Generalized gradient-augmented harmonic Fourier beads method

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THE JOURNAL OF CHEMICAL PHYSICS 127, 124901 共2007兲

Generalized gradient-augmented harmonic Fourier beads method with multiple atomic and/or center-of-mass positional restraints Ilja V. Khavrutskiia兲,b兲 and J. Andrew McCammona兲,c兲 Howard Hughes Medical Institute, University of California, San Diego, La Jolla, California 92093-0365, USA; Center for Theoretical Biological Physics, University of California, San Diego, La Jolla, California 92093-0365, USA; and Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, California 92093-0365, USA

共Received 8 June 2007; accepted 19 July 2007; published online 24 September 2007兲 We describe a generalization of the gradient-augmented harmonic Fourier beads method for finding minimum free-energy transition path ensembles and similarly minimum potential energy paths to allow positional restraints on the centers of mass of selected atoms. The generalized gradient-augmented harmonic Fourier beads 共ggaHFB兲 method further extends the scope of the HFB methodology to studying molecule transport across various mobile phases such as lipid membranes. Furthermore, the new implementation improves the applicability of the HFB method to studies of ligand binding, protein folding, and enzyme catalysis as well as modeling equilibrium pulling experiments. Like its predecessor, the ggaHFB method provides accurate energy profiles along the specified paths and in certain simple cases avoids the need for path optimization. The utility of the ggaHFB method is demonstrated with an application to the water permeation through a single-wall 共5,5兲 carbon nanotube with a diameter of 6.78 Å and length of 16.0 Å. We provide a simple rationale as to why water enters the hydrophobic nanotube and why it does so in pulses and in wire assembly. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2771172兴 I. INTRODUCTION

Important biological processes such as protein folding and assembly, enzyme catalysis, molecular transport through membranes and various channels, ligand binding to a target system, and many more are governed by the free-energy differences between the various states involved. From the thermodynamics point of view, only the free-energy differences between the initial 共reactant兲 and the final 共product兲 states are important as they determine the equilibrium distribution of the states. Thus, the thermodynamic point of view does not require specification of the path by which the system transforms from the initial to the final state, as the final freeenergy differences must be independent of the path. However, from the point of view of kinetics, these reaction paths become essential as they determine the efficiency and hence the rate with which the system can alternate between the specified states. In general, multiple pathways exist for any given transition and they contribute differently to the overall minimum free energy transition path ensemble, which includes the most probable transition path. Such a path provides the ultimate description of a particular transition from both thermodynamics and kinetics view. Experimental studies of the transition path ensembles are difficult due to the transient nature of the involved intermediates and transition states and, therefore, computational studies are often the only alternative. However, exploring the minimum free-energy transition path ensembles with coma兲

Authors to whom correspondence should be addressed. Electronic mail: [email protected] c兲 Electronic mail: [email protected] b兲

0021-9606/2007/127共12兲/124901/12/$23.00

puter simulations represents a considerable challenge.1,2 This task requires not only computing the free-energy gradient, which is challenging by itself, but also efficient means to search the free-energy space to locate the minimum free-energy transition path with the help of the computed gradient. Previously, one of us extended the formalism derived by Kaestner and Thiel for computing unbiased mean forces3 to multiple dimensions.4 This extension provided a practical way to compute multidimensional Cartesian mean forces for any N-atom system using molecular dynamics 共MD兲 or Monte Carlo 共MC兲 simulations with mass-weighted harmonic restraints on atomic positions 共APs兲. We then used this solution to augment the harmonic Fourier beads method for finding reaction or transition paths and path ensembles,5 establishing a superior, gradient-augmented harmonic Fourier beads 共gaHFB兲 method.4 In brief, the gaHFB method is a practical technique that provides complete steepest descent 共SD兲 paths on either potential energy or free-energy surfaces along with accurate, respective energy profiles as the system changes from one state 共reactant兲 to another state 共product兲. The SD path in the free-energy space is a useful generalization of Fukui’s intrinsic reaction coordinate approach to describe chemical reactions.6–9 Nevertheless, the original gaHFB method operated with AP restraints and thus was incapable of treating mobile collections of atoms or molecules such as those forming lipid membranes. Here we remove this limitation by further generalizing the gaHFB method to allow the use of multiple positional harmonic restraints and corresponding free-energy gradients of two kinds. Specifically, in addition to the AP restraints, we consider center-of-mass 共c.m.兲 restraints–total-

127, 124901-1

© 2007 American Institute of Physics

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J. Chem. Phys. 127, 124901 共2007兲

I. V. Khavrutskii and J. A. McCammon

mass-weighted harmonic restraints on the position of the c.m. of collections of atoms. The generalization allows multiple restraints of both kinds and their nonoverlapping combinations to be used in the context of the gaHFB method. Using the generalized positional gradients in the context of the HFB method yields the generalized gradientaugmented harmonic Fourier beads 共ggaHFB兲 method. The ggaHFB method further extends the scope of applicability of the HFB method family to simulations of various pulling experiments5 as well as transport of molecules through different media barriers such as lipid membranes and various channels. Advantageously, the new formulation in some cases bypasses the need for path ensemble optimization and, thus, allows constructing relevant free-energy profiles or potentials of mean force 共PMFs兲 immediately. This paper is organized as follows. Initially, we establish the ggaHFB methodology and then provide a simple test case of a single water molecule passing through an armchair 共5,5兲-type uncapped single-wall carbon nanotube, first, in gas phase and then in explicit water.

II. THEORY AND METHODOLOGY A. Mean forces under stationary state condition

Consider an N-atom system with 3N Cartesian coordinates described by a configuration vector ¯X = 共x1 , . . . , x3N兲. To define a PMF for this system, partition the system into the reaction coordinate space 共RCS兲 and spectator coordinate space 共SCS兲 by assigning each individual degree of freedom either to RCS or to SCS. Previously, we partitioned the system based on atoms, each having three Cartesian coordinates. More generally, for each atom, its three Cartesian coordinates could be considered independently, and further split across the RCS and SCS, providing additional flexibility for directional pulling experiments. For example, one can restrain only the z component of an atom in the RCS, leaving its x and y components unrestrained in the SCS. Thus, the partitioning defines the RCS configuration vector ¯x = 共x1 , . . . , xn兲 that is simply a subset n of the total 3N coor¯ 兲 that dedinates of the system. For a biasing potential Vb共x pends exclusively on the RCS configuration, the standard definition of the PMF for the biased ensemble obtained with MD or MC simulations follows:10–13 兰 exp共− 共U + Vb兲/kBT兲d⌫⬘ ¯ 兲 = − kBT ln F 共x . ¯ 兰 exp共− 共U + Vb兲/kBT兲d⌫⬘dx b

共1兲

Here, we express the complete 3N-dimensional configuration ¯ , using a shorthand notation dx ¯ space element d⌫ as d⌫⬘dx n = 兿 j=1dx j. The U is the potential energy of the system in the absence of the bias, kB is the Boltzmann constant, T is the MD simulation temperature, and superscript b indicates the bias. Note that in the numerator integral of Eq. 共1兲, we do not ¯ but over the complementary SCS integrate over the RCS dx d⌫⬘. The denominator integral is just a constant that does not depend on ¯x. The unbiased PMF is defined similarly to the biased one as

