Multiconfiguration thermodynamic integrationT. P. Straatsma and J. A. McCammon Citation: J. Chem. Phys. 95, 1175 (1991); doi: 10.1063/1.461148 View online: http://dx.doi.org/10.1063/1.461148 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v95/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Multiconfiguration thermodynamic integration T. P. Straatsma and J. A. McCammon Chemistry Department, University of Houston, 4800 Calhoun Road, Houston, Texas 77204-5641

(Received 27 August 1990; accepted 10 April 1991 )

A modified thermodynamic integration technique is presented to obtain free energy differences from molecular dynamics simulations. In this multiconfiguration thermodynamic integration technique, the commonly employed single configuration (slow growth) approximation is not made. It is shown, by analysis of the sources of error, how the multiconfiguration variant of thermodynamic integration allows for a soundly based assessment of the statistical error in the evaluated free energy difference. Since ensembles of configurations are generated for each integration step, a statistical error can be evaluated for each integration step. By generating ensembles of different lengths, the statistical error can be equally distributed over the integration. This eliminates the difficult problem in single configuration thermodynamic integrations of determining the best rate of change of the Hamiltonian, which is usually based on equally distributing the free energy change. At the same time, this procedure leads to a more efficient use of computer time by providing the possibility of added accuracy from separate calculations of the same Hamiltonian change. The technique is applied to a simple but illustrative model system consisting of a monatomic solute in aqueous solution. In a second example, a combination of multiconfiguration thermodynamic integration and thermodynamic perturbation is used to obtain the potentials of mean force for rotation of the sidechain dihedral angles for valine and threonine dipeptides with restrained backbones in aqueous solution.

I. INTRODUCTION

The calculation of free energies from molecular simula­tions presents special problems as a result of the direct rela­tion between free energy and the partition function, which is prohibitively difficult to determine from simulation calcula­tions. The origin of this difficulty is the inability to sample the entire accessible phase space within any reasonable amount of simulation time, even for rather simple systems. To circumvent this problem, methods have been developed to evaluate free energy differences rather than absolute free energies. Two such methods, thermodynamic perturbation and thermodynamic integration, have gained relatively wide usage.

The method of thermodynamic integration best corre­sponds to the general idea of how to measure a change in the free energy of a system when some parameter specifying the thermodynamic state of the system is slowly varied. When carefully conducted, thermodynamic integration is an accu­rate and straightforward method that does not suffer from the problems encountered in many other methods in dealing with systems that are large or have high densities.

The thermodynamic integration technique has become one of the most frequently used procedures to obtain free energy differences in a wide variety of systems. Unfortunate­ly, the technique is often used without a proper error analy­sis. Recent work has led to the notion that, in order to obtain reliable free energy differences, even for relatively simple systems, far longer 'simulations are necessary than is com­monly assumed to be sufficient. 1.2 The purpose of this article is to present a multiconfiguration thermodynamic integra­tion (MCTI) technique in which statistical errors can be determined straightforwardly by avoiding the usual single

configuration (SCTI) approximation. With this method the reliability of calculated free energy differences is greatly im­proved, and application of the method allows for more effi­cient use of computer time. The method is illustrated in the calculation of the free energy of hydration difference be­tween neon and xenon, showing the most important aspects of this new approach. Also, it is shown how a combination of multiconfiguration thermodynamic integration and thermo­dynamic perturbation can be used to obtain potentials of mean force, illustrated for the rotation of the sidechain dihe­dral angles for valine and threonine dipeptides with re­strained backbones in aqueous solution.

II. THEORY

A. Thermodynamic integration

The most common application of thermodynamic inte­gration calculations is the determination of the free energy difference between two states of a system that differ in inter­molecular or intramolecular interaction potentials. The in­teraction potential is expressed as a function of some control variable A that determines the state of the system. As a con­sequence of the Hamiltonian being a function of this control variable A, the partition function for the system is a function of A as well. For an isothermal isobaric ensemble the parti­tion function is3

il(A) =-l-fffexp{ - H(A) +PV}dVdpN dqN h 3NN! kBT '

(1)

where N is the number of particles, h is Planck's constant, H(A) is theA-dependent Hamiltonian,p is the pressure, Vis

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1176 T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration

the volume, kB is Boltzmann's constant, T is the absolute temperature, and pN and qN are the momenta and positions of the N particles.

