DESCRIPCIÓN DE RESULTADOS

4.2

MD-Simulationen mit der modifizierten

Kontinuumsmethode

Es folgt nun ein Abdruck1 _{des Artikels}

Martina Stork and Paul Tavan:

„Electrostatics of proteins in dielectric solvent continua: II. First applications in molecular dynamics simulations.“ Journal of Chemical Physics 126, 165106/1-13 (2007),

der den Folgeartikel zu der in Abschnitt 4.1 abgedruckten Ver¨offentlichung darstellt. Hier wird der Einsatz der Kontinuumsmethode in MD-Simulationen mit dem Pro- gramm ego-mmii beschrieben. Nach einer Parametrisierung mit Hilfe des Prion-

Protein-Systems aus Abschnitt 2 wird die Methode am Alanin-Dipeptid getestet. Da- bei wird das effizienteExtended Lagrange-Verfahren zur Berechnung der RF-Dipole eingesetzt.

Electrostatics of proteins in dielectric solvent continua. II. First applications

in molecular dynamics simulations

Martina Stork and Paul Tavana兲

Theoretische Biophysik, Lehrstuhl für BioMolekulare Optik, Ludwig-Maximilians-Universität München, Oettingenstrasse 67, D-80538 München, Germany

共Received 13 November 2006; accepted 2 March 2007; published online 27 April 2007兲

In the preceding paper by Stork and Tavan,关J. Chem. Phys. 126, 165105共2007兲兴, the authors have reformulated an electrostatic theory which treats proteins surrounded by dielectric solvent continua and approximately solves the associated Poisson equation关B. Egwolf and P. Tavan, J. Chem. Phys.

118, 2039 共2003兲兴. The resulting solution comprises analytical expressions for the electrostatic reaction field共RF兲and potential, which are generated within the protein by the polarization of the surrounding continuum. Here the field and potential are represented in terms of Gaussian RF dipole densities localized at the protein atoms. Quite like in a polarizable force field, also the RF dipole at a given protein atom is induced by the partial charges and RF dipoles at the other atoms. Based on the reformulated theory, the authors have suggested expressions for the RF forces, which obey Newton’s third law. Previous continuum approaches, which were also built on solutions of the Poisson equation, used to violate the reactio principle required by this law, and thus were inapplicable to molecular dynamics共MD兲simulations. In this paper, the authors suggest a set of techniques by which one can surmount the few remaining hurdles still hampering the application of the theory to MD simulations of soluble proteins and peptides. These techniques comprise the treatment of the RF dipoles within an extended Lagrangian approach and the optimization of the atomic RF polarizabilities. Using the well-studied conformational dynamics of alanine dipeptide as the simplest example, the authors demonstrate the remarkable accuracy and efficiency of the resulting RF-MD approach. ©2007 American Institute of Physics.关DOI: 10.1063/1.2720389兴

I. INTRODUCTION

As outlined in the Introduction of the preceding paper, which we will refer from now on as “Paper I,”1much longer time scales could be covered by molecular dynamics共MD兲 simulations of biological macromolecules in solution if ac- curate and computationally tractable implicit solvent models were available. In such hybrid approaches, only the solute molecule is treated at atomic resolution, whereas the solvent is represented by a dielectric and possibly also ionic con- tinuum. Within corresponding MD simulations, one has to solve a partial differential equation 共PDE兲, like, e.g., the Poisson equation or the linearized Poisson-Boltzmann equa- tion, on the fly with the integration of the protein dynamics. If one tries to avoid this complicated task by resorting to the simple “generalized Born” approximation,2 one can cover impressive time scales3 with MD but one gets free energy landscapes, which drastically differ from those obtained with explicit solvent.4,5If one tries to tackle this task by standard numerical methods for solving PDEs, one is confronted with an intractable computational effort.6