¯ 兲 = − kBT ln Fu共x

兰 exp共− U/kBT兲d⌫⬘ . ¯ 兰 exp共− U/kBT兲d⌫⬘dx

共2兲

In the unbiased PMF the denominator is also a constant that does not depend on ¯x, although it is different from that in the biased case. The corresponding biased and unbiased mean forces at the RCS configuration ¯x are as follows: ¯ 兲 兰共⳵U/⳵xi + ⳵Vb/⳵xi兲exp共− 共U + Vb兲/kBT兲d⌫⬘ ⳵Fb共x = ⳵xi 兰 exp共− 共U + Vb兲/kBT兲d⌫⬘ 共3兲 and ¯ 兲 兰共⳵U/⳵xi兲exp共− U/kBT兲d⌫⬘ ⳵Fu共x = . ⳵xi 兰 exp共− U/kBT兲d⌫⬘

共4兲

Because by construction the biasing potential Vb depends only on ¯x, in Eq. 共3兲 we can pull the exponential terms that involve the biasing potential outside the integrals, in both the numerator and the denominator. Canceling the exponential terms with the biasing potential Vb in front of the integrals out, we obtain ¯ 兲 ⳵Fu共x ¯ 兲 ⳵Vb共x ¯兲 ⳵Fb共x = + . ⳵xi ⳵xi ⳵xi

共5兲

As expected, the unbiased and biased mean forces are equal in the minimum of the biasing potential. This property might be useful in assessing if the configuration corresponds to the free-energy minimum and also can be used to estimate the unbiased mean force.3,4 Instead of using Eq. 共5兲 to derive the value of the unbiased mean force at some configuration ¯x in the RCS, consider the following quantity Ii.4 Ii = =

¯ 兲/⳵xi兲exp共− 共U + Vb共x ¯ 兲兲/kBT兲d⌫⬘dx ¯ 兰共⳵U/⳵xi + ⳵Vb共x ¯ 兲兲/kBT兲d⌫⬘dx ¯ 兰 exp共− 共U + Vb共x

冓 冔 冓 冔 ⳵U ⳵xi

+

b

¯兲 ⳵Vb共x ⳵xi

.

共6兲

b

The quantity Ii corresponds to the sum of the averaged forces of the potential energy and the biasing potential that simultaneously act in the biased system along the coordinate xi. Running biased MD or MC simulations provide estimates of these quantities for each coordinate xi of ¯x. Importantly, these quantities should identically equal zero for each coordinate xi at an equilibrium or stationary state 共i.e., Ii = 0兲 to ensure that the averaged biased configuration is stationary as well. Furthermore, the biased MD or MC simulation, unlike the unbiased one, should reach an equilibrium or stationary state relatively quickly after sampling only a small fraction of the configuration space due to restrictions from the applied bias. From the definitions of the biased and unbiased mean forces and their potentials, it can be shown that the property Ii can be alternatively expressed as follows:

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Ii =

J. Chem. Phys. 127, 124901 共2007兲

Generalized gradient-augmented harmonic Fourier beads

¯ 兲/⳵xi + ⳵Vb共x ¯ 兲/⳵xi兲exp共− 共Fu共x ¯ 兲 + Vb共x ¯ 兲兲/kBT兲dx ¯ 兰共⳵Fu共x = u ¯ b ¯ ¯ 兰 exp共− 共F 共x兲 + V 共x兲兲/kBT兲dx

Note that Eq. 共7兲 involves the biased average of the unbiased mean force and is isomorphous to Eq. 共6兲 expressed in terms of the full potential energy and the same biasing potential. The subscript ¯x , b indicates that the average is computed over ¯x using the biased simulation. Equation 共7兲 can also be derived directly from Eq. 共5兲 by taking the appropriate averages. Below we demonstrate how Eq. 共7兲 can be used to obtain the desired value of the unbiased mean force at a particular point in the configuration space.

冓 冔 冓 冔 ¯兲 ⳵Fu共x ⳵xi

¯x,b

n

n

b ¯兲 ⳵Vc.m. 共x b,ref b,ref b b = 2kc.m. 共xc.m. − xc.m. 兲 = 2kc.m. ␣ j共x j − xc.m. 兲 兺 ⳵xc.m. j=1 n

n

b,ref 2 b b ¯ 兲 = 兺 kAP,j 共x 共x j − xAP,j 兲 . VAP

共8兲

j=1

Another important biasing potential type is the restraint on the position of a center-of-mass of a particular collection of atoms, xi coordinates of which comprise ¯x—the c.m. restraint. As mentioned above, the c.m. restraint is very useful in dealing with mobile phases, for example, a lipid bilayer in which lipid molecules can move without altering the overall shape.

冉兺 n

b,ref ␣ j共x j − xc.m. 兲

j=1



i=1

冓 冔 b ¯兲 ⳵VAP 共x ⳵xi

¯x,b b,ref b = 2kAP,i 共具xi典¯x,b − xAP,i 兲

共10a兲

j=1

and

␣i =

mi 兺nj=1m j

,

共10b兲

such that n

␣ j = 1. 兺 j=1

共10c兲

The force component associated with the AP restraint is as follows: b ¯兲 ⳵VAP 共x b,ref b = 2kAP,i 共xi − xAP,i 兲. ⳵xi

Similarly, the force component for the c.m. restraint is

共11兲

共13兲

and by the same token



b ¯兲 ⳵Vc.m. 共x ⳵xc.m.



b,ref b = 2kc.m. 共具xc.m.典¯x,b − xc.m. 兲 ¯x,b n

=

b 2kc.m. n

=兺

共9兲

xc.m. = 兺 ␣ jx j

共12b兲

b,ref b = 2kAP,i 具共xi − xAP,i 兲典¯x,b

,

n

b ¯兲 ⳵Vc.m. 共x . ⳵xi

Using Eqs. 共11兲 and 共12兲 we find that the averages of the biasing forces over the biased ensemble equal the biasing ¯ 典¯x,b: forces at the corresponding averaged positions ¯x = 具x

2

where

共12a兲

It is useful to note that the force corresponding to moving the center of mass can be expressed as the sum of individual atomic forces

=兺

Let us now construct harmonic biasing potentials of the ¯ 兲 that could be used with either MD or MC simuform Vb共x lations in various applications. The first type of biasing potential that we use is the familiar restraint on the atomic positions in Cartesian space—the AP restraint.

共7兲

. ¯x,b

b ¯兲 ⳵Vc.m. 共x b,ref b = 2kc.m. ␣i 兺 ␣ j共x j − xc.m. 兲. ⳵xi j=1

B. Harmonic approximation of the potential of mean force

b,ref 2 b b b ¯ 兲 = kc.m. 共x 共xc.m. − xc.m. 兲 = kc.m. Vc.m.

¯兲 ⳵Vb共x ⳵xi

+

i=1



b,ref ␣ j共具x j典¯x,b − xc.m. 兲 兺 j=1 b ¯兲 ⳵Vc.m. 共x ⳵xi



.