The fundamental thermodynamic state function for the isothermal isobaric ensemble is the Gibbs free energy G,

I

aG(A) = _ kBT [aa(A) ] aA a (A) aA

which also is a function of A,

(2)

Differentiating G(A) with respect to A then gives4•5

(3)

s s s (aH(A)laA)exp{ - [H(A) + pV IkBT ]}dV dpN dqN =----------------~~------------~~------- (4)

S S Sexp{ - [H(A) +pVlkBT]}dVdpNdqN

= (aH(A) ) aA ....

which is an ensemble average of the mechanical quantity aH(A)laA for a system with Hamiltonian H(A). This en­semble average is readily obtained from molecular simula­tions.

The Gibbs free energy difference between two states of a system, described by the Hamiltonians H(A = 0) and H (A = 1) can be obtained by integration of Eq. (5),

aG = G(A = 1) - G(A = 0) = t [aG(A)] dA Jo aA A

= t (aH(A) ) dA. (6) Jo aA A

Since molecular simulations are performed with dis­crete steps, this integral has to be evaluated as a sum of en­semble averages (MeT!),

aG=GI-Go = -kBTln~ ao

(7)

(5)

where i counts over the number of different values of A, and aA; is the difference between successive values of A. In gen­eral the aA; do not have to be constant during a thermody­namic integration.

In a very common approximation of thermodynamic integrations, the sum over ensemble averages is replaced by a sum over single configuration values for the derivative (SCTI) ,

aG = I (aH) aA;. ; aA A

(8)

B. Thermodynamic perturbation

Thermodynamic perturbation is based on the expres­sion, derived by Zwanzig,6 for the free energy difference be­tween two systems with Hamiltonians Ho and HI =Ho+aH,

(9)

= -kBTln S S Sexp{ - (HI +pVlkBT)}dVdpNdqN

S S Sexp{ - (Ho+pVlkBT)}dVdpNdqN (10)

k S S Sexp{ - (Ho + pV IkBT )}exp{ - (aH IkBT )}dV dpN dqN

= - B TIn ---------------------------------------------- (11 ) S S Sexp{ - (Ho+pVlkBT)}dVdpNdqN

= -kBTln(exp{ -::r })o'

where the averaging is performed over the ensemble of con­figurations obtained for Hamiltonian H o, the reference en­semble. As was pointed out by several authors/'s the term perturbation is somewhat misleading, since this expression is exact and does not correspond to a perturbation theory in the usual sense.

In multistep thermodynamic perturbation (MSTP) an ensemble of configurations is generated for a range of values for A, where A has the same function as in seT! or MeT!. A perturbation free energy contribution is evaluated from each ensemble generated at A; of 9

(12)

aG; = kB Tln(exp{ _ H;_ 112 - H; }' kBT /A,

- k B Tln( exp { - H; + ~;; H; } t, (13)

where H; = H(A; ),H;-1I2 = H(A; - HA.; - A.;_I})' and H;+ 112 = H(A; + HA; - A;_I})' For the first value 11.1 =0,H;_1I2 =H1/2 =HI,andforthelastvalueAN = 1, H N + 112 = H N' so that the first and last aG; have only one term in Eq. ( 13). The totalfree energy difference can then be found from

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T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration 1177

(14)

and the total statistical error from

( )

1/2

E(flG) = ~ E; , (15)

where Ei = E(A.G;) is the statistical error of the ith pertur­bation step.

c. Systematic, statistical, and sampling errors

Analyses of the applicability and limitations of thermo­dynamic integrations using Eq. (8), in which such factors as total simulation time, the use of cutoff radii, and the sizes of the system and the time step were considered, have been reported.2,7.10.11 The Hamiltonian time-lag problem, caused by the strong correlation of successive dynamics steps,12 is a major concern in SCT!.