In that respect it was fortunate that Egwolf and Tavan

共ET兲 succeeded in deriving, by means of analytical math- ematics, a conceptually different approach for solving the Poisson and linearized Poisson-Boltzmann equations,6,7 which led to highly efficient numerical algorithms. For a

given protein configuration R⬅共r1, . . . ,rN兲, the method

yields the electric fieldE共ri兲at the positionsriof the protein

atomsiand the total electrostatic energyW共R兲of the system. In addition to the Coulomb interactions among the protein charges, the total electrostatic field E and energy W also cover the so-called reaction field共RF兲 contributionsE␧and

W␧, respectively, which describe the effect of the continuum polarization on the protein charge distribution. As is well known, for each protein configurationR, the RF contribution

W␧共R兲to the total electrostatic energyW共R兲is, up to surface terms associated with the hydrophobic effect and with van der Waals interactions, the free energy of solvation.6 With respect to changes of that configuration, W␧共R兲 is thus a potential of mean force.1

According to the ET theory6,7 and its reformulation in Paper I,1which, for the sake of clarity, was restricted to the formally somewhat simpler Poisson case, the polarization of the dielectric continuum surrounding a protein can beexactly

represented by dipole densitiesP_iE共r兲localized at the protein atoms. Approximating these densities by Gaussian models with amplitudespiand widths␴ileads to a “linear response”

equation system for the amplitudes

pi= −␣iEpol q,p_共

ri兲, 共1兲

which has to be self-consistently solved, because the electric fieldE_polq,p polarizing the protein atoms through their RF po- larizabilities␣idepends not only on the chargesqj but also

on the amplitudespjof the Gausssian dipole densities at the

a兲_{Author to whom correspondence should be addressed. Electronic mail:} [email protected]

other atoms j⫽i 关cf. Eqs. 共50兲 and 共51兲 of Paper I兴. As explained in Paper I, the RF polarizabilities ␣_iare functions of the dielectric constants ␧s and ␧c chosen for the protein

volume and for the surrounding dielectric continuum, respec- tively共e.g., with the choices␧s= 1 and␧c= 80 for a protein in

water at room temperature and ambient pressure兲. The ␣i

furthermore depend on the Gaussian widths ␴i through so-

called effective atomic volumes vi, which also have to be

determined by a self-consistency iteration. This volume itera- tion is defined by vi共 n兲_=v i 共n−1兲

冋兺

j v共j n−1兲 G共rij;␴j兲

册

−1 , 共2兲

where G denotes a normal distribution and rij the distance

between the protein atomsiand j. Here, the Gaussian “over- laps” G共rij;␴j兲 measure the values of normal distributions

centered around positions rjat the position of atomi.

Inspection of Eq.共1兲immediately shows that the ampli- tudes pi, which are called “RF dipoles,” follow from the

electric fieldE_polq,pnearly in the same way as induced dipoles in a polarizable force field 共see, e.g., Ref. 8兲. The important difference is the negative sign, which reflects the fact that the atomic RF dipoles are images of the polarization within the surrounding continuum. Once the effective volumes vi and

the RF dipolespiare known, the RF contributionsE␧to the

field and the solvation free energyW␧can be approximately calculated from analytical formulas关cf. Eqs.共52兲and共60兲of Paper I兴.

Because the resulting approximate solvation free energy

W˜␧ is composed of pair interactions among the atomic charges and RF dipoles depending on the distance vectorsrij,

the corresponding mean RF forces −⵱r_iW˜␧ act both on

charged and uncharged protein atoms and obey Newton’s third law. However, this approach to the force computation leads neither to an efficient nor to a very accurate algorithm. In contrast, the resulting approximation E˜␧ for the electric field is quite accurate, can be evaluated much more effi- ciently, and can be used for a construction of RF forces com- patible with Newton’s third law.1Such a construction is nec- essary, because the simple idea of obtaining the mean RF forces by multiplying the atomic charges qi with the local

fields E˜␧共ri兲 yields instead the so-called qE forces, which

solely act on the charged atoms and violate Newton’sreactio

principle.9According to that principle, the surrounding con- tinuum exerts a “dielectric surface pressure” on the protein counterbalancing the qE forces. Thus, our construction of approximate mean RF forces关cf. Eqs.共70兲–共75兲of Paper I兴, which obey Newton’s third law and act both on charged and uncharged atoms, automatically accounts for the required counterbalance.