共14兲

¯x,b

For practical reasons it is imperative that selections between different types of biasing potentials do not overlap, i.e., atoms that enter the c.m. restraint should not enter any AP restraints and vice versa. Up to this point the equations we have seen are exact from the statistical mechanical point of view. However, to compute the unbiased mean forces we have to make some approximations. The trivial fact that the averages of linear forces equal forces at the averaged positions provides a hint. Suppose that when we use the AP restraint, the PMF of the unrestrained ensemble at some configuration ¯x can be approximated by a quadratic form centered at another configuu,ref u : and with its own force constant vector ¯kAP ration, say, ¯xAP n

u ¯兲 共x FAP

u,ref 2 u u u ¯ 兲. ⬇ 兺 kAP,j 共x j − xAP,j 兲 + CAP = ˜FAP 共x

共15兲

j=1

Similarly, for the c.m. restraint we suppose that the corresponding unbiased PMF can be approximated as

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J. Chem. Phys. 127, 124901 共2007兲

I. V. Khavrutskii and J. A. McCammon

u,ref 2 u u u ¯ 兲 ⬇ kc.m. Fc.m. 共x 共xc.m. − xc.m. 兲 + Cc.m.



u = kc.m.

n

u,ref 兲 兺 ␣ j共x j − xc.m. j=1



2 u u ¯ 兲. + Cc.m. = ˜Fc.m. 共x

共16兲 Here, tilde indicates the harmonic approximation made. We emphasize that these quadratic forms match the quadratic forms of the corresponding restraints but have completely different force constants and biasing configuration centers. u u and kc.m. , the corresponding Although the force constants ¯kAP u,ref u,ref u biasing centers ¯xAP and xc.m. , and the offset values CAP and u Cc.m. that relate the unbiased PMFs to the biasing potentials are unknown, they provide straightforward means to carry out the required derivation and are trivially eliminated at the end. The mean forces derived from these quadratic PMF approximations are linear,14 just like the corresponding bias forces. u ¯兲 ¯兲 ⳵FAP 共x ⳵˜Fu 共x u,ref u,ref ⬇ 2kAP,i 共xi − xAP,i 兲 = AP , ⳵xi ⳵xi u ¯兲 ⳵Fc.m. 共x

⳵xi

n

u,ref u ⬇ 2kc.m. ␣i 兺 ␣ j共x j − xc.m. 兲=

⳵xc.m.

⳵xi

,

共18a兲

u,ref u,ref u u = 2kc.m. 共xc.m. − xc.m. 兲 = 2kc.m. 兲 兺 ␣ j共x j − xc.m. j=1

i=1

u ¯兲 ⳵˜Fc.m. 共x . ⳵xi

共18b兲

Now let us insert the approximations 共15兲 and 共17兲 or 共16兲 and 共18兲 for the unbiased PMF and corresponding mean force into the integrals of Eq. 共7兲. Regardless of the restraint type, we have a product of two multivariate Gaussians that is well known to reduce to a single multivariate Gaussian with a positive semidefinite quadratic form in the exponent.15 Such a Gaussian corresponds to a multivariate normal distribution. Hence, we can evaluate the integrals in Eq. 共7兲 analytically. We find that the coordinates averaged over the actual biased trajectory give us the approximate configuration for which we can compute the estimate of the unbiased mean force from the averaged bias force 共refer to Appendix for additional information兲:



u ¯兲 ⳵FAP 共x ⳵xi



¯x=具x ¯ 典b

⬇−



b ¯兲 ⳵VAP 共x ⳵xi



¯x=具x ¯ 典b

b,ref b = − 2kAP,i 共具xi典b − xAP,i 兲,



u ¯兲 ⳵Fc.m. 共x ⳵xc.m.



xc.m.=具xc.m.典b

⬇−



b ¯兲 ⳵Vc.m. 共x ⳵xc.m.

s=1

p+r

b ¯ t兲. Vc.m. 共x 兺 t=p+1

共21兲

The same methodology is applicable to the potential energy as was demonstrated previously and provides practically exact potential energy gradients.4 One has to simply replace the averages with the corresponding minimized structures in Eqs. 共19兲 and 共20兲 and the PMFs with the potential energies.4 In the following sections, we briefly review the gaHFB methodology for path optimization and computing the corresponding energy profiles.

1. The Fourier path

The family of the HFB methods optimizes the Fourier path,5,16 P

n

=兺

p

b ¯ s兲 + 共x = 兺 VAP

C. The overview of the gaHFB methodology

u ¯兲 ⳵˜Fc.m. 共x

j=1

n

¯ 1,x ¯ 2, . . . ,x ¯ p ;x ¯ p+1,x ¯ p+2, . . . ,x ¯ p+r兲 Vb共x

共17兲

and, consequently, similar to Eq. 共12b兲, u ¯兲 ⳵˜Fc.m. 共x

equations are well known. These approximations are useful and allow computing the unbiased mean force of the system at the average configuration by using simple averages from the biased simulations. In the ggaHFB method the biasing potentials of different types can be combined provided their RCS configuration basis vectors ¯x j do not overlap:



共19兲

xc.m.=具xc.m.典b

b,ref b 共具xc.m.典b − xc.m. 兲. = − 2kc.m.

共20兲

In this case, all the values on the right hand side of the above

xi共␣兲 = xi共0兲 + 共xi共1兲 − xi共0兲兲␣ + 兺 bin sin共n␲␣兲,

共22兲

n=1

that is an analytical function of a progress variable ␣ 苸 关0 ; 1兴, which defines a curve in multidimensional coordinate space between given reactant 共␣ = 0兲 and product 共␣ = 1兲 configurations. In Eq. 共22兲 xi is the ith component of the configuration vector ¯X = 共x1 , . . . , x3N兲, also called a bead, describing the positions of all N atoms in Cartesian space; , 3N 兵bin其i=1 n=1 , P are the Fourier amplitudes; and P is the series truncation index. In the case of ggaHFB, Eq. 共22兲 as well as equations in the following sections also apply to the components of the centers of mass of collections of atoms. For brevity, we do not make this connection explicit.

2. The path evolution

To optimize the path, the HFB method discretizes the curve into a finite set of beads and then evolves each bead k independently of all the other beads. The evolution simply entails running relatively short MD or MC simulations or energy optimization subject to the AP or c.m. restraints. For example, the AP restraint 共8兲 confines the absolute positions xi of only n degrees of freedom comprising the RCS to the b,ref = xi共␣ref kth reference configuration xAP,i k 兲. In general, ggaHFB employs restraints of the form 共21兲. Thus, for each reference bead k: ¯X共␣ref兲 = 共x 共␣ref兲, . . . ,x 共␣ref兲兲, 1 k 3N k k

共23兲

the evolution returns either the averaged bead

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J. Chem. Phys. 127, 124901 共2007兲

Generalized gradient-augmented harmonic Fourier beads

¯ 典 = 共具x 典 , . . . ,具x 典 兲 具X b,k 1 b,k 3N b,k

共24a兲

or the corresponding energy minimized bead ¯ 兴 = 共关x 兴 , . . . ,关x 兴 兲. 关X b,k 1 b,k 3N b,k

共24b兲

We call configurations 共24a兲 and 共24b兲 “raw evolved beads.” The subscript b , k indicates that the evolution is performed for the kth bead in the presence of the bias such as Eq. 共8兲 or in general Eq. 共21兲. 3. Gradient-augmented HFB optimization