The criterion for a correctly performed thermodynamic integration is that for each value of A the ensemble of config­urations [MCTI, Eq. (7)] or the single configuration [SCTI, Eq. (8) ] is representative for the system with Ham­iltonian H (). ). Several sources of error need to be consid­ered. A systematic error is introduced by the fact that, after each change of A, the configuration used in the next molecu­lar dynamics step will not be representative for the Hamilto­nian H(A). This error decreases with decreasing step size in A. In SCTI the number of steps in A can be made quite large. In practical calculations, however, the step size of A can not be made infinitesimally small, and SCTI will lead to this systematic error. To assess the magnitude of this error the SCTI is usually performed in a forward and, after some equi­libration of the final state, in a reverse direction. The differ­ence between the two results obtained, the hysteresis, is usually taken as a measure of this systematic error, assuming that the systematic error in the forward direction does not cancel the systematic error in the reverse direction. The ra­tionale behind this assumption is that the free energy evalu­ated from a configuration that is not representative for the Hamiltonian used will be higher than the value obtained from a representative configuration. Because this is true for the SCTI in the reverse direction also, the systematic errors from both directions do not cancel. In MCTI an ensemble of configurations is generated for each value of A. To keep cal­culations feasible, the number of steps in A must be chosen considerably smaller than is possible in SCTI. Consequently, the step size of A is larger. Generation of each ensemble should therefore be preceded by equilibration.

The fluctuations in properties obtained from equilibri­um molecular dynamics simulations lead to statistical er­rors. In MCTI calculations the derivative in Eq. (7) for each value of A is obtained as an ensemble average. Consequently, a statistical error Ei = E [(aH)(A)laA >;] can be evaluat­ed for the ensemble average (aH(A)laA >; at each value of A;. Assuming that the statistical errors from successive steps are independent, an estimate for the statistical error for the entire MCTI can be found from

( )

112

E(A.G) = ~ E;A.A.; . (16)

Since SCTI simulations do not involve the calculation of ensemble averages, a statistical error can not be evaluated. There is no reason, however, to expect statistical errors not to be important in SCT! simulations. This is a serious disa?­vantage, since this not only affects the final free en~rgy dif­ference that is obtained, but also the hysteresis. ThIS makes the hysteresis a very unreliable measure of the error in SCT!. In a recent publication Mitchell and McCammon2 discuss the difficulties in obtaining precise free energy values from SCTI and the limited value of hysteresis as a measure of error.

Adequate sampling of phase space is a fundamental problem in any molecular simulation. Unfortunately, unl~ke systematic and statistical errors, errors due to poor sampltng are difficult to determine quantitatively. A general case in which inadequate sampling presents a serious problem is when mUltiple minima exist on the energy surface, separated by energy barriers. A specific example is any molecule that has distinct rotational isomeric states (e.g., the gauche and trans conformations of butane). Since only one configura­tion is used to obtain the derivative of the Hamiltonian, im­portant restrictions have to be applied to the systems that can be studied using SCT!. If more than just one stable con­formation of the system can exist at any point during the integration, a single configuration value for the derivative will be a poor approximation for the value that would have been obtained from a well sampled ensemble. The entropic contribution due to the existence of other conformations of the molecule would be completely neglected. But even if there is just one valley through phase space from initial to final state of the system, this valley may change in breadth. This change in accessible phase space would be reflected in the value obtained from an ensemble average, but is not in the value obtained from a single configuration. Although such contributions would be properly accounted for in an extremely long SCTI calculation in which a fully representa­tive sample is generated for every small interval in A, such calculations will be prohibitively impractical in practice. Where multiple minima occur, special techniques can be ap­plied to combine all of their contributionsY-15 MCTI, how­ever, has the advantage that the statistical error estimates may provide indications of poor sampling and, as shown below, allows for systematic improvements in sampling.

In MCT! simulations not all ensembles need to be of equal size. Each ensemble can be made as large as needed to make the statistical error roughly the same for each value of A. Each part of the free energy curve obtained then has the same statistical accuracy. Due to the strong correlation in molecular dynamics simulations statistical error will have to be evaluated using a correlation analysis approach.

