on time scalesranging from nanoseconds to milliseconds, sev-eral internal water molecules exchange with thebulk (39, 40), a number of aromatic rings rotate(38, 41), and a disulfide bridge isomerizes (42, 43).We used a 1-ms MD simulation at a temperatureof 300 K to reproduce and interpret the kineticsof folded BPTI.

Table 1. Computational f-value analysis of FiP35. In columns 2 and 3, the f values for six selectedmutants, calculated from the folding and unfolding rates obtained from reversible folding simulations,are compared with the values estimated from a contact approximation. In columns 4 and 5, the calculatedfree-energy changes upon mutation are compared to the values measured experimentally at the meltingtemperature of the hPin1 WW domain (49).

Mutation f Value DDGmut (kcal mol−1)MD Contact approx. MD (±SEM) Experiment

Leu4 → Ala −0.6 −0.1 0.5 (0.4) 1.5Trp8 → Phe −0.1 0.4 1.6 (0.4) 1.8Arg11 → Ala 0.2 0.8 1.8 (0.5) 1.7Ser13 → Ala 1.1 0.9 0.4 (0.5) n/aTyr19 → Leu 0.3 0.7 1.1 (0.4) 1.1Phe21 → Leu 0.4 0.5 2.4 (0.5) 1.4

Fig. 3. Folding kineticsacross a low energy barrier.(A) Free-energy profile alongan optimized reaction coor-dinate. The profile exhibitstwo minima, centered at 0.1and 0.7, corresponding to thefolded and unfolded basins,respectively. The folding andunfolding free-energy bar-riers are 2kBT and 3.5kBT,respectively. (B) Langevin sim-ulation of WW folding in aone-dimensional model. Thesimulation was based on the one-dimensional free-energy profile in (A) and a position-dependentdiffusion coefficient, both derived from the MD simulation data.

0 2Free energy (kBT) Time (µs)

0

0.2

0.4

0.6

0.8

Rea

ctio

n co

ordi

nate

0 20 40 60 80 100

A B

1 3

15 OCTOBER 2010 VOL 330 SCIENCE www.sciencemag.org344

RESEARCH ARTICLES

state 2

(s)

normalized histogram of s(t)

• Distribution functions can thus give an idea of most probable values of the coordinates (and thus the structure)

s

P(s)

P (s0) =1

Q

ZdNR dNP exp

⇥��H(RN ,PN

)

⇤�(s(RN

)� s0)

Q =

ZdNR dNP exp

⇥��H(RN ,PN

)

⇤

Z =

ZdNR exp

⇥��U(RN

)

⇤

) P (s0) =1

Z

ZdNR exp

⇥��U(RN

)

⇤�(s(RN

)� s0)

⌘ h�(s� s0)i

Radial Distribution Function g(r)

• Many body interactions→ equilibrium structure is complicated to understand looking at the snapshots

• Pair-Correlation function or distribution function (for spatial correlation in structural motions)

Definition: g(r) =(N � 1)

4⇡⇢r2h�(r � r0)i

count number of atoms within the thin shell

at a given distance r and average over atoms and

frames

r

normalizationg(r) = 1

for ideal density

Radial Distribution Function g(r)

D. ChandlerStat. Mech.

density 4 times higher than uniform

density

density close to the uniform density

long-range order

Importance of g(r) (contd.)

Bhargava, Balasubramanian and Klein, Chem Comm. 2008g(r) is related to

• structure factor (Fourier transform)

• thermodynamic properties

• Real gases: e.g. constant “a” in VdW EOS

Simulation of soft condensed matter systems

• Solvent effect & finite temperature effect

potential energy→free energy