With the energy gradients 共19兲 and 共20兲 we can now step some distance away from the raw evolved beads along either the full energy gradients or their components orthogonal to the path. The former is beneficial in cases where the endpoints are allowed to evolve or if the density of beads is small. To achieve the maximum accuracy in computing the energy gradients, we substitute the “as is” coordinates of the original reference and of the corresponding evolved beads in the analog of Eq. 共19兲 and 共20兲:





˜ ¯ ˜ ¯ ˜ 共具X ¯ 典 兲 = ⳵Fu共具X典b,k兲 , . . . , ⳵Fu共具X典b,k兲 . ⵜF u b,k ⳵具x1典b,k ⳵具xn典b,k

共25兲

To accurately compute the orthogonal component of the gradient to the evolved path, we, first, analytically compute its tangent following Fourier transform of the evolved beads: n共␣兲 = 共x1⬘共␣兲, . . . ,xn⬘共␣兲兲

共26兲

and then utilize standard projection technique:17 ˜ ¯ ¯ 典 兲 = ⵜF ˜ 共具X ¯ 典 兲 − n共␣ 兲 n共␣k兲 · ⵜFu共具X典b,k兲 . ⵜ⬜˜Fu共具X b,k u b,k k n共␣k兲 · n共␣k兲 共27兲 To step along the estimates of the mean force further away from the evolved beads, we use the SD-like approach. Note, however, that because we apply the step to the raw evolved bead and not the reference bead, this procedure corresponds to an enhanced SD method. Thus, the enhanced SD optimization step using either the full or the orthogonal component of the gradient is as follows: ¯XSD = 具X ¯ 典 + ␥ ⵜ ˜F 共具X ¯典 兲 b,k k u b,k k

共28兲

or ¯XSD = 具X ¯ 典 + ␥ ⵜ ˜F 共具X ¯ 典 兲. b,k k ⬜ u b,k ⬜k

共29兲

Here ␥k is the parameter that controls the step size for the kth bead and the superscript SD indicates that the configuration corresponds to the bead generated with the enhanced SD step. In the present paper we use the uniform step size parameter ␥ for all the beads and all the individual AP and c.m. restraints that comprise the generalized biasing potential 共21兲 for simplicity. The SD step provides beads substantially more evolved than in the original gradient-free HFB implementation where we used the raw evolved beads to generate the next path.5 Following the Fourier transform of the evolved beads to ob-

tain new sets of the amplitudes,5 redistribution of the beads along the evolved path provides new reference beads. These reference beads are realigned to maintain the coordinate system during the ggaHFB optimization. For this purpose a mass-weighted best-fit procedure in the RCS is invoked, treating the centers of mass as super atoms, that enforces the Eckart conditions on the beads.4,18–20 In cases where only few centers are available for the best fit or if their geometric arrangement breaks down the standard best fit procedure, simpler alignment methods could be used. If desired the realignment can be followed by another reparametrizationredistribution step to produce the final set of new reference beads. The described overall ggaHFB optimization procedure is iterated until convergence. Noteworthy, the HFB optimization provides a very useful general approach for a tough problem of finding saddle points or transition states14 because the harmonic biasing potential renders the modified energy surfaces along the path strongly convex, even in the vicinity of transition states. As the convergence rate of the gaHFB method to some extent depends on the employed bias force constant it is useful to devise an optimization strategy to achieve the fastest convergence possible.4 Although we have provided initial discussion of the enhanced convergence rate and simple optimization strategies,4 we feel this question would benefit from additional studies and the corresponding work is currently in progress in our laboratory. 4. Energy profiles from reversible work line integral with Fourier transform

Importantly, with the help of the free-energy gradients 共19兲 and 共20兲, ggaHFB method allows computing accurate estimates of the PMF along the Fourier path in multidimensional RCS. To this end we find it optimal to perform a Fourier transform of the computed energy gradients the exact same way as the corresponding evolved beads. With the continuous Fourier representation of the forces and of the path normal we can then trivially compute the corresponding reversible work along the path passing through the evolved beads as the generalized line integral of the second order in the RCS: n

˜F 共␣兲 = 兺 u i=1

冕冋 ␣

0



⳵˜Fu共␣兲 x ⬘共 ␣ 兲 d ␣ . ⳵xi i

共30兲

This procedure differs from a previously proposed method21 in that the global Fourier interpolation of both the path and the forces is performed prior to integration, thus providing the energy profile as an analytical function of the progress variable. Note that the analytical form of the energy profile and that of the corresponding path renders pinpointing the energy extrema and their accurate RCS coordinates particularly trivial. When used with potential energy gradient as opposed to the free-energy gradient this procedure should give exact potential energy profiles and thus provides a perfect opportunity for benchmarking the generalized gradient-augmented HFB method. Below we discuss some illustrative applications of this methodology.

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124901-6

J. Chem. Phys. 127, 124901 共2007兲

I. V. Khavrutskii and J. A. McCammon

FIG. 1. The armchair 16 Å long 共5,5兲 nanotube with water molecule in the center 共reactant兲. The three RCS basis vectors R1, R2, and W1 are labeled and the corresponding atoms are depicted in ball and stick.

III. COMPUTATIONAL DETAILS A. General setup

The coordinates for the uncapped 共5,5兲 armchair type single-wall carbon nanotube, hereafter called nanotube, were generated using the coordinate builder from Jaffe.22 The final nanotube comprised 14 rings of 10 carbon atoms each, with total of 140 atoms. Standard CA-type parameters with zero partial charge from the CHARMM22 共Refs. 23 and 24兲 force field were used for the carbon atoms of the nanotube, and CHARMM22 TIP3P 共CTIP3P兲 water model was used for water. Note that the CTIP3P model differs from the original TIP3P water model in that hydrogen atoms carry nonzero van der Waals 共vdW兲 parameters.25–27 Adiabatic optimization 共see below for details兲 of the nanotube in gas phase yielded predominantly 1.42 Å long C–C bonds away from the uncapped tips, and 1.39, 1.41, or 1.43 Å long C–C bonds at the tips and in their immediate vicinity. Considering the equilibrium C–C bond distance for the CA atom type of 1.375 Å, the optimized bonds hold significant strain energy, biasing the nanotube fluctuations toward more compact configurations. Based on the optimized C–C bond distance of 1.42 Å, the diameter of the nanotube in the gas phase should be 6.78 Å and its length should be 15.99 Å. Unless otherwise stated, all calculations in the gas phase employed nonbonded interactions with a 22 Å cutoff truncated by switching functions between 22 and 24 Å.