Another advantage of MCTI simulations is that the lim­ited number of values of A allows for recording of the final configuration and some additional information for each val­ue of A. The accuracy of the free energy difference result can then easily be improved by increasing the size of generated ensembles, without loss of CPU time spent in earlier calcula­tions. For SCTI simulations entire runs have to be redone at a longer total simulation time, essentially wasting the CPU time spent in the shorter runs.

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1178 T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration

There is no need for a non-linear change of A. in MCT! simulations if the length of each ensemble is determined by the statistical error. A nonlinear change of A. is often needed in SCT!, because regions with large fluctuations will require more densely spaced values of A. than regions with relatively smaller fluctuations in the value of the derivative. In MCTI this problem is taken care of automatically by continuing to generate configurations for each ensemble until the statisti­cal error, which is determined by the fluctuations, becomes less than a certain preset value.

In SCTI simulations, evaluating a hysteresis from a re­verse simulation is a way to get at least some crude measure of the error, but this makes it necessary to perform exactly the same integration again in the reverse direction, thereby doubling the CPU time needed. Another option is to per­form multiple runs with equal integration lengths. This will give an estimate of the statistical error, but the systematic error is expected to be roughly the same in all runs because the change in the Hamiltonian is made in the same simula­tion time. Considering the fact that the use of the hysteresis as a measure of error appears more questionable as the SCTI technique is further analyzed, the MCT! technique is pro­viding a more reliable estimate of the error.

D. Free energy calculations using restraining potentials

Since free energy is a thermodynamic state function the path followed through Hamiltonian space in thermodynam­ic integrations is in principle irrelevant for the resulting total free energy difference. The particular choice of the func­tional form may affect the error in the free energy difference considerably in actual calculations, and a particular choice is usually made in an attempt to decrease the error in the final free energy difference result. This is true for free energy dif­ference calculations between real states of the systems, but also between states that include A-independent restraining potentials to keep one or both systems in a fictitious state. The thermodynamic perturbation method can be used to evaluate the free energy difference between restrained and unrestrained system. 13, 14 An example of a particular choice of the path followed, in order to decrease the error in the final result, is electrostatic decoupling.16

In certain cases one is interested in the intermediate states during a thermodynamic integration in which A-de­pendent restraining potentials are used. The change in the system during the thermodynamic integration can, for ex­ample, describe a postulated reaction coordinate. In that case, the Hamiltonian H used for the thermodynamic inte­gration consists of the A-independent Hamiltonian Ho de-

W(x) = - RT lnp(x)

S exp[ - (H /RT) ]dr' - RT In ~---=-~-----=-­

S exp[ - (H /RT) ]dr

I

scribing the normal interactions in the system and a A-de­pendent restraining potential U that determines the desired path through Hamiltonian space that describes the reaction coordinate

H(p,q,/l.) = Ho(p,q) + U(q,A). (17)

Analogous to the result ofthe derivation in Sec. II A, the free energy as a function of A is given by

aG(A) = r). (au(q~ '») dA', (18) J.I.u aA ).'

where the integration is from some initial value A ' = Ao to A'=A.

The free energy difference found using Eq. (18) repre­sents the free energy of the system described by the Hamilto­nian including the restraining potential. The free energy of the unrestrained system when in the phase space defined by the simulation using the Hamiltonian including the restrain­ing potential can be found by correcting for the use of the restraining potential by means of a thermodynamic pertur­bation calculation

aG(A) = r' (au(q,~ '») dA' J)." aA ).'

+RTln(exp[U~*) ]L. (19)

The free energy difference given by Eq. (19) represents the free energy difference of the unrestrained system between different parts of phase space, namely those parts as obtained from the simulations in which the restraining potential was used.

Since two ensemble averages are needed for each value of A in this expression, MCT! can be used to evaluate this free energy difference. Note that the first term is an integral and in a practical calculation will be an accumulated value, while the second term in Eq. (19) is a single correction term for each value of A. The SCTI approximation can be made here also, leading to

aG(A = r' [au(q,~ ')] dA' - U(A). (20) L" aA ).'

Obviously, the discussion in the previous sections of the ap­plicability and reliability of this approximation is valid here as well.