With the thus accomplished “marriage” between New- ton’s third law and RF forces obtained from an approximate but numerically efficient solution of the Poisson equation, the main hurdle toward MD simulations with implicit solvent appears to be surmounted. However, as outlined in Sec. III F of Paper I, a few technical points still need to be addressed before the viability of the approach can be evaluated. These points cover the following questions: 共i兲 How should one

include the required self-consistent evaluations of the effec- tive atomic volumes vi 关Eq. 共2兲兴 and of the RF dipoles pi 关Eq. 共1兲兴into the integration of the Newtonian dynamics of atomic motion?共ii兲How should one choose the widths␴iof

the Gaussian atomic dipole densities P˜_iE共r兲 for the various “atom types” occurring in a protein? Here, as is common in molecular mechanics 共MM兲force fields, the notion of atom type refers to an atom of given chemical identity covalently linked to specific other atoms in a well-defined stereochemi- cal arrangement. This question is important, because the RF polarizabilities ␣i of the atoms depend on the chosen ␴i

through the vi 关cf. Eqs. 共46兲–共48兲 of Paper I兴. Considering

the local character of the interactions among thevirevealed

by Eq.共2兲leads to the conclusion that the RF polarizabilities

␣i are shaped for given␴i by the respective local covalent

bonding patterns. Therefore, the ␴_i represent atom type pa- rameters of the method and one has to provide procedures for an optimal choice.

In the following, we will first address the algorithmic question共i兲and will subsequently characterize in Sec. III the methods employed for the optimization of ␴i. Because this

optimization will rely on comparisons with MD simulations of proteins and peptides embedded in explicit water, we will also sketch the corresponding simulation systems and tech- niques. Additionally, this section will explain the procedures employed in our testing scenario for the implicit solvent MD approach, which mainly deals with the free energy landscape of alanine dipeptide in solution.

II. EXTENDED LAGRANGE METHOD FOR UPDATING RF DIPOLES

Using Eq. 共2兲 the effective atomic volumes vi entering

the ET approach have to be self-consistently determined for each protein configuration R. Because this configuration changes at each MD integration step T 共T= 1 , 2 , . . .兲, thevi

have to be continuously updated. Experience has shown that the volume iteration 关Eq. 共2兲兴exhibits a rapid convergence and that thevivary only a little with the change fromR共T兲to

R共T+ 1兲. Therefore, only one iteration step suffices to keep the vivery close to their self-consistent values, if one uses

thevi共T兲resulting at integration stepTas starting values for

the volume computations at stepT+ 1. Furthermore, Eq.共2兲 shows that the viare determined by Gaussian overlaps be-

tween covalently bound or neighboring atoms. Due to the local character of the Gaussians, only a small set of neigh- boring atoms has to be considered for the computation of eachviand this algorithmic step can be easily integrated into

distance class schemes for the calculation of electrostatic forces 共like the ones employed in particle-mesh Ewald10 or in structure-adapted multipole11 methods兲. As a result, the computational effort required for the calculation of thevican

be kept very small and the atomic RF polarizabilities ␣iare

readily determined from the vi through Eqs. 共46兲–共48兲 of

Paper I.

In contrast, the iteration of the RF dipoles p_i, which is required by the self-consistent field 共SCF兲condition共1兲for each configuration R共T兲, exhibits a much slower conver- gence. Moreover, the differences兩p_i共T+ 1兲−p_i共T兲兩are not al- 165106-2 M. Stork and P. Tavan J. Chem. Phys.126, 165106共2007兲

ways small. Therefore, at each MD integration step, many iterations may be necessary to reach convergence, even if the results pi共T兲 of the preceding MD integration step are em-

ployed for the computation ofpi共T+ 1兲. The attempt of keep-

ing the RF dipoles at their SCF values is, thus, computation- ally quite expensive.