We then performed conventional gaHFB optimization combining the three basis vectors R1, R2, and W1 into the RCS for a single AP restraint, comprising 21 out of 143 total atoms. The gaHFB optimization employed P = 24, the force constant of 10.0 kcal g−1 Å−2 weighted by the corresponding atomic masses, and the step size parameter of ␥ = 0.004 Å2 kcal/ mol−1. We used the tandem of the SD and the adaptive basis Newton-Raphson optimizers with 200 and 1000 steps, respectively. To monitor the path convergence we simply computed RMSDs with respect to the original reactant and product for each reference bead. During the gaHFB optimization, we kept the product endpoint fixed at its initial coordinates, while allowing the reactant and all the intermediate beads to be optimized. It took only 40 gaHFB steps to converge the path to within 10−5 Å in any of the two RMSDs for any bead, with only minute changes observed at the reactant bead during optimization. For all practical purposes the initial path is as good as the final path, as can be seen from the corresponding potential energy profiles that agree to within 0.0005 kcal/ mol 共data not shown兲. The gaHFB-optimized path was used as the reference to benchmark the performance of the newly implemented ggaHFB method. In particular, we computed potential energy profiles along the reference path by employing different combinations of AP and c.m. restraints. More specifically, we used 共a兲 a single AP restraint as in gaHFB, with the RCS comprising R1, R2, and W2; 共b兲 three individual AP restraints for R1, R2, and W2 separately; 共c兲 two individual c.m. restraints for the R1 and R2 with an additional AP restraint on W1; and, finally, 共d兲 three individual c.m. restraints for the R1, R2, and W1. The ggaHFB profiles 共a兲 and 共b兲 must be identical to the gaHFB benchmark. Profiles 共c兲 and 共d兲 must at least be identical to each other and within numerical accuracy of the ggaHFB method to the gaHFB benchmark. We also compare the potential energy profiles computed with the ggaHFB method to the exact potential energies of the beads. This comparison is provided in terms of the relative energy RMSD over 128 quadrature points along the path.

B. Gas phase „T = 0 K…

To perform path optimization, we first prepared the reactant coordinates using the c.m.-restraints on the nanotube and water oxygen atom both centered at the origin 共0.00, 0.00, 0.00兲 with the force constant of 10 000 kcal mol−1 Å−2. The optimized coordinates were reoriented to align principal axes with the laboratory coordinate system. To generate the product coordinates we reoptimized the aligned reactant structure imposing three c.m. restraints on the positions of the centers of mass of the two outermost ten-atom rings 共R1 and R2, refer to Fig. 1兲 at their respective positions of 共−7.93, 0.00, 0.00兲 and 共7.93, 0.00, 0.00兲, and also the oxygen atom of water 共W1, see Fig. 1兲 at 共−16.50, 0.00, 0.00兲 with the same force constant. The initial path with water translating 16.5 Å away from the center of the nanotube along its principal symmetry axis was generated by linear interpolation between the endpoints using 32 beads. For the RCS, we defined three 共nonoverlapping兲 RCS basis vectors, namely, the two carbon rings R1 and R2 and the oxygen of water W1 共see Fig. 1兲.

C. Gas phase „T = 298…

Similar to the potential energy benchmark, we computed the free-energy benchmark in the gas phase. For this purpose, we employed Langevin dynamics 共LD兲 to evaluate the unbiased mean forces. Unless otherwise stated, SHAKE 共Ref. 28兲 constraints were imposed on all covalent bonds involving hydrogen atoms in all of the dynamics simulations. We obtained the minimum free-energy transition path ensemble by running the gaHFB optimization with step size parameter of 0.001 Å2 kcal/ mol−1. The product configuration was held fixed during the optimization. For all practical purposes, the initial path was as good as the final optimized path. Due to the noise in the unbiased free-energy forces computed using stochastic LD, the positions of individual beads experience minute perturbation on the order of 0.0005 Å in terms of RCS RMSDs during the gaHFB free-energy optimization. The PMF collection was performed in batches of 8 ns long runs for the total of 320 ns. Despite the great number of

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124901-7

Generalized gradient-augmented harmonic Fourier beads

steps, the final free-energy profile for the water translocation across the nanotube could only be converged to within 0.05 kcal/ mol for the free-energy differences between the endpoints, which hold the greatest errors. We used the gaHFB optimized path and the corresponding free-energy profile to benchmark the ggaHFB method. Because generating an accurate PMF is much more time consuming than computing the potential energy, we only used the case 共d兲 with the three c.m. restraints for this benchmark. D. Explicit water „T = 298…

Finally, encouraged by the ggaHFB performance in the gas phase, we performed a much more challenging PMF calculation for the water molecule permeation through the nanotube in explicit water using a truncated octahedron box of 2639 water molecules with periodic boundary conditions. This simulation was run in the NPT ensemble at T = 298 K and P = 1 atm using the Nose-Hoover thermostat with a thermal piston mass of 250 kcal/ mol ps2 to maintain the temperature.29,30 The pressure was maintained using the Langevin piston method, with the piston mass of 250 amu and Langevin collision frequency of 20 ps−1.31,32 For the explicit water simulations, we employ the particle mesh Ewald electrostatics33–35 with a real space electrostatic interaction cutoff of 12 Å. The vdW interactions employed the same cutoff distance but were truncated by switching functions between 12 and 14 Å. The beads from the gas phase optimized minimum free-energy path were solvated and prepared for the ggaHFB PMF collection via a 200 ps long multistep equilibration procedure. The truncated octahedron box of initial size of 48 Å equilibrated to a smaller size of 47 Å throughout all the beads. At the final 10 ps long step of this procedure, a single mass-weighted AP restraint was imposed on the RCS coordinates 关case 共a兲兴 with the force constant of 10.0 kcal/ g Å−2. The resulting restart files were used to initiate the ggaHFB PMF data collection with the three c.m. restraints as in case 共d兲. The minimum free-energy gas phase path was used to compute the reference positions of the respective c.m. restraints for the R1 and R2 as well as the W1 basis vectors. The force constants used were as follows: 160.0 kcal/ mol Å−2 for the W1 and 1200.0 kcal/ mol Å−2 for R1 and R2. Note that we did not attempt to refine the reference path by optimization in water because we did not anticipate significant changes in the nanotube length upon solvation. The ggaHFB method with the c.m. restraints naturally allows radial relaxation of the nanotube. The PMF data were collected in sets of 200 ps 共100 000 steps兲 saving the solute coordinates every 50 steps. These data were then combined to produce the cumulative PMFs in 200 ps increments. IV. WATER PERMEATION THROUGH THE SINGLE-WALLED CARBON NANOTUBE

Nanotubes have been widely used to study the behavior of water in confined environments22,36–59 and are considered a good model of such biological systems as aquaporins and