If A can be uniquely associated with the value of some internal coordinate in the system, the free energy difference evaluated using Eq. (19) or Eq. (20) is identical to the po­tential of mean force (PMF) Walong this internal coordi­nate. In that case, the free energy difference expression Eq. (19) is a good approximation for the PMF I7

(21)

(22)

(23)

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T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration 1179

S exp( - [H + U(x)/RT] )dr' S exp[ - (H IRT) ]dr = -RTln -U(x)+RTln---------- (24)

S exp( - [H + U(x)/RT] )dr S exp( - [H + U(x)/RT] )dr

= - RTlnp*(x) - U(x) - RTln(exp( - ~;) )), (25)

where p is the probability density for the internal coordinate in the unrestrained system and p* is the probability density for the internal coordinate in the restrained system. If Ax is the value for A that results in the value x for the internal coordinate, then W(Ax)::::: W(x). The phase space integra­tion S dr = S dr' + S dx is separated into an integration along the dimension of the reaction coordinate S dx and the remainder of phase space S dr'. For the exact one-to-one correspondence between Ax, and x, the correction needed for the use of the restraining potential is equal to this poten­tial,

- U(.·{x) = - U(x).

(26)

As an example of a restraining potential applied to an internal coordinate, consider a proper dihedral potential function with single multiplicity,

U(if;,A) = C,,{l + cos [if; - £S(A) n, (27)

where £S(A) is the A-dependent angle determining the angle if; = £S(A) + 1T oflowest energy. The function form of £S(A) could be a function such as

(28)

so that 8(A) changes from the initial value 8; to the final value 8[ if A changes from 0 to 1. If this is the only change in the thermodynamic integration, the derivative is found from

/au(q,~'») =C,,(£S[-8;)(sin[if;-8(A')]),\,' (29) \ aA ,t.

In order to obtain a small spread in the internal coordinate if; around the average value if;x for the simulation at A = Ax, the restraining force constant C", has to be chosen as large as possible without causing numerical problems in the integra­tion of the equations of motion. 18 It has also been suggested, in addition to using a large force constant, to only count contributions to the ensemble averages ofEq. (19) for which the internal coordinate is equal to the desired value. 19 In the example given above, only contributions would be counted for which <P = 8(A) + 1T with a very small tolerance. In practice this may prove to be inefficient, since many configu­rations generated may have to be skipped. Also, with this procedure the nature of the ensemble generated becomes un­clear, and the validity of the ensemble average uncertain.

The use of high force constants for restraining potentials presents a special problem in the choice of the time step. Clearly, the time step should not be taken too large in order to prevent the simulation from becoming unstable. Too small a time step, however, will not lead to representative ensembles of configurations, from which the ensemble aver­ages have to be evaluated, within a reasonable amount of computer time. Due to numerical errors, caused by the use of finite time steps, the use of high force constants may lead to

I configurations with high energies being over-represented in the generated ensembles. This has a small effect on the en­semble average of the derivative of the Hamiltonian as given in Eq. (18). High restraint energies will have a large effect on the ensemble average (exp[ U(A)/RT]) needed for the free energy correction in Eq. (19). The large fluctuations in the restraint energies are found to have a smaller influence on this free energy connection if only the first order approxi­mation to this correction is used,

aG(A) = ((au(q,~'») dA'-(U(A»,t. (30) J,tO) aA ,t'

The potential of mean force evaluated using Eq. (30) or ( 19) will be an approximation due to the spread of the reac­tion coordinate for each value of A, which is caused by the fact that, in order to obtain stable dynamics, the force con­stant of the applied restraining potential can not be chosen too large. Fortunately, this spread is expected to be small in the wells of potentials of mean force since there is a natural tendency for the reaction coordinate to remain at the posi­tion of the well, even without the use of the restraining poten­tial. Consequently, the uncorrected free energy difference between wells can be expected to be very similar to the cor­rected result.

The procedure outlined in this section is an alternative to the calculation of free energy differences along a reaction coordinate, in which constraints I5 ,20.21 are used rather than a restraining potential.

