Quasi-harmonic analysis of mode coupling in fluctuating native proteins

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Phys. Biol. 7 (2010) 046006 (12pp) doi:10.1088/1478-3975/7/4/046006

Quasi-harmonic analysis of modecoupling in fluctuating native proteinsMert Gur and Burak Erman

Center for Computational Biology and Bioinformatics, Koc University, Istanbul 34450, Turkey

E-mail: berman@ku.edu.tr

Received 4 June 2010Accepted for publication 27 September 2010Published 26 October 2010Online at stacks.iop.org/PhysBio/7/046006

AbstractMode coupling and anharmonicity in a native fluctuating protein are investigated in modalspace by projecting the motion along the eigenvectors of the fluctuation correlation matrix.The probability distribution of mode fluctuations is expressed in terms of tensorial Hermitepolynomials. Molecular dynamics trajectories of Crambin are generated and used to evaluatethe terms of the polynomials and to obtain the modal energies. The energies of a few modesexhibit large deviations from the harmonic energy of kT/2 per mode, resulting from couplingto the surroundings, or to another specific mode or to several other modes. Slowest modeshave energies that are below that of the harmonic, and a few fast modes have energiessignificantly larger than the harmonic. Detailed analysis of the coupling of these modes toothers is presented in terms of the lowest order two-mode coupling terms. Finally, the effectsof mode coupling on conformational properties of the protein are investigated.

S Online supplementary data available from stacks.iop.org/PhysBio/7/046006/mmedia

1. Introduction

The expectation that the fluctuations of residues of a proteinshould be strongly coupled in order for the protein to performits function is leading to growing interest in protein physics(Hawkins and McLeish 2006, Juanico et al 2007b, Piazza andSanejouand 2008). Recently, several papers addressed thisissue and gave specific examples of mode coupling: Moritsuguet al (2000, 2003) were the first among those who studied modecoupling due to intramolecular vibrational energy transfer inmyoglobin at near zero temperature by molecular dynamicssimulations. Calculations were performed by adding a smallamount of energy to one mode and monitoring the transferof this energy into three other modes. The latter wereselected according to a Fermi-resonance-related argumentamong frequencies. Sanejouand and collaborators (Juanicoet al 2007a, Piazza and Sanejouand 2008, 2009) studiedthe consequences of energy transfer at room temperature bycooling specific residues at the surface of dimeric citratesynthase and observing the transfer of energy to differentmodes. They explained the localization of energy at specificresidues located in the stiff parts of the protein and itstransfer to other modes by employing an anharmonic potential.

The interplay between the surroundings of the protein andintraprotein dynamics has also been investigated (Moritsuguand Smith 2005, Reat et al 2000). Of particular relevanceto our work is the paper by Moritsugu and Smith in whichintra-protein dynamics is investigated at 300 K. The energeticinteractions of a protein with its surroundings and thedistribution of the energy absorbed by the protein to differentresidues are the major issue concerning protein function.

Interest in modal decomposition in protein dynamics is notnew. Decomposition of trajectories into an essential subspaceand a remaining Gaussian subspace has been employed widely(Ichiye and Karplus 1991, Horiuchi and Go 1991, Teeter andCase 1990, Perahia et al 1990, Dinola et al 1984, Karplusand Kushick 1981, Amadei et al 1993). In general, theinterest has been focused on retaining the essential subspaceof anharmonic modes for the analysis of correlated motionsand ignoring the remaining Gaussian subspace that consistsof the faster modes of motion. The interplay betweenharmonicity and anharmonicity in proteins was studied byHayward et al (1995), by employing molecular dynamicssimulations, reaching the conclusion that the motion withina local minimum is mainly harmonic and the anharmoniccomponent arises from transitions from one minimum to

1478-3975/10/046006+12$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

the other. Although it is generally true that the essentialsubspace of anharmonic modes are the ones that lead toimportant couplings between modes, higher modes also playimportant roles in protein fluctuation dynamics. The findingsof Moritsugu et al (2000, 2003) showing energy transferbetween modes 1589, 1860, 5858 out of a total of 7418modes at zero temperature of myoglobin are an example ofthe importance of faster modes. In the interest of observingthe effects of quasi-harmonic motions, we used the covarianceprojection technique (Amadei et al 1993) and analyzed thefluctuations of a small protein, Crambin at 273.15 and 310 K.The principal aim of this paper is to understand how modesin proteins are coupled with each other, whether they persistthroughout the full trajectory, how the energy of the proteinis distributed to these modes, and what the observableconsequences of such coupling are. Our calculations showthat, among the 1971 modes of the system, the slowest fewmodes are strongly coupled among themselves throughout thefull 40 ns trajectory. The time-averaged energies of the slowestmodes are always below the average energy of the protein.Strong and long-lasting couplings are also observed in thefaster modes. The time-averaged energies of the faster modesthat exhibit strong coupling are always larger than the averageenergy. The faster modes that exhibit coupling are few innumber and they appear and disappear at different stretchesof the trajectory. The coupling that we identified betweentwo of the modes suggests a distinct mechanism where theresidue fluctuations of one of the modes drive the motions ofthe other mode. The transfer of energy among different modes,including slow as well as fast modes, and the resulting affinityand conformation changes in a protein are of importance forunderstanding allostery (Hawkins and McLeish 2006).