J. Chem. Phys. 127, 124901 共2007兲

other channels.2,42,60–65 Nanotubes are also the simplest models that provide an opportunity to study a variety of welldefined regular pores in a systematic way. Numerous studies of the nanotubes have already provided important insights into the behavior of water at the nanoscale.22,36,37,40,41,44–48,50–53,58,59,66–68 Among the most interesting observations made on nanotubes was that of the pulsating character of the nanotube filling with water or the wetting-dewetting transition.52 In particular, water permeates the nanotube concertedly by forming a chain of hydrogenbonded water molecules—the molecular wire. The water wires can have two opposing orientations and can translate through the nanotube in two possible directions. These and other properties of nanotubes continue to fuel interest in these systems. It has been suggested that hydrophobic pores with diameter less than 9.0 Å are completely impenetrable to water.66 However, one of the better studied nanotubes, namely, the 共6,6兲-type nanotube with the diameter of 8.14 Å was shown to be penetrable to water.52 The water permeation through the smaller size 共5,5兲 nanotube with the diameter of 6.78 Å remains somewhat controversial. In particular, some simulations find the diameter of the 共5,5兲 nanotube too small and hence the nanotube impenetrable to water,56 whereas others observe water permeation and measurable conduction rate.37 One must bear in mind that simulations of the same nanotube type by different authors often differ in the nanotube length, capping, force field parameters, and even the extent of flexibility of the nanotube. All of these factors are likely to affect the calculated free energies to some extent. Another aspect of simulations involving nanotubes is including the effects of polarizability. The polarizability has been found to substantially influence binding molecules with permanent dipole and/or charge at T = 0 K inside the typically fixed nanotubes.69 The importance of polarizability at higher temperatures and with fully flexible nanotubes is far from being understood. Thus, for the water-nanotube interactions, the polarization effects at T = 298 K have been found to be small.70 Using partial charges from quantum mechanical potentials creates a characteristic charge profile along the axis of the nanotube with prominent boundary effects. As expected distribution of water molecules inside the nanotube adjusts to the charge profile.71–74 The nanotube model used in the present study does not include polarizability effects. However, we stress that because the goal of this paper is to demonstrate the utility of the ggaHFB method that is independent of the potential used, we feel it would be appropriate to employ simple nonpolarizable model instead of computationally more expensive polarizable models. Therefore, the following discussion pertains to nonpolarizable nanotube models only. Based on the frequency of wetting-dewetting transitions, it has been inferred that the barrier height to water penetration from the bulk into the nanotube also increases with the decreasing diameter of the nanotube.37,41 In the case of the hydrogen-capped 14.2 Å long 共5,5兲 nanotube embedded into an engineered slab, the free-energy barrier for the water entering the nanotube was estimated to be 6 kcal/ mol.37 However, the computed values might provide only approximate

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J. Chem. Phys. 127, 124901 共2007兲

I. V. Khavrutskii and J. A. McCammon

barrier height because free MD simulations only seldom visit the high-energy barrier region thus providing only poor statistics in the region of interest. There are also state-based estimates of the free-energy difference between the empty and 共partly兲 filled states of the nanotube, which provide reasonable free-energy differences between water inside the nanotube and in the bulk. These free-energy differences have been shown to linearly depend on the diameter of the nanotube.36 For the 8 Å long 共5,5兲 nanotube fragment with an engineered mouth region and embedded in a slab, the linear relation gives the free-energy difference of 4.0 kcal/ mol in favor of the filled state. Importantly, in the free simulations, water conduction through the nanotube interior is bidirectional. Averaging of the permeation events in the two possible directions might present a problem as the time scale for the water reorientation along the single file water wire was found to be on the order of 2 ns at least in the 共6,6兲 nanotube. It is conceivable that with the smaller diameter of 共5,5兲 nanotube such reorientation would take even longer. Thus, one can anticipate difficulties converging the free-energy profiles for the water permeation in the nanotubes of subnanometer size. Despite the great deal of literature on the water permeation through various carbon nanotubes, certain aspects of the energetics of this process remain poorly characterized. Hence, we compute a complete PMF that would provide the free-energy difference between the water inside the carbon nanotube and out in the bulk along with the corresponding desolvation barriers. In the present study we use an uncapped 16.0 Å long 共5,5兲 nanotube as the model. We first study interaction of this nanotube with a single water molecule in the gas phase and then in explicit water. The free-energy profile is computed for the water translating from the center of the nanotube into the bulk water by 16.5 Å along the principal symmetry axis of the nanotube. Computing such a path and the corresponding PMF without the need of setting up a slab is made possible by the ggaHFB method. A. Gas phase

Gas phase calculations are extremely useful to establish that the ggaHFB methodology developed here is correct and able to provide accurate energy estimates in practice. The adiabatic potential energy profile in the gas phase can be computed exactly and provides direct information into the energetics of interaction between the nanotube and a single water molecule. The potential energy differences help explain why filling the nanotube with water is a favorable process not only in the gas phase but also in the bulk water. 1. Potential energy „T = 0 K…

The potential energy profile for the gaHFB-optimized transition path of water passing through the nanotube is depicted in Fig. 2. The potential energy difference between the endpoints that is independent of the path is provided in Table I together with the RMSD from the exact energies along the whole path. Note that the RMSD involves 128 grid points used to compute the work quadrature as opposed to just 32 bead points.

FIG. 2. The potential energy profile for the water translocation 16.5 Å away from the center of the nanotube in gas phase. The inset shows a closeup of the profiles at the region approaching the product. The legend is as follows: 共a兲 R1, R2, and W2 in a single AP restraint; 共b兲 R1, R2, and W2 separately in three individual AP restraints; 共c兲 R1 and R2 in two individual c.m. restraints, and W1 in a single AP restraint; 共d兲 three individual c.m. restraints for the R1, R2, and W1. The truncation parameter P was set to 24.

As is clearly seen from the potential energy profile the water molecule is stabilized in the center of the nanotube by at least 5.81 kcal/ mol. There is no barrier for the water entering the nanotube through the opening in the gas phase; however, the profile displays an inflection point near the entrance. The inflection point is likely due to constriction at the opening and the qualitative change in the mode of interaction of the water molecule with the carbon atoms of the nanotube. In particular, because the hydrogen atoms of the CHARMM22 TIP3P 共CTIP3P兲 water molecule have nonzero vdW parameters, the OH vectors orient themselves to point toward the nanotube opening, thus aligning the water dipole with the principal axis. Upon entering the nanotube, the OH vectors flip 90° to point perpendicular to the axis toward the nanotube walls. Inside the nanotube the CTIP3P water molecule is most comfortable slightly off center from the principal axis due to vdW repulsion between the hydrogen and the wall. 2. Binding energy for water clusters „H2O…n with n = 1, 2, and 3

Before we move on to the free-energy profiles in the gas phase, we consider interactions of the water molecules with TABLE I. The gas-phase potential and free-energy differences for the water in the center and 16.5 Å away from the center of the nanotube.

Simulation

⌬E 共kcal/mol兲

Adiabatic potential energy 共T = 0 K兲 Exact 5.81 gaHFB: benchmark 5.81 ggaHFB: 共a兲 5.81 ggaHFB: 共b兲 5.81 ggaHFB: 共c兲 5.81 ggaHFB: 共d兲 5.81 Potential of mean force 共T = 298 K兲 gaHFB: benchmark ggaHFB: 共d兲

5.60 5.57

ERMSD⫻ 103 共kcal/mol兲

0.00 5.42 5.42 5.42 7.58 8.09

na na

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124901-9

FIG. 3. A schematic representation of the potential energy calculations performed to compute binding energies of water clusters. Circles represent water and the elongated rectangle represents the nanotube. The corresponding energies are provided in kcal/mol.