III. COMPUTATIONAL DETAILS

The MCT! technique is the standard free energy ther­modynamic integration technique in the molecular dynam­ics program package ARGOS,22 with which the simulations described in the following sections were carried out. SCT! is just a limiting case of MCT!, in which the minimum and maximum number of configurations per step in A are both 1. To be able to perform SCTI with nonlinear change in A this option is available too, although this is not needed in MCT!.

The simulation volume is a cubic box, subject to period­ic boundary conditions. To keep temperature and pressure at 298 K and 105 Pa, respectively, coordinates and velocities were scaled every dynamics step, using a method to weakly couple the system to an external heat and pressure bath.23

Relaxation times used for the weak coupling are 0.4 and 0.5 ps for temperature and pressure, respectively.

The potential model used for water is the SPC/E water model developed by Berendsen et al.24 The Lennard-Jones parameters for the interaction

Cl2 C6 V,u(r.) = __ 'J ---" (31)

l} 12 6 rij rij

between the noble gases and the water oxygen atoms 10 are given in Table I. The intramolecular distances are fixed, us-

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1180 T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration

TABLE I. Lennard-Jones potential parameters for the noble gas interac­tions with SPC/E oxygen.

Interaction

Ne-O Xe-O

~ ~2 10- 3 kJnm6 mol-! 10- 6 kJnm!2mol-!

1.12457 0.99806 8.92289 18.472 I

ing the coordinate resetting procedure SHAKE.25-

27

The GROMOS28 force field was used to describe the inter­actions of the dipeptide.

The statistical errors reported in this article have been evaluated using a method by Straatsma et al. 7

IV. FREE ENERGY OF HYDRATION OF Ne AND Xe

The difference in free energy of hydration for Ne and Xe provides a simple but illustrative problem to analyze the

AG I kJ mol-1

6.

4.

2.

o.

-2.

O. 5.

(llG/Il"-) I kJ mol-1

400.

300.

200.

100.

O.

o.

-100.

-200.

-300.

-400. O. 5.

r

10. 15.

10. 15.

TABLE II. SCTI free energy difference results for the mutation Ne-Xe in aqueous solution, using the functional form A = (t IT )'.

Time IlG- IlG- Hysteresis ps kJ mol-! kJ mol-! kJ mol-!

25 -2.9 2.4 -0.5 50 -0.7 3.5 2.8

100 1.1 1.9 3.0 250 -2.2 1.8 0.4 500 - 1.4 2.3 1.0 750 - 1.0 1.6 0.6

thermodynamic integration technique. Simulations were performed in the isothermal isobaric ensemble of the nonpo­lar monatomic solute and 128 SPC/E water molecules.

Intuitively, the most accurate final free energy differ­ence result from a scn calculation is expected if the total change in the free energy during the process is evenly distrib-

20.

20.

a

b

c

FIG. I. Single configuration thermody­namic integration free energy Ne-Xe,

25. withA = (tIT)2in25ps. (a) IlGvstime, forward (solid) and reverse (dashed); (b) (aG laA) for forward SCT!; (c) (aG I aA) for reverse SeT!.

25.

Time I ps

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T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration 1181

AG / kJ mol-1

6.

4.

2.

o.

-2.

O.

(BG/BX) / kJ mol-1

400.

300.

200.

100.

o.

o.

-100.

-200.

-300.

-400. O.

250. 500.

250. 500.

a

FIG. 2. Single configuration thermody­namic integration free energy Ne .... Xe.

750. with A = (t IT)2 in 750 ps. (a) t:.G vs time. forward (solid) and reverse (dashed); (b) (aG laA) for forward SCTI; (c) (aG laA) for reverse SCT!.

b

c

750.

Time / ps

uted over the molecular dynamics time steps. This means that the slope of the free energy change versus simulation time should be as uniform as possible over the whole range. By appropriately choosing the time dependence of the con­trol variable A., steep slopes can often be avoided. For the process of changing Ne into Xe in aqueous solution this can be accomplished by a functional form

A.(t)=(~r. (32)

where t is the running simulation time, and T is the total simulation time for the thermodynamic integration. With this functional form