In this paper, we present results for Crambin in water at273.15 K. The results for 310 K, which are similar to thosefor 273.15 K, are presented in the supplementary material(available at stacks.iop.org/PhysBio/7/046006/mmedia).

This paper is organized as follows: in section 2, wepresent the model and the simulations. In section 3, we firstevaluate the marginal energies of the modes in order to seewhich modes deviate strongly from the harmonic. Havingidentified these modes, we then use the lowest order two-mode coupling analysis for their detailed analysis. The paperends with section 4.

2. The model and simulations

2.1. Molecular dynamics simulations

Crambin, Protein Data bank code 1EJG.pdb, was selected asan example since it is a relatively small, 46 residue proteinand its dynamics is widely studied (Levitt et al 1985, Langeand Grubmuller 2006, Teeter and Case, 1990). All moleculardynamics simulations were performed for an N, V, T ensemblein explicit solvent (water) using NAMD 2.6 package withCHARMM27 force field (NAMD 2009). Two differentsimulations were performed at temperatures of 273.15 and310 K. The protein was solvated in a waterbox of 12 Acushion and periodic boundary conditions were applied. Ions

were added in order to represent a more typical biologicalenvironment. Langevin dynamics was used to control thesystems’ temperature and pressure. All atoms were coupledto the heat bath. A time step of 1 fs was used. Non-bondedand electrostatic forces were evaluated at each time step. Inorder to keep all degrees of freedom, no rigid bonds wereused. Three minimization-equilibration cycles were applied:the first one was applied under N, P, T conditions to relax thewater in the first place and the second and third ones wereapplied under N, V, T conditions to find a local minimum ofthe whole system’s energy (NAMD 2009). The energy ofthe initial system was first minimized for 20 000 steps. Thesystem was then equilibrated first by keeping the protein fixedfor the first 0.1 ns. At 273.15 K, it took 0.01 ns for the volumeto converge whereas at 310 K it took 0.02 ns for the volumeto converge. During the remaining time, volume fluctuatedaround 155 500 A3 at 273.15 K and around 159 000 A3 at310 K. Then, the protein was released stepwise by applyingharmonic constraining forces to every backbone atom of 1, 0.5and 0.25 kcal/(mol∗Angstrom2) in magnitude each for 0.05 ns.Finally, the simulation was performed for an additional 0.05ns without applying any force. Having finished the first cycle,the second minimization-equilibration cycle was performed;this time the protein was free to move. Again, 20 000 stepsof minimization were applied and the system was equilibratedfor 1 ns. After a final 20 000 steps of minimization, the proteinwas equilibrated for an additional 1 ns at both temperatures.

After equilibration, several stretches of the trajectory ofdifferent lengths were used for analysis. In the main part ofthe work, 2.75 ns long trajectories from different stretches ofa full 40 ns simulation were used. The first 2.75 ns stretch wastaken 10 ns after the final equilibration.

At every 500th time step of the 2.75 ns trajectory, theinstantaneous atomic coordinates R of all atoms, the velocities,the pressures and the energies were recorded. In orderto eliminate all the rotational and translational motions, allstructures were aligned with respect to the structure observedat a time instant of 1.5 ns of the production phase. Alignmentswere performed using the transformation matrix which showsthe best fit of the backbone atoms. All transformation matriceswere constructed via Tcl commands in VMD.

The 46-residue Crambin consists of 657 atoms. Hence, aset of 1971 modes are obtained. Then, the overall rotationaland translational motions were eliminated since they areirrelevant for the internal motions [13]. Thus, an overall of1965 non-zero modes were obtained.

2.2. Fluctuations, correlations and principal componentdecomposition

The fluctuations �R of atoms are defined by �R = R − R,where R are the mean atomic coordinates and hence are time-independent quantities, which define an average configurationobtained by the protein during the part of the trajectory thatwe use for calculations.

The covariance matrix C is defined in terms of �R as

C = 〈�R�RT 〉. (1)

2

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

Figure 1. The normalized histograms of the modal coordinates �r for the first 12 slow modes. Filled points are the calculated values andthe lines through them are evaluated using equation (14), up to the 17th-order terms.

Here, the angular brackets denote the average, taken over thetrajectory. The instantaneous fluctuations are transformed intomodal space using principal component decomposition withthe covariance matrix as (Amadei et al 1993, Yogurtcu et al2009)

�r = C−1/2�R, (2)

where �r are the dimensionless transformed fluctuations inmodal space. We let e represent the eigenvector matrixthat diagonalizes C, and λ represent the eigenvalues. Then,〈�R�RT 〉−1/2 = diagλ−1/2eT and the fluctuations �r arethe fluctuations in mode space spanned by the eigenvectors, e(Yogurtcu et al 2009). The components of the real trajectorythat correspond to a given mode are obtained simply bykeeping the eigenvalue of interest, equating all the others tozero, followed by a back transformation of equation (2).

In figure 1, the distributions W(�r) of the modalcoordinates �r are shown for the first 12 slowest modes. The

range of values taken by �r is divided into 25 stations, andthe normalized frequency of observing a given �r is shown inthe figures by the filled circles. The lines through the pointsare drawn using a 17th-order Hermite series expansion of thefluctuation probability distribution function (see equation (14)below). The distributions of the 1st, 2nd, 3rd, 4th, 8th and 12thmodes depart strongly from a harmonic distribution. Thesemodes mainly contain information about the anharmonicitiesin the system. The overall behavior is that the distributionsconverge to single peak Gaussians with increasing modenumbers.

2.3. Tensorial Hermite series approximation andthermodynamic analysis

In order to describe the behavior of the protein in its fullgenerality, we use a 3n-dimensional tensorial Hermite series,

3

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

where n is the number of residues. Originally, a three-dimensional Tensorial Hermite series was used by Flory andYoon to describe the statistical behavior of short polyethylenechains (Flory and Yoon 1974a, Yoon and Flory 1974).The method was then generalized and applied to proteinfluctuations by Yogurtcu et al (2009) in 3n-dimensional space.We adopt the notation of Yogurtcu et al and Flory and Yoon(Flory and Yoon 1974b, Yoon and Flory 1974, Yogurtcu et al2009).

In terms of the actual coordinates �R, the probabilityfunction W(�R) may be approximated by the tensorialHermite series (Yogurtcu et al 2009):

W (�R) = (2π)−N/2(det〈�R�RT 〉)−1/2

× exp

[−1

2�RT 〈�R�RT 〉−1�R

]

×[1 +

∞∑ν=3

(ν!)−1〈Hν〉 · Hν(〈�R�RT 〉−1/2�R)

].

(3)

Here, N is the number of nonzero modes after the eliminationof translational and rotational degrees of freedom. Theleading term of the distribution function is the Gaussian andthe terms in the last square brackets show the deviationsfrom the Gaussian, introduced as corrections in terms of theHermite polynomials. The expression W (�R) is valid forany amplitude of fluctuations. These correction terms becomeunimportant as the fluctuations become small. The first fewpolynomials, Hν , are

H1(�R) = �Ri

H2(�R) = �Ri�Rj − δij

H3(�R) = �Ri�Rj�Rk − (�Rδ)ijk

H4(�R) = (�R4 − �R2δ + δ2)ijkl

H5(�R) = (�R5 − �R3δ + �Rδ2)ijklm

H6(�R) = (�R6 − �R4δ + �R2δ2 − δ3)ijklmn,

(4)

where δij is the Kronecker delta, and (�Rδ)ijk in theexpression for H3 is a shorthand notation for �Riδjk+�Rj δik+�Rkδji , with similar expressions for the remaining termsin equation (3). For example, (�R2δ)ijkl = �Ri�Rj δkl +�Ri�Rkδjl + �Ri�Rlδkj + �Rj�Rkδil + �Rj�Rlδik +�Rk�Rlδij .

The expression given by equation (3) contains all theinformation on the behavior of the system in its full generality.However, being an infinite series, it is complicated, andbeing a moment-based expansion it may have problems ofslow convergence (Flory and Yoon 1974a, Yoon and Flory1974). If the higher moments of the system can be calculatedeasily, such as from a molecular dynamics trajectory, thenequation (3) may suitably be used to extract informationon the system at different levels of approximation. Thedeviations from harmonicity are due to two different sources:(i) anharmonicity of pure modes and (ii) coupling of twoor more modes. The first is the presence of higher orderterms �rm

i , where m > 2 of pure modes. The second is thecoupling of two modes represented by a nonzero value of theaverage

⟨�rm

i �rkj

⟩. Coupling of three different modes such

as⟨�rk

i �rlj�rm

k

⟩and coupling of more than three modes also

contribute to coupling. However, keeping track of such highorder coupling is a formidable task, and we will consideronly the lowest order coupling in this work. It is to benoted that all orders of the moments that appear in 〈Hν〉 inequation (3) can easily be evaluated from molecular dynamicstrajectories. The Hermite series are generally known as slowconverging series (Flory and Yoon 1974b, Yoon and Flory1974, Yogurtcu et al 2009). In this work, we adopt the Hermiteseries up to the 17th-order moments that ensures convergencewithin a few percent of simulation results. A general schemeand a computer script for easily computing tensorial Hermitepolynomials from molecular dynamics simulation trajectoriesup to any desired order will be available soon (Kabakcıogluet al 2010)

In terms of the modal coordinates, �r, where �r is thevector with elements {�r1,�r2, . . . .,�rN }, the probabilityfunction reads as

W(�r) = (2π)−N/2 exp

[−1

2�rT �r

]

·[

1 +∞∑

ν=3

(ν!)−1〈Hν〉 · Hν(�r)

], (5)

where the average Hermite polynomials are defined as

〈Hv〉 =∫ ∞

−∞Hv(�r) W(�r) d�r. (6)

The elements of 〈Hv〉 now contain products of modalcoordinates. For example, the third-order terms are now〈�ri�rj�rk〉 and are measures of the extent of modecoupling. The second-order modes are decoupled since〈�ri�rj 〉 = δij .

Equation (5) may be rewritten as

W(�r) = C exp

[−1

2�rT �r

+ ln

(1 +

∞∑ν=3

(ν!)−1 〈Hν〉 · Hν(�r)

)], (7)

where C = (2π)−N2 is the normalization constant. Writing

equation (7) as

W(�r) = Z−1 exp

[−1

2�rT �r

+ ln

(1 +

∞∑ν=3

(ν!)−1 〈Hν〉 · Hν(�r)

)− βE0

], (8)

where βE0 = − ln Z − ln C and comparing it with thenormalized distribution for a T, V, N ensemble

f (q) = e−βE(q)

Z(9)

lead to the energy of the system at a specific microstate havingmodal coordinates �r as

E(�r) = 1

β

[1

2�rT �r

− ln

(1 +

∞∑ν=3

(ν!)−1 〈Hν〉 · Hν(�r)

)]+ E0. (10)

4

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

Equation (10) can be considered in two parts: (i) the fluctuationpart, which depends on the microstate �r, and (ii) thereference energy Eo which depends on the free energy as

Eo = F + 12NkT ln(2π). (11)

Using equations (9) and (10), the thermodynamic energy, U ,is written as

U =∫

�rE(�r)f (�r) d�r

= kT

2N − kT

⟨ln

(1 +

∞∑ν=3

(ν!)−1〈Hν〉 · Hν(�r)

)⟩+ E0.

(12)

Substituting Eo = U − T S + 12NkT ln(2π) into equation (12)

leads to the following expression for the entropy:

S = k

2N(1 + ln(2π))

− k

⟨ln

(1 +

∞∑ν=3

(ν!)−1〈Hν〉 · Hν(�r)

)⟩. (13)

Thus correlations, depending on their signs, decrease orincrease the entropy with respect to that of the harmonic model.The leading term on the right-hand side of equation (12) is theenergy of the N modes in the uncoupled case. The second termis the contribution of anharmonicity and coupling. The thirdterm is the reference energy, which depends on the macro stateof the system as was defined in equation (11).

The marginal energy of a mode is defined as the energyobtained by considering only one mode, equating all the othermodal variables to zero. Equation (10) is now written as

Ei = E(0, 0, . . . ,�ri , . . . , 0, 0)

= 1

β

[1

2�r2

i − ln

(1 +

1

v!

∞∑v=0

〈Hv(�ri )〉 • Hv(�ri )

)]+E0.

(14)

Here, the Hermite series expansion in the parentheses may beexpressed as a vth-order polynomial of the modal coordinate�ri ,

1

v!

∞∑v=0

〈Hv(�ri )〉 • Hv(�ri ) = c1 + c2�ri

+ c3�r2i + · · · + cv�rv

i . (15)

Here, �rvi is the νth power of the ith modal coordinate only

and cv is a function of the various moments of the Hermitepolynomials. Since the reference energy E0 is constantthroughout all calculations, for simplicity it will be set to zeroin all calculations and figures.

2.4. The lowest order coupling of two modes

The results of MD simulations to be reported below showthat the energies of few of the modes deviate strongly fromthe harmonic resulting from coupling, obtained by the 17th-order Hermite series. A mode may be coupled to thesurrounding water, or to another mode, or to several othermodes simultaneously. We are specifically interested in the

coupling of two modes to each other. In order to understandthe details of coupling between two modes, we perform a first-order approximation keeping the third-order moments only,which read as

⟨�ri�r2

j

⟩and

⟨�r2

i �rj

⟩. In the lowest order,

coupling of three modes,⟨�ri�rj�rk

⟩where i and j and k

are different from each other, may also contribute to coupling,which we also discuss in some detail.

If both of the terms⟨�ri�r2

j

⟩or

⟨�r2

i �rj

⟩have the

same sign, the fluctuations of these two modes are positivelycorrelated. If they have different signs, fluctuations of modei and j are negatively correlated. In terms of energy, largevalues of

⟨�ri�r2

j

⟩and

⟨�r2

i �rj

⟩indicate the presence of

large energy coupling among them, but the type of coupling,i.e. whether i gains or loses energy while j loses energy andvice versa, cannot be concluded.

Expanding equation (10) up to the third-order terms andsetting all modes other than i and j to zero, the marginal energyof modes i and j together is obtained as

Ei+j = 1

β

[1

2�r2

i +1

2�r2

j − ln

(1 +

1

6

⟨�r3

i

⟩�ri (�r2

i − 3)

+1

6

⟨�r3

j

⟩�rj

(�r2

j − 3)

+1

2

⟨�ri�r2

j

⟩ [�ri�r2

j − �ri

]+

1

2

⟨�r2

i �rj

⟩ [�r2

i �rj − �rj

] )]. (16)

The marginal energy of the ith mode reads as

Ei = 1

β

[1

2�r2

i − ln

(1 +

1

6

⟨�r3

i

⟩�ri

(�r2

i − 3))]

. (17)

The energy coupling �Eij ≡ Ei+j − Ei − Ej among modesi and j at the microstate �r is obtained from equations (16)and (17) as

�Eij = − 1

βln

[1 + �i + �j + ij

(1 + �i)(1 + �j)

], (18)

where

�i = 16

⟨�r3

i

⟩�ri

(�r2

i − 3)

(19)

is the single mode anharmonicity term, and

ij = 12

⟨�ri�r2

j

⟩ (�ri�r2

j − �ri

)+ 1

2

⟨�r2

i �rj

⟩ (�r2

i �rj − �rj

)(20)

contains the joint distribution of modes i and j .We now expand equation (18) up to the second-order terms

in �i , �j , and ij

�Eij = − 1

β

[ij − 1

22

ij − (�i + �j

)ij − �i�j + · · ·

].

(21)

The function ij is of the third order in the modal coordinates.Thus, the first term on the right-hand side of equation (21)is of the third order. The remaining ones are all of the sixthorder. The energy coupling in the presence of the third-orderterm only reads as �Eij = −ij

β, or

�Eij = − 1

2β

[ ⟨�ri�r2

j

⟩ (�ri�r2

j − �ri

)

+1

2

⟨�r2

i �rj

⟩ (�r2

i �rj − �rj

) ]. (22)

5

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

Equation (22) is the lowest order contribution to modecoupling. It is a function of the third-order moments

⟨�r3

i

⟩,⟨

�r3j

⟩,⟨�ri�r2

j

⟩,⟨�r2

i �rj

⟩, as well as their microstate values

�r3i , �r3

j , �ri�r2j , �r2

i �rj , and of the linear terms �ri

and �rj . At a given time, if ij > 0, then �Eij < 0,indicating that the combined energy of modes i and j is lessthan the sum of the single mode energies. Similarly, whenij < 0, the coupling energy is positive, i.e. �Eij > 0.Thermodynamically, �Eij < 0 implies that the couplingof modes i and j gives off energy to the surroundings, and�Eij > 0 absorbs energy from the surroundings. Thus, thesign of ij can differentiate the case when two modes gain orlose energy simultaneously. Another possibility of couplingis obtained when one mode loses energy and the other onesimultaneously gains energy. This type of coupling cannotbe determined by the sign of ij . One should examine theabsolute energies Eiand Ej over the full trajectory for thispurpose.

The average value⟨ij

⟩of ij reads as

〈ij 〉 = 12

[⟨�ri�r2

j

⟩2+

⟨�r2

i �rj

⟩2](23)

and the energy coupling is

〈�Eij 〉 = − 1

2β

[⟨�ri�r2

j

⟩2+

⟨�r2

i �rj

⟩2]. (24)

Since 〈ij 〉 is always positive, equation (24) indicates thatin the third-order coupling,

⟨�Eij

⟩� 0, i.e. coupling of two

modes i and j gives off energy to the surroundings.The cumulative coupling ci of a given mode i to all of

the other modes is obtained by summing 〈ij 〉 over all othermodes j

ci =∑

j

〈ij 〉 = 1

2

∑j

[⟨�ri�r2

j

⟩2+

⟨�r2

i �rj

⟩2]. (25)

Comparing equations (24) and (25), we conclude that withinthe third-order approximation the cumulative coupling energyof mode i to the rest of the modes is always negative.

3. Results

In order to have a clear picture on the temperature dependenceof coupling effects around physiological temperatures,simulations were performed at 273.15 K and at 310 K. Here,we present results for 273.15 K. Full data for the 310 Kcase are presented in the supplementary material (availableat stacks.iop.org/PhysBio/7/046006/mmedia).

The RMSD plots for all non-hydrogen atoms with respectto their average are shown for 310 and 273.15 K in figure 2for a stretch of 2.75 ns of the trajectory. Results indicate thatan increase of about 40 K approximately doubles the RMSDvalues.

3.1. Marginal energies of modes as a function of time andmode index

The energy of each mode is evaluated using the marginalenergy formula via equation (14) up to the v = 17th order.As stated before, almost full convergence of the probability

0,0 0,5 1,0 1,5 2,0 2,5

0,5

1,0

1,5

RM

SD

(Å

2 )

Time (ns)

Figure 2. The RMSD values at 310 K (black solid line) and at273.15 K (gray solid line).

0 500 1000 1500 2000-0.2

-0.1

0.0

0.1<

Ei>

/ kT

Mode Index i

Figure 3. Energy 〈Ei〉 of each mode at 273.15 K relative to theharmonic energy kT /2 per mode. The mode index corresponds toan increasing frequency order. Mode 1 is the slowest mode whereasmode 1965 is the fastest mode.

function is obtained when 17 terms are taken. The meanmarginal energy of each mode is evaluated in the sameway by equation (12) and are shown in figure 3 for 273.15K. E0 is set to zero, and the results are presented relativeto kT /2, the harmonic component, as may be seen fromequation (12). The figure shows that the majority of themodes have the harmonic energy. However, a few of theslower modes exhibit a significant negative deviation from theharmonic energy. The largest ten negative deviations fromthe harmonic approximation in the decreasing order are formodes 1, 2, 8, 12, 3, 4, 18, 5, 6 and 10. In figure 1, the firstsix of them can be identified as highly unharmonic. Twoof the faster modes, modes 310 and 445, exhibit positivedeviation from the harmonic. In the following discussion, theharmonic component of the energy kT /2 will be subtractedand the corresponding modes will be named ‘positive modes’or ‘negative modes’ depending on the sign of the energy.

In figure 4, energies at different time intervals arepresented. Each of these stretches was aligned separately.

6

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

(a)

(b)

(c)

(d )

Figure 4. Energy 〈Ei〉 of each mode at 273.15 K relative to the harmonic energy kT /2 per mode at different stretches of the trajectory. (a)Stretch 1.25–4.25 ns, (b) stretch 5.75–11.25 ns, (c) stretch 8.25–11.5 ns and (d) stretch 32–38 ns.

After alignment principal component analysis was performedfor each stretch. Using the obtained modes �r, the energies〈Ei〉 which are depicted in figure 4 were obtained. One seesfrom figures 3 and 4 that modes with high positive couplingfall into four regions in the mode spectrum: (i) 310, (ii) 445,(iii) 792 and (iv) 1319. As the trajectory extends to longer andlonger times, new modes are not introduced, but the relativemagnitudes of the coupled modes vary.

The energy jumps observed in figures 3 and 4 result fromstrong jumps in modal coordinates. In figure 5, the third-ordermodal coordinates of the modes 310 and 445 are shown as afunction of time. Both of these modes make large-scale jumpsduring the simulation. The jumps take place during a shorttime interval as shown in the insets of the figures. However,a strong jump in the modal coordinate is not a necessarycondition for coupling between modes. We will show belowthat mode 310 is strongly coupled with mode 148 although themarginal energy of mode 148 does not deviate from the meanenergy.

3.2. Third-order moments and coupling of modal coordinates

The largest 500⟨�ri�r2

j

⟩values are shown for 237.15 K in

figure 6 in a scatter plot with axes ri , rj , rk , with j = k. Thelargest 500 〈�ri�rj�rk〉 terms make a total of 2346 points dueto the multiple presence of one type of coupling, i.e.

⟨�r1�r2

2

⟩occurs as 〈�r1�r2�r2〉 , 〈�r2�r1�r2〉 and 〈�r2�r2�r1〉.

Figure 5. Third-order moment �r3i of modes 310 and 445 at

273.15 K.

As can be clearly seen, the third-order couplings among thetop 500 are clustered in the slow mode regime. The largest750 000 third-order couplings 〈�ri�rj�rk〉 were determined.There are a total of 140 520 values of

⟨�ri�rj�rk

⟩when

multiplicities are removed. In figure 7, we present the absolutevalues of

⟨�ri�rj�rk

⟩in decreasing order as a function of

rank, where the latter goes from 1 to 140 520. It is clearly

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Phys. Biol. 7 (2010) 046006 M Gur and B Erman

0

50

100

0

50

100

0

50

100

ModeIndexj

ModeIndexk

Mode Index i

Figure 6. The scatter diagram of the largest 500⟨�ri�r2

j

⟩terms at

237.15 K. There are 2346 points due to the multiple presence of onetype of coupling.

100 101 102 103 104 105 10610-2

10-1

100

<Δr

iΔ rjΔr

k>

Rank

Figure 7. Distribution of⟨�ri�rj�rk

⟩terms at 237.15 K. Rank

goes from 1 to 140 520.

seen from this log–log plot that there is an extended power lawregion.

Figure 6 shows that the third-order coupling is confinedmostly to slower modes. As an example, we show here thethird-order coupling of mode 1 with other modes. The third-order coupling of mode 1 with all other modes j occurs intwo forms:

⟨�r2

1�rj

⟩and

⟨�r1�r2

j

⟩. In figure 8, both of these

coupling terms are shown as a function of index j . It canbe seen that the third-order coupling for mode 1 in the formof

⟨�r2

1�rj

⟩decreases with increasing mode number. Most

important couplings are observed with modes smaller than200. For the couplings of type

⟨�r1�r2

j

⟩, on the other hand, an

increase in the coupling terms with increasing mode numbersis observed after mode 100, which has its peak betweenmodes 200 and 250. Subsequent to mode 300, coupling againdecreases up to mode 400. A small but non-negligible couplingof the type

⟨�r1�r2

j

⟩is present for all modes j as can be seen

from the lower panel of figure 8.

Figure 8. Third-order coupling terms⟨�r2

1�rj

⟩and

⟨�r1�r2

j

⟩of the

first mode to other modes.

Figure 9. 〈�r1�r2�rk〉 /⟨�r1�r2

2

⟩as a function of the mode

index k.

In this paper, we have not gone into a detailed analysis ofthe coupling of three different modes, such as

⟨�ri�rj�rk

⟩which is also of the third order. Our calculations show thatmixed three mode terms are generally larger for slow modes.For example, one such term is 〈�r1�r2�r8〉 which is 50%of the correlation

⟨�r1�r2

2

⟩. This is shown in more detail in

figure 9, where the ratio 〈�r1�r2�rk〉 /⟨�r1�r2

2

⟩is presented

as a function of mode index k.

3.3. Time-averaged third-order coupling〈ij 〉The form of equation (23) shows that the values of 〈ij 〉 arepositive. The largest coupling terms 〈ij 〉 are among the slowmode pairs, among modes 1–31. The magnitude of couplingdecreases with increasing mode number. However, among thefaster modes, a strong coupling between modes 148 and 310exists. We compare the relative magnitude of the couplingsof these two modes with the couplings of the slowest mode infigure 10. In figure 10(a), the coupling energy 〈ij 〉 betweenmode i = 1 and all other modes is shown. Strong couplings are

8

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

mainly observed among mode 1 and other slow modes. Thisplot is essentially the sum of the two parts given by figure 8.In figures 10(b) and (c), the coupling of modes i = 148 and310 with all other modes is presented. The strong couplingbetween modes 148 and 310 is evident from an inspection offigures (b) and (c).

The second panel of figure 10 shows that mode 148couples strongly with mode 310, and the third panel shows thatmode 310 couples strongly with mode 148. Weaker couplingof these two modes to other modes also exists as can be seenfrom the figures.

In figure 11, we present the cumulative correlation ci ofmode i with all other modes, defined as ci = ∑

j 〈ij 〉.In this figure, the peak at mode 310 is visible. More

precisely, the peak consists of several closely spaced spikesaround mode 310. The second group of modes around 148can also be seen in the figure. Modes 148 and 310 are stronglycoupled as will be discussed in more detail below.

3.4. Third-order coupling ij as a function of time

There may be instances when the values of 〈ij 〉 become smalldue to the cancellation of positive and negative values alongthe trajectory when the average is taken. For this reason, wepresent the non-averaged values of ij as a function of time.Again, the slower modes exhibit pronounced coupling withsmall modes. In figure 12(a), for example, the coupling term1,2 among modes 1 and 2 is shown. Slow variation of positiveand negative couplings along the trajectory is apparent.Overall, the coupling is positive when the full trajectory isconsidered. Coupling shows positive characteristic in the first0.3 ns. After 0.3 ns, coupling switches between positive andnegative values. In figure 12(b), the coupling term 2,31

is depicted. These two modes are selected because theircoupling is strong compared to other slow modes. Couplingis strong for the first 0.29 ns and the period 0.86–1.17 ns.Other than these periods, an additional peak is observed at0.660 ns. In figure 12(c), the coupling term 148,310 is shown.An outstanding peak is observed at a time instant of 2.0375 ns.The maximum magnitudes of 1,2 and 2,31 are 1.5 and 3.5,respectively, whereas that of 148,310 is 18 as can be seen fromthe comparison of figures 12(a)–(c). The coupling amongmodes 1–2 and 2–31 is continuous and generally larger thanthe coupling among 148–310, except for the peaks. The strongcoupling among modes 148 and 310 in the period 2.0375–2.0450 ns is shown in the inset of figure 12(c). Additionalsmaller peaks outside these periods are observed, mainly at0.9975 ns and 0.4125 ns.

3.5. Coupling of conformations

The couplings discussed above are all in modal space. Here,we investigate the effects of the stated couplings on the three-dimensional conformations of the protein. The dot product ofthe eigenvectors of modes i and j gives the correlation of thepositional fluctuations of these modes since⟨

�Ri�RTj

⟩√

λiλj

= eieTj . (26)

(a)

(b)

(c)

Figure 10. (a) Time-averaged coupling of energy values⟨1j

⟩among the first mode and all other modes j , (b) among mode 148and all other modes j

⟨148j

⟩and (c) among mode 310 and all other

modes j ,⟨310j

⟩.

Here, �Ri is the fluctuation vector obtained by the backtransformation from the modal space by keeping only the ithmode. For simplicity, only the carbon alpha atoms will beconsidered.

9

Phys. Biol. 7 (2010) 046006 M Gur and B Erman

0 500 1000 1500 2000

0,8

1,2

1,6

2,0

310148Σ j<

Φi,j>

Mode Index i

Figure 11. Cumulative coupling of mode i to all other modes.

In figure 13(a), the correlations of the residues in mode310 e310eT

310 are shown. The most active residues in this modeare 3, 4, 31 and 36. Especially residue 31 is correlated toa wide range of other residues. The five largest negativelycorrelated residue pairs of mode 310 are CYS4-SER6, CYS3-SER6, CYS4-GLY31, ARG17-GLY31 and CYS3- ILE33.

CYS3 and CYS32 are connected together by two parallelhydrogen bonds, resulting in a strong interaction. Bothresidues GLY31 and ILE33, which are negatively correlatedto CYS3, are neighbor residues of CYS32. The constraintimposed by the CYS3-CYS32 bridge against the anticorrelatedfluctuations of both GLY31 and ILE33 is expected to storeenergy into the system, which is observed as the positivedeviation of the energy of mode 310 from the harmonic infigure 3.

In figure 13(b), the correlations of the residues in mode148 e148eT

148 are shown.The most active residue in this mode is GLY 37 and it is

anticorrelated with PRO5, LEU18 and ALA45.In figure 13(c), the correlations of the residues in modes

148 and 310 e148eT310 are shown. GLY 37 in mode 148 is the

key residue that is correlated with a large number of residuesfluctuating in mode 310 such as GLY37- ARG17, GLY37-VAL8 and GLY37-ILE33.

4. Discussion

A direct method of identifying mode coupling from moleculardynamics trajectories would be to project the trajectory ontothe eigenvectors obtained from principal component analysisand calculate the energies for each mode. If modes i andj are dependent, then, E (i + j) �= E (i) + E (j), whereE (i + j) is the energy calculated in the presence of the twomodes. Software packages such as NAMD can calculate theenergy of the system when the trajectory is given. Althoughthis approach would lead to the energies of the modesfor small fluctuations, such as that obtained at very lowtemperatures, it becomes unreliable when applied to proteinsaround physiological temperatures where fluctuations aredominant. For this reason, in this paper, we take an alternate

(a)

(b)

(c)

Figure 12. (a) Coupling term among modes 1 and 2, (b) couplingterm among modes 2 and 31 and (c) coupling term among modes148 and 310, all as a function of time.

route where we search for signs of coupling by expanding theprobability function of residue fluctuations into the tensorialHermite series expressed in principal components. This

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Phys. Biol. 7 (2010) 046006 M Gur and B Erman

(a) (b)

(c)

Figure 13. (a) Contour plot of e310eT310, (b) contour plot of e148eT

148 and (c) contour plot of e148eT310.

approach is valid for all fluctuation levels, and therefore acts asa good method of identifying coupling of all orders in modesat physiological temperatures. For clarity and brevity, weconcentrate mostly on the third-order coupling terms.

If the system were harmonic, the total energy would be3NkT/2. This energy would be distributed to the 3N modesequally, with an energy of kT/2 per mode. Our calculationsshow that the slowest modes have energies below the harmonic.Slow modes exhibit coupling with other slow modes. Few fastmodes have energies above the harmonic energy. It is thesemodes that exhibit strong coupling. When the third-ordercoupling term is plotted as a function of time, we see that thecoupling of fast modes is about an order of magnitude strongerthan that of the slow modes.

One of the modes, mode 310, results from the presenceof the CYS-CYS bridge that restricts the fluctuations of theprotein significantly. In mechanical terms, such a restrictionwould increase the strain energy of the protein, and this excessenergy should transfer to another mode. In this case, this isachieved by coupling to mode 148 via the residue GLY37. Thepresent approach makes it possible to identify the importantmodes and the residues that take part in these modes, andto estimate their contributions to the behavior of the nativeprotein.

It is interesting to note that the coupling of modes isnot active continuously throughout the full trajectory. Thecoupling of two modes is ephemeral; it persists for severalpicoseconds, and then disappears, but reappears at a later time

during the trajectory. However, it is to be noted that thecoupling involves only the same few modes throughout thetrajectory.

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