the nanotube in the hope to provide an explanation as to why water fills the nanotube and why it does so in a pulsating manner. For this purpose, in addition to the single water molecule, we consider two and three water molecule clusters inside the nanotube. Nonbonded interactions were not truncated in these calculations. Although the total number of water molecules the model nanotube can accommodate is 6, we stop at 3 as this is sufficient to show the trend. We construct an energy decomposition ladder shown in Fig. 3 to assess the stabilization energies of water clusters inside the nanotube in gas phase. Specifically, we employ a supermolecule approach in which we pull the designated water molecules 16.5 Å away from the center of the nanotube along specified directions. For the case with a single water molecule we simply translate that water from the center to 16.5 Å away along the nanotube axis. As demonstrated above, this gives ⌬E10 = 5.8 kcal/ mol binding energy. For the cases with more than one water molecule we translate the two outermost water molecules in the opposite directions, each 16.5 Å away from the nanotube center also along its principal symmetry axis, leaving the residual water cluster in the center of the nanotube. Thus computed bistabilization energies for the dimer and trimer are ⌬E20 = 17.5 and ⌬E31 = 24.4 kcal/ mol, respectively. Now we can compute the total stabilization energy for the trimer ⌬E30, which is the sum of the binding energy for the single water molecule ⌬E10 and the trimer bistabilization energy ⌬E31. The corresponding values are summarized in Table II. We use the literature value of 9.5 kcal/ mol for the CTIP3P average water potential energy26,27,59 and consider on average four hydrogen bonds per water molecule. This approximation suggests that an average hydrogen bond costs TABLE II. Energetics of water clusters binding by the 16 Å long 共5,5兲 nanotube in the gas phase. 共We assume 9.5 kcal/ mol in average energy penalty for moving a single water molecule from the bulk into the gas phase, and 2.4 kcal/ mol for the average cost of a single hydrogen bond in the bulk. The cluster binding potential energies ⌬Eij are computed using a supermolecule approach as described in the text with 16.5 Å maximum separation of water from the center of the nanotube.兲 No. of Water i→ j 1→0 2→0 3→0

J. Chem. Phys. 127, 124901 共2007兲

Generalized gradient-augmented harmonic Fourier beads

No. of H bonds

⌬Eij 共kcal/mol兲

⌬Ebulk 共kcal/mol兲

⌬Ebulk-⌬Eij 共kcal/mol兲

0 1 2

5.8 17.5 30.2

9.5 16.6 23.7

3.7 −0.9 −6.5

FIG. 4. The free energy profile at T = 298 K for the water translocation 16.5 Å away from the center of the nanotube in gas phase. The inset shows a closeup of the profiles at the region approaching the product. Letter 共d兲 in the legend indicates ggaHFB setup with three individual c.m. restraints for the R1, R2, and W1, and time in nanoseconds corresponds to the cumulative time of the data collection in each bead. With these restraints the nanotube is free to spin about its principal axis. The truncation parameter P was set to 24.

2.4 kcal/ mol to break. As seen from Table II, a single water molecule inside the nanotube pays 9.5 kcal/ mol penalty and gains only 5.8 kcal/ mol. A water dimer pays 16.6 kcal/ mol penalty and gains 17.5 kcal/ mol. Water trimer pays 23.7 kcal/ mol and gains 30.2 kcal/ mol. Thus, it is seen that with this simple gas phase model it is more favorable to fill the nanotube with more than just one water molecule at once. For comparison, the water dimer energy is only 5.1 kcal/ mol. Interestingly, the hydrogen bonding along the wire grows stronger with more water molecules in it. These simple calculations assume that solvation energy of the nanotube is independent of its filling state, which is a crude approximation. Furthermore, computed binding and bistabilization energies do not include the entropic contribution, which will further reduce these values.75–78 Nevertheless, we feel such a discussion provides an intuitive explanation as to why water enters the nanotube and why it does so in a concerted/ pulsating fashion. 3. Free-energy „T = 298 K…

Unlike in the case with weighted histogram analysis 共WHAM兲 used to reconstruct the free-energy profiles from umbrella sampling simulations, we do not have to use overlapping windows with HFB method.3,4,79 Therefore, ggaHFB method allows us to cover a 16.5 Å long translation range with only 32 beads to provide an accurate PMF. Furthermore, the ggaHFB method, unlike WHAM, does not require an iterative procedure to calculate the PMF, which is an additional bonus. The PMFs for the optimized transition path ensemble of water permeating through the nanotube in the gas phase are depicted in Fig. 4. In particular, Fig. 4 shows the benchmark gaHFB profiles, as well as profiles for case 共d兲 ggaHFB at 160 and 320 ns long simulations. The free-energy difference between the endpoints is also provided in Table I. These free-energy profiles resemble the potential energy profiles. Water is again stabilized in the center of the nanotube by at least 5.6 kcal/ mol. It is difficult to converge the PMF to within 0.01 kcal/ mol accuracy, hence we omit the second

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J. Chem. Phys. 127, 124901 共2007兲

I. V. Khavrutskii and J. A. McCammon

TABLE III. The cumulative PMF extrema for the water molecule permeation through the nanotube in the explicit water. 共The cumulative PMF 共run I + II兲 was computed with P = 31 using 32 beads, uniformly spanning the 16.5 Å distance R from the center of the nanotube, along its principal axis. The data for both runs I and II was collected for 10.0 ns using the force constants for the R1, R2, and W1 c.m. restraints as described in the text. The progress variable ␣ and the distance R are linearly related. TS stands for transition states, Min for the minima, and End for the fixed product configuration. The PMFs have been shifted up by 0.22 kcal/ mol.兲



R 共Å兲

PMF 共kcal/mol兲

0.000 0.084 0.168 0.252 0.327 0.413 0.718 0.818 0.924 0.977 1.000

0.0 1.4 2.8 4.2 5.4 6.8 11.9 13.5 15.2 16.1 16.5

0.22± 0.00 0.00± 0.03 0.27± 0.04 0.07± 0.01 0.35± 0.05 0.03± 0.07 3.38± 0.06 1.92± 0.09 2.19± 0.05 2.12± 0.03 2.13± 0.03

Extremum

FIG. 5. The free-energy profile at T = 298 K for the water translocation 16.5 Å away from the center of the nanotube along its principal axis in explicit water. The ggaHFB setup for this calculation employed three individual c.m. restraints for the R1, R2, and W1 RCS basis vectors 共case d兲 and truncation parameter P = 31. Runs I and II are the PMF profiles from two independent collection data sets with 10 ns long runs in each bead, and the run I + II is the combined PMF profile. The nanotube is free to rotate about its principal axis during data collection.

digit. Thus the contribution from the configurational entropy to the free-energy difference between the endpoints is only 0.2 kcal/ mol. The decrease is likely due to the averaging over the water rotational degrees of freedom that introduce configurations with the OH vectors pointing away from the nanotube at the product endpoint, thus contributing reduced interaction energy and forces to the corresponding averages. Interestingly, there is some difference in the free-energy profiles in the region of the inflection point at the opening of the nanotube, with case 共d兲 profile slightly disfavoring water entry compared to that of the gaHFB benchmark. This difference is most likely due to the distortion of the nanotube opening that makes its effective diameter smaller compared to perfectly cylindrical opening. With the AP restraints on the R1 and R2 ring used in gaHFB benchmark, such distortions are greatly penalized reducing the configurational entropy of the system. B. Explicit water

We computed the PMF for the water translocation away from the center of the nanotube into the bulk water along the principal axis of the nanotube by 16.5 Å. Employing the ggaHFB method, we do not need to use a slablike model setup,37,80 and the flexible nanotube is fully solvated in water. The computed PMF provides accurate estimates of the freeenergy difference between the water inside the nanotube and in the bulk. Furthermore, the PMF provides accurate barriers for the water entering and exiting the nanotube. In collecting the PMF, we do not concern ourselves with the specific behavior of the water conduction through the nanotube. 1. Free-energy „T = 298 K…

The PMF is shown in Fig. 5 and the corresponding extrema are summarized in Table III. As seen from Fig. 5 and Table III, water is stabilized inside the nanotube by about 2.1 kcal/ mol compared to bulk. Unlike in the gas phase, the PMF in explicit water displays nontrivial structure. In particular, there is a tiny barrier going from the bulk toward the nanotube at around 15.2 Å, which leads to about

TSគ0 Minគ1 TSគ1 Minគ2 TSគ2 Minគ3 TSគ3 Minគ4 TSគ4 Minគ5 End

0.3 kcal/ mol deep local minimum. Escaping that minimum toward the nanotube has a barrier of about 1.5 kcal/ mol. There are three minima within each half of the nanotube positioned 1.4, 4.2 and 6.8 Å away from the center. Past the water desolvation barrier, the water molecule slides downhill all the way toward the third 共counting from the center of the nanotube兲 internal minimum. The inflection point at the nanotube entrance persists. The internal minima are only 0.2– 0.3 kcal/ mol deep and appear asymmetric. Note that the very center of the nanotube now corresponds to a local freeenergy maximum. The asymmetry is likely due to the intrinsic difficulty to average the configurations over two possible orientations of the single file water wire permeating the nanotube. It has been estimated that the water chain reorients inside the nanotube once a few nanoseconds.52,55 In addition, the wire can spontaneously disassemble, vacating the nanotube. Such events are difficult to get proper statistics for, as the barrier for water escaping the nanotube is about 3.4 kcal/ mol. Therefore, it is impossible to obtain a fully converged PMF on the time scale of our simulations. Clearly, the constriction inside the nanotube provides a particularly difficult test case for any PMF calculations. We feel that using two independent simulations each 10.0 ns long to compute the PMF achieves the goal of demonstrating the utility of the ggaHFB method with c.m. restraints. V. CONCLUSIONS

In brief, we have described a generalization of the gaHFB method that permits the use of positional restraints of two types, namely, individual atomic and group centers of mass, as well as their various combinations. We demonstrated the new methodology for water permeation through the nanotube both in gas phase and in explicit water. The generalized gaHFB method with the added group center of mass restraints makes possible addressing problems that involve mobile phases such as molecule permeation across lipid membranes. This was not previously possible with the

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124901-11

J. Chem. Phys. 127, 124901 共2007兲

Generalized gradient-augmented harmonic Fourier beads

冓 冔 冉 b ¯兲 ⳵VAP 共x ⳵xi

gaHFB that exclusively employed atomic positional restraints. Finally, the ggaHFB method provides additional flexibility for finding minimum free-energy transition path ensembles and computing corresponding free-energy profiles as well as their adiabatic counterparts for various other problems of interest.



This work was in part supported by grants from NIH, NSF, NBCR, CTBP, and HHMI. Computer time at CTBP, and NBCR clusters is greatly appreciated. We also thank A. Babakhani and Dr. A. A. Gorfe for critical reading of the manuscript, and Dr. D. Hamelberg for useful comments.

b ¯兲 ⳵Vc.m. 共x ⳵xi

¯x,b



u,ref b,ref u b kAP,i xAP,i + kAP,i xAP,i u b kAP,i + kAP,i

u,ref u = 2kAP,i 共xai vr − xAP,i 兲=

冓 冔 u ¯兲 ⳵˜Fc.m. 共x ⳵xi







¯x,b

u,ref b,ref u b kc.m. xc.m. + kc.m. xc.m.







b ¯兲 ⳵Vc.m. 共x ⳵xc.m.

冓 冔 冓兺 冔 冔 兺冓 u ¯兲 ⳵˜Fc.m. 共x ⳵xc.m.

n

=

¯x,b

j=1

n

=

j=1 n

=兺 j=1

=





u ¯兲 ⳵˜Fc.m. 共x ⳵x j

¯x,b

u ¯兲 ⳵˜Fc.m. 共x ⳵x j

¯x,b

u ¯兲 ⳵˜Fc.m. 共x ⳵xi

u ¯兲 共x ⳵˜Fc.m. ⳵xc.m.







n

=

¯x,b

j=1

j=1 n

=兺 j=1

=

, ¯x=x ¯ a vr

u,ref − xc.m.

b ¯兲 ⳵VAP 共x ⳵xi



冊 ,

¯x=x ¯ avr

u,ref b,ref b u kc.m. xc.m. + kc.m. xc.m. u b kc.m. + kc.m.

b ¯兲 ⳵Vc.m. 共x ⳵xi

冔 冓兺 兺冓



b,ref − xc.m.









b ¯兲 ⳵Vc.m. 共x ⳵x j b ¯兲 ⳵Vc.m. 共x ⳵x j

b ¯兲 ⳵Vc.m. 共x ⳵xi

b ¯兲 ⳵Vc.m. 共x ⳵xc.m.



共A4a兲

, ¯x=x ¯ a vr







¯x,b

¯x,b

¯x=x ¯ a vr

. avr xc.m.=xc.m.

共A4b兲

Both the unbiased mean force and the bias force averaged over the biased simulation focus at the same configuration point that is the configuration averaged over the biased trajectory:



avr = xAP,i

avr xc.m. =

共A2a兲

, ¯x=x ¯ a vr

¯x,b

=

u,ref u avr = 2␣ikc.m. 共xc.m. − xc.m. 兲 u ¯兲 ⳵˜Fc.m. 共x = ⳵xi



b = 2␣ikc.m.

u,ref − xAP,i

u ¯兲 ⳵˜FAP 共x ⳵xi

u b kc.m. + kc.m.





b,ref − xAP,i

共A3兲

n

共A1兲

u = 2␣ikc.m.

¯x,b

=

One can verify analytically that ˜Ii = 0, regardless of the type of the restraint, and even for the hybrid involving multiple restraints of both types. Moreover the following simple relations hold: u = 2kAP,i

u b kAP,i + kAP,i

b,ref b avr = 2␣ikc.m. 共xc.m. − xc.m. 兲

APPENDIX: POSITIONAL AVERAGES AND MEAN FORCES

u ¯兲 ⳵˜FAP 共x ⳵xi

u,ref b,ref u b kAP,i xAP,i + kAP,i xAP,i

b,ref = 2kv共xai vr − xAP,i 兲=

ACKNOWLEDGMENTS

冓 冔

= 2kv

u,ref b,ref u b kAP,i xAP,i + kAP,i xAP,i u b kAP,i + kAP,i u,ref b,ref u b xc.m. + kc.m. xc.m. kc.m. u b kc.m. + kc.m.

.

,

共A5兲

共A6兲

Thus, although we do not know many of the constants involved in these averages, we can always eliminate them by using the configurations averaged over the actual simulation trajectory: xai vr ⬇ 具xi典b ,

共A7兲

avr ⬇ 具xc.m.典b . xc.m.

共A8兲

1

¯x=x ¯ avr

. avr xc.m.=xc.m.

共A2b兲

Thus, within the approximation made, the average of the unbiased mean force over the biased ensemble provides the unbiased mean force at a particular configuration ¯x. As expected, for the bias forces we get

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