Stavros C. Farantos- Periodic orbits in biological molecules: Phase space structures and selectivity in alanine dipeptide

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Periodic orbits in biological molecules: Phase space structuresand selectivity in alanine dipeptide

Stavros C. Farantos Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, Iraklion 71110,Crete, Greece and Department of Chemistry, University of Crete, Iraklion 71110, Crete, Greece

Received 2 March 2007; accepted 19 March 2007; published online 1 May 2007

Small and large molecules may localize their energy in specic bonds or generally in vibrationalmodes for extended periods of time, an effect which may have dramatic consequences in reactiondynamics. Periodic orbits offer the means to identify phase space regions with localized motions.The author demonstrate that techniques to locate periodic orbits developed for small molecules canbe applied to large molecules such as alanine dipeptide. The widely used empirical force elds areemployed and principal families of periodic orbits associated with local-type motions and emanatedfrom the lowest energy minima and saddle points are investigated. Continuation of these families athigh energies unravels the stable and unstable regions of phase space as well as elementarybifurcations such as saddle nodes. ©2007 American Institute of Physics .DOI:10.1063/1.2727471

I. INTRODUCTION

Many body complex systems are studied by two differ-ent approaches. Either by using statistical mechanics meth-ods or by the systematic methods of nonlinear mechanics.1 Inthe latter case, models of complex dynamical systems areexplored by locating hierarchically classical mechanical sta-tionary objects, such as equilibrium pointsminima, maxima,and saddles of the potential function, periodic orbits andtheir bifurcations, tori, reduced dimension tori, as well asstable and unstable manifolds.2 These multidimensional sta-tionary objects reveal the structure of phase space and theyassist us to understand and elucidate nonlinear effects. The

progress of nonlinear mechanics in the last decades is im-mense and the mathematical theories and numerical tech-niques which have been developed are now powerful toolsfor the computer exploration of realistic systems.

Molecules are complex many-particle systems and theyare usually studied by quantum andsemiclassical mechani-cal theories. Chemical reactions involve the break and theformation of chemical bonds after the excitation of the mol-ecule at energies above potential barriers. The appearance of nonlinear phenomena, such as resonances and chaos, is in-evitable and such phenomena have been observedspectroscopically.3 Selectivity and specicity are well estab-lished concepts in elementary chemical reactions when the

role of mode excitation in the reactant molecules and theenergy disposal in the products are investigated. Triatomicmolecules have been used as prototypes to develop theoriesas well as to build sophisticated experimental apparatus tostudy elementary chemical reactions at the molecular leveland at the femtosecond time scales.4 The small number of degrees of freedom in these systems has allowed a detailedanalysis of the correspondence between quantum and classi-cal theories.5

Studying larger molecules such as biological ones, theapplication of systematic methods becomes a challenge,

since, not only more computer power is needed but also development of concepts and techniques to extract the phics from the calculations. It is not surprising that up to nstatistical mechanics methods have mainly been used, impmented either by averaging over phase space points or transition paths.6 The latter method is promising for studyinrare events in large dynamical systems. On the other hathe systematic approach to explore polyatomic moleculesusually exhausted by the location of equilibrium poiminima and saddles, to be followed with the calculation o

phase space averages.7

The hierarchical detailed exploration of the molecuphase space requires rst the location of the equilibriupoints of the potential function and then the location of riodic orbits POs, the tori around stable POs, stable andunstable manifolds for the unstable POs, and even transitstate objects such as the normally hyperbolic invariamanifolds.8 Such a program has been implemented up tnow to two andthree degrees of freedom models for triatomic molecules.9–11 This work has revealed the importancof periodic orbits in elucidating nonlinear effects in spectrcopy and the good correspondence between classical aquantum mechanics. They have also motivated the develment of semiclassical theories.

Efforts to nd localized motions in innite periodic

random anharmonic lattices have led to the concept of discrete breathers.12,13 The initial observations of localized motions in the work of Sievers and Takeno14 triggered the dis-covery of signicant mathematical theorems for texistence of periodic orbits in innite dimensional latticHowever, most of the potential functions employed in numerical studies were rather simple to describe realistic stems.

In this article, we apply the methods of locating POs twe have developed for small molecules to biological mecules, such as peptides described with empirical potent

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functions. Using the alanine dipeptide as a prototype system,we show how we can systematically trace regions in phasespace where the trajectories stay localized in specic vibra-tional modes of a minimum or of a transition state. Withcontinuation techniques we obtain families of periodic orbitsfor an extended energy range and we nd elementary bifur-cations such as saddle node and Hamiltonian-Hopf type.15,16

In this way, the road is opened for investigatinglocalization

phenomena and selectivity in biological systems.13,17

II. COMPUTATIONAL METHODS

To locate periodic orbits in a dynamical system isequivalent of nding the roots of the nonlinear equationswhich describe the return of the trajectory to its initial pointin phase space after the time periodT . If q1,q2, . . . ,q N arethe generalized coordinates of a dynamical system of N de-grees of freedom andp1, p2, . . . , p N their conjugate momenta,we dene the column vector

x = q ,p+

, 1where denotes the transpose matrix. Usingx we can writeHamilton’s equations of motion in the form

d x t

dt = J H x t 0 t T , 2

where H is the Hamiltonian function, andJ is a 2 N 2 N dimension matrix which is used to dene the symplecticsymmetry of a Hamiltonian system

J =0 N I N

− I N 0 N . 3

0 N and I N are the zero and unitN N matrices, respectively. J H x is a vector eld, and a matrixM which has the sym-plectic property satises the relationJMJ += M .

If x 0 denotes the initial conditions of a trajectory attime t =0, then this trajectory is periodic if it returns to itsinitial point in phase space after the timet =T

B x 0 ;T = x T − x 0 = 0. 4

Thus, to nd periodic solutions of periodT , it is necessary tosolve Eq. 2 subject to the two-point boundary conditions,Eq. 4 .

The roots of Eq.4 are usually found by iterative meth-ods, such as Newton-Raphson, which require the examina-tion of the time evolution of neighboring trajectories withrespect to a reference one. Taking the difference of two ini-tially neighboring trajectories in time, ˙ t =x t −x t andexpanding it as a Taylor series with respect tox t , the linearterms result in what is known asvariational equations VEs

˙ t = J 2 H x t t 0 t T , 5

where the second derivatives of the Hamiltonian with respectto coordinates and momenta of the reference trajectory areneeded, and thus, they depend on the curvature of the poten-tial function. The general solution of the linear equationEq.5 with time dependent coefcients can be expressed as

t = Z t 0 , 6

where 0 describes the initial displacement from the refeence trajectoryx, andZ is thefundamental matrix

Z t = x t x 0 . 7

The fundamental matrix is also a solution of the variatio

equations as can be seen by substituting Eq.6 into Eq. 5 .In other words,Z satises the equation

Z ˙ t = J 2 H x t Z t . 8

Obviously, att =0, Z is the unit 2 N 2 N dimension matrixand it can be integrated in time simultaneously with Hamton’s equations.

Solving the VEs helps us not only to locate POs but ato calculate the Lyapunov exponents which determine stability of a trajectory. Particularly, for a periodic orbitperiodT the fundamental matrix,M = Z T , is calledmono-dromy matrix from the eigenvalues of which we can detemine the stability of the trajectories around the periodic bit. An initial displacement 0 after k periods will become

kT = M k 0. Therefore, the eigenvalues of the monodrommatrix, , dictate the stability of the nearby trajectories the linearized system. Usually,is written as

= exp T . 9

For conservative Hamiltonian systems the eigenvalues of monodromy matrix appear as complex conjugate pa

, * , and one pair is always equal to 1.18 When all eigen-values lie on the unit complex circleare pure imaginarynumbers the PO is stable. If one pair of eigenvalues lies othe real axis and out of the unit circle the orbit is sing

unstable, if two pairs lie on the real axis the PO is calldoubly unstable and so on. For systems with larger than tdegrees of freedom it may happen four eigenvalues are outhe unit circle on the complex plane,, *, −1, * −1 . Inthis case we call the periodic orbit complex unstable.

Once we have located one member of the family of pe-riodic orbits we can use continuation techniques19 to ndtrajectories for different periodsT . This is done by usingT asthe control parameter. Usually, for small increments ofT linear extrapolation methods are sufcient. By varyingT andthus the energy, the eigenvalues of the monodromy matmove on the unit complex circle, collide, and may come of the unit circle rendering the PO unstable and vice ver

At every periodT for which one pair of eigenvalues becomeequal to 1, a bifurcation takes place and new POs aborn.20,21

Powerful existence theorems for POsRefs.22 and 23guarantee that the predicted bifurcations in the linearizsystem will also remain in the nonlinear system. From eaminimum of a molecular potential energy surface we expat leastN stable families of periodic orbits, which are callprincipal or fundamental. They are associated with theN normal modes of the molecule. At a saddle point, the normmodes with pure imaginary eigenvaluesin phase spacegive birth to principal families with unstable periodic orbThe number of unstable directions is equal to the rank

175101-2 Stavros C. Farantos J. Chem. Phys. 126 , 175101 2007




instability of the saddle point. Admittedly, the well devel-oped theory of periodic orbits and their bifurcations has con-verted the art of solving nonlinear equationsEq. 4 toscience.

We use multiple shooting techniques and the algorithmsand computer codes have been described in previouspublications.24 However, the challenge to extend these meth-ods to many degrees of freedom systems such asalanine dipeptide 2-acetamido- N -methylpropanamide,CH3CONHCHCH3 CONHCH3 , the molecule that we usein this study, and even larger biomolecules requires the adop-tion of new practices. We use Cartesian coordinates and em-pirical force elds to describe the forces among the atoms.We have adopted the molecular mechanics suite of programs,TINKER,25 to our computer code for locating periodic orbits,POMULT,26 in order to calculate the potential and its rstinHamilton’s equationsand second in variational equationsderivatives analytically.

III. RESULTS

Alanine dipeptide has served as a prototype molecule fortesting new algorithms in numerous studies in the past.27–29

We also use this molecule by employing the parameters of CHARMM27 for the force eld,30 Morse functions for the bondstretches,31 and harmonic potentials for the angles.

A. Equilibrium points

The hierarchical approach for studying the dynamics of this molecule starts with the location of minima and saddlepoints in the potential energy surface. For a 60 internal de-grees of freedom molecule such as alanine dipeptide, the

number of stationary points found is large. The lowest thminima are tabulated in TableI together with the saddlepoints among them. Energies and the distance of the tnitrogen atoms in the molecule are shown in this tablewell as the harmonic frequencies of two characteristic vibtional modes to be discussed and studied thoroughly beloFigure 1 depicts the minimum potential energy pathwayalong a generalized isomerization coordinate. They havebeen calculated with the method of Czerminski and Elbe27

implemented inTINKER. As we can see in Fig.1, the twolowest minima are separated by a small barrier of abo0.6 kcal/mol. To open the other reaction channels potentbarriers of about 6.5 kcal/mol should be surmounted. In tstudy we concentrate in the rst isomerization pathway awe show that, contrary to our expectation that the small brier will have negligible inuence in the dynamics, domaof phase space where trajectories are trapped for tens of coseconds even at high excitation energies can be tracfrom these stationary points. The geometries of the thequilibrium conformations of the dipeptide are shownFig.2.

For the isomerization reaction min 1↔ min 2 we ndthat the distance between the two nitrogen atomsd NN is amonotonic function of the reaction coordinateFig. 3 . InFig.4 we plot the minimum energy pathway as a functiond NN. The nitrogen-nitrogen distance varies by 0.57 Å in ttwo minima and it is used for assigning isomerization eveDuring this process the peptide folds and unfolds and ttime scales of such reactions are important in biology.

B. Periodic orbits

The method that we have proposed to discover domain phase space withde localized trajectories is by locatingfamilies of periodic orbits associated with equilibriupoints. In this article we investigate in detail the princifamilies emanating from the minima min1 and min2, and transition state, ts1. The principal families generated frothe minima are initially stable. However, because of the nlinearity they may turn to unstable at higher energies. Tprincipal families of POs are the natural extensions of harmonic normal modes, which are valid at energies close

TABLE I. Energies in kcal/mol, the distance of the two nitrogen atoms in Å,and the harmonic frequencies in cm−1 for the 23rd and 24th normal modesof stationary points in the potential energy surface of alanine dipeptide.

Energy N–N distance h.f. 23 h.f. 24

min1 −16.53 3.071 661.87 736.05min2 −15.59 3.641 668.65 704.07min3 −14.48 3.108 656.11 735.62ts1 −15.00 3.566 655.12 700.66ts2 −8.18 3.211 652.25 695.97ts3 −7.91 2.900 654.32 713.00

FIG. 1. Color onlineMinimum energy pathways connecting three minimaof alanine dipeptide along a generalized reaction coordinate.

FIG. 2. Color onlineThe geometries of the two minima and the transitiostate for the lowest energy isomerization reaction of alanine dipeptide. two squares drawn on the transition state enclose the atoms which exethe largest motions in thef 23 left and f 24 right periodic orbitssee text.Quenching the energy from congurations of thef 23 andf 24 periodic orbitsspecically leads to the minima min1 and min2, respectively. From leright the tubes correspond to the atoms of the chemical structCH3CONHCHCH3 CONHCH3.

175101-3 Periodic orbits in biological molecules J. Chem. Phys. 126 , 175101 2007




the minimum, to high energies where anharmonicity andcoupling of the degrees of freedom are unavoidable. Semi-classical quantization of these periodic orbits provides agood approximation to the overtone quantum states of themolecule as previous studies have shown.3,5,11 The periodicorbits which emerge from the transition state start as unstable

with the same rank of instability as the transition state. ts1has rank-1 instability.Among the 60 vibrational normal modes, we have cho-

sen to study the 23rd and 24th frequencies of which are alsogiven in TableI. The numbers used to assign the families of periodic orbits are the same as the enumeration of the har-monic normal modes by increasing frequency. Other princi-pal families have been calculated and presented in Ref.31.The 23rd and 24th normal modes have approximately local-ized motions that involve the atoms enclosed in the squaresof Fig.2. The NH and CO bonds oscillate in phase executingthe largest displacements. Our interest to these particular nor-mal modes came from their specicity. Starting with initial

congurations from these oscillations and minimizing the en-ergy we approach a specic minimum, thef 23 mode leads tomin1 and thef 24 to min2.

In Fig.5 the continuation/bifurcationC/B diagram forthe f 23 andf 24 families coming out from the three equilibriaof the moleculemin1, min2, and ts1is shown. The anhar-monic behavior of the vibrational modes is evident. For the f 24 families of min1 and the saddle point, ts1, an earlysaddle-node bifurcation is observed. This means that at aspecic energy the continuation line levels off, decreasing itsanharmonicity, and a new pair of families of periodic orbits

emerge, one of them with stable periodic orbits and the otwith unstable oneswe show the stable branch. The mecha-nism of appearance of such bifurcations has been describbefore.15 It is worth noting the higher frequency of the 24mode of min1 compared to the other two equilibrium poiof the potential function. After the appearance of the saddnode bifurcationfamily min 1− f 24−sn1 it was very dif-cult to continue this branch at higher energies. We expecacascade of saddle-node bifurcations as we go up in energ3

Plots of some representative periodic orbits are shown in F6. We project these POs in the plane of nitrogen-nitrogdistance and its relative velocity. The lines in the graph doubly drawn for the complete periodic orbits.

After locating a periodic orbit we carry out a linear sbility analysis to nd the eigenfrequencies and the eigenvtors of the monodromy matrix from which we can determnethe behavior of the trajectories in the near neighborhood15

Those POs which originate from the minima remain stabl

most of the degrees of freedom in the examined enerrange. However, at the energy of about −15.6 kcal/mol min1 and −14.1 kcal/mol for min2 we nd one quadrupleeigenvalues which come out of the unit complex circle. call this kind of instability as complex and we have examined it in the past32 with respect to the quantum mechanicaconsequences. The complex instability is associated wwhat is called Hamiltonian-Hopf bifurcation which leads tothe appearance of new POs and tori.16

At the saddle pointrank 1 we have always one pair of

FIG. 3. The variation of the nitrogen-nitrogen distance along the minimumenergy isomerization pathway.

FIG. 4. The minimum energy pathway connecting the two lowest minima of alanine dipeptide as a function of nitrogen-nitrogen distance.

FIG. 5. Continuation/bifurcation diagrams of the principal families of podic orbitsf 23 andf 24 originated from the equilibria min1 and min2, anthe saddle point, ts1.

FIG. 6. Plots of representative periodic orbits projected in the planenitrogen-nitrogen distance and its relative velocity.





real eigenvalues, thus the POs born are singly unstable. Forthe ts1− f 24 family and at the energy of about −13 kcal/molwe nd a saddle-node bifurcation. The new family emergedfrom the saddle-node bifurcation is also singly unstable andthe frequency continuous to decrease as the total energy in-creases Fig.5 . The calculated instability parameter is about

=1.2. From this we can deduce a characteristic time,−1

see Eq. 9 , for energy randomizationabout 0.3 ps, whichdetermines thetime scale for the molecule to develop statis-tical behavior.33,34 This time may be interpreted as a lowerbound of the lifetime of the molecule at a particular confor-mation.

In Fig.7 we depict the instability parameter for the pe-riodic orbits born at the saddle point. We can see that theinstability for the ts1− f 23 family decreases with energy andonly for positive energies starts increasing. ts1− f 24 showsmore complex behavior whereas the saddle-node family in-creases with energy.

C. Localization and selectivity

The location of periodic orbits and their continuation inenergy allow one to select trajectories from regions of phasespace that are associated with the normal modes of the mol-ecule. This method is free of approximations, such as normalform expansions of the Hamiltonian. Numerically exact pe-riodic orbits are located using the fully coupled anharmonicpotential energy surface. At a chosen energy we can sampletrajectories from the neighborhood of the periodic orbit tocalculate correlation functions or to study isomerization re-actions. For example, in Fig.8 we plot the NN distance as afunction of time for 1000 trajectories selected from a Gauss-

ian distribution centered at a PO of the saddle point, ts1− f 23, and at energy of −10 kcal/mol. The probability toreach min1 or min2 is about 1/2 with a lifetime distributionin the range of 0.5,2.5 ps. We propagate the trajectoriesforward and backwards in time, a technique which speeds upthe calculations by carrying the computations parallelly. Attimes longer than 6 ps the system starts having frequent jumps from one minimum to the other. Contrary to that, trajectories selected from periodic orbits of the same type,f 23,of the minima min1 and min2 stay localized for 40 ps as canbe seen in Fig.9.

We determine the regularand localizedor chaotic be-havior of the system by calculating autocorrelation functions.

By sampling 1000 trajectories from a Gaussian distributcentered on the periodic orbit we calculate the autocorretion function and from it the power spectra. Examples shown in Fig.10. The frequency of the highest peak in thpower spectra is that of the periodic orbit whereas the speaks is the result of the nonlinear coupling among the nmal modes. Complexity increases from min2 to the ts1.

Although for a few degrees of freedom systems we cvisualize the POs by projections on coordinate planes thinot practical with many degrees of freedom systems. Instethe motions of the atoms along the periodic orbit are bvisualized by using the graphics available for molecular mchanics. We have visually examined the motions of the oms for all families of POs at several energies. We conrmthat the f 23 and f 24 modes are mainly local-type motioninvolving the atoms enclosed in the squares of Fig.2, even athigh excitation energies. Furthermore, by minimizing the ergy starting from phase space points along the periodic

bits, we found that every point in the region of f 23 leads tomin1, whereas by quenching from the region of f 24 the sys-tem converges to min2.

IV. DISCUSSION

Localizationin complex systems is currently a subject ointense research.13,17 For example, energy localization andthe theory of breathers have been utilized to argue for existence of long, nonexponential excited state relaxationmyoglobin.17 In these studies the authors used simple mode

FIG. 7. The instability parameterfor the periodic orbits at the saddlepoint. FIG. 8. Color onlinePlots of the NN distance with time obtained from

trajectories sampled around a periodic orbit of thef 23 family of the saddlepoint at energy of −10 kcal/mol.

FIG. 9. Color onlinePlots of the NN distance with time obtained fromtrajectories sampled around a periodic orbit of thef 23 families of theminima min1 and min2 at energy of −10 kcal/mol.





to argue that localized states may be responsible for the ob-served long relaxation times. The present study unequivo-cally demonstrates the existence of stable periodic orbits for

substantial energy ranges in alanine dipeptide described withan empirical potential function. Such potentials are widelyused in simulations of biomolecules. However, the extentionto larger molecules and even the introduction of a solventlikewater are necessary to conrm the existence of local-typemotions under experimental conditions. The advantage of searching for stationary classical objects such as periodic or-bits and their bifurcations is the expected structural stability;in other words small perturbations either in the environmentor in the potential function will not introduce major topologi-cal changes but only small quantitative differences. Al-though, one has also to prove that localization remains inquantum calculations, the previous work on small molecules3

supports our expectations that such phenomena will remainin the quantum world.

We nd different time scales in the isomerization processdepending on the excitation of specic vibrational modes butfrom different conformations. In spite of exciting similarmodes in the three conformations their dynamics differ sub-stantially. Controlling chemical reactions at such a level isone of the goals of chemical dynamics. However, novel spec-troscopic methods have indeed appeared which study smallpeptides in subpicosecond time scale. In a recent investiga-tion of alanine tripeptide in water by two-dimensional vibra-tional spectroscopy conformational uctuations at the timescale of 0.1 ps have been reported.35

V. CONCLUSIONS

Families of periodic orbits associated with equilibriumpoints, the principal ones, of an empirical force eld poten-tial function for alanine dipeptide have been calculated withshooting and multiple shooting techniques and propagated inenergy by using analytical rst and second derivatives. Lin-ear stability analysis of the POs allows one to predict local-ized trajectories and upper bounds for isomerization rate con-stants. We have demonstrated that, with the periodicsolutions of the classical equations of motion, we can climbup to high energy regions of phase space and select system-

atically trajectories from specic anharmonic modes, whlead the molecule to specic conformations. The preswork has demonstrated that we can systematically expl

the dynamics of a small peptide. Currently, we study petides with ten aminoacids as well as the stability of exciconformations in a solutionlike water. The results will presented in future publications.

ACKNOWLEDGMENTS

The author is grateful to Dr. Reinhard Schinke and DSergy Grebenshchikov for their comments and stimulatidiscussions. Financial support from the Ministry of Edution and European Union in the frame of the progrPythagoras IIEPEAEK is kindly acknowledged.

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Stavros C. Farantos- Periodic orbits in biological molecules: Phase space structures and selectivity in alanine dipeptide

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