More importantly, however, the attempt of keeping the RF dipoles at their SCF values is hardly compatible with the physical meaning of these variables: According to the ET theory, the RF dipoles are images of the orientational polar- ization induced by the protein charges within a surrounding polar solvent. As is well known, the dielectric relaxation times of polar liquids range from a few hundred femtosec- onds up to several picoseconds.12 These times are much longer than the integration time steps of about 1 fs, which are usually employed in MD simulations for a smooth sam- pling of the very fast bond vibrations. Correspondingly, the orientational polarization of the liquid represents an average over the structures sampled by a protein on a femtosecond time scale and can respond only to much slower conforma- tional changes. RF dipoles, which by the SCF requirement are forced to follow the femtosecond protein motions, exhibit correspondingly fast fluctuations and, therefore, do not ad- equately model the more slowly fluctuating orientational sol- vent polarization.

For all these reasons, we have additionally selected a second approach to dynamically adjust the dipoles within MD, which is the extended Lagrangian 共EL兲 method.13 By adopting this method, the RF dipoles can be treated as dy- namical variables whose speed of adjustment can be steered by assigning suitable “masses.” The EL method is frequently used in polarizable force fields to update the variables de- scribing the polarization.14–19If these variables are inducible atomic dipoles共see, e.g., Ref. 16兲, the dipoles are allowed to fluctuate within a potentialU, whose minimum is marked by their SCF values.

In analogy to the latter case, we thus define the potential function U=

兺

i

再

p_i2 2␣i +piEpol q,p_共 ri兲

冎

, 共3兲

whose gradients are given by ⳵U

⳵pi

=pi

␣i

+E_polq,p共ri兲. 共4兲

Whenever the gradients vanish, the SCF condition共1兲is ful- filled. The equations of motions for the dipoles are then

mˆ_id 2_p i dt2 = −

冋

pi ␣i +E_polq,p共ri兲

册

, 共5兲

where themˆiare the above mentioned masses of the reaction

field dipoles. The mˆ_i have the dimension amu/e2_{. Because}

these masses merely determine how quickly the dipoles react to a change of the coordinates, they can be replaced by the global relaxation time ␶p=

冑

mˆi␣i, which should be chosen

such that it matches the average dielectric relaxation time of

The equations of motion 共5兲 can now be numerically integrated by the Verlet algorithm20

pi共t+⌬t兲= 2pi共t兲−pi共t−⌬t兲−关pi共t兲+␣iEpol q,p_共 ri,t兲兴 ⌬t2 ␶p 2 共6兲 in parallel with the integration of the atomic motion. The overall movement of the dipoles can be monitored by defin- ing a dipole temperatureTp. For this purpose, we define by

E_kin=␶p 2 2

兺

_i 1 ␣i冉 p_i共t兲−p_i共t−⌬t兲 ⌬t

冊

2 共7兲 the kinetic energy of the dipoles. Making use of the equipar- tition theorem

Ekin= f_p

2NkBTp, 共8兲

wherefp= 3 is the number of degrees of freedom per dipole

andk_Bis the Boltzmann constant, we obtain the dipole tem- perature T_p= ␶p 2 3NkB

兺

i 1 ␣i冉 pi共t兲−pi共t−⌬t兲 ⌬t

冊

2 . 共9兲

Due to the coupling of the atomic and dipolar motions, the associated dynamical subsystems will have, on the average, the same temperature. For temperature control, one can, thus, either couple the atoms or the dipoles or both to a heat bath using, e.g., a Berendsen thermostat.21 If one decides to couple also the dipoles to the heat bath, one may choose a coupling time constant ␶p,T different from the one selected

for the atoms 共␶T兲 to keep the system near the joint target

temperatureT₀.

The SCF and EL approaches outlined above have been implemented in our parallelized MD program EGO-MMII.22 The Appendix sketches the flow of computational steps char- acterizing this implementation. With such an implementation at hand, a MM/RF-MD simulation approach is, in principle, available. In practice, however, the question as to how one should choose the widths␴icharacterizing the Gaussian RF

dipole density remains to be answered关cf. question共ii兲fur-

In document Análisis musical de la “suite popular brasileña” de Heitor Villalobos : Un diálogo entre lo tradicional y lo clásico (página 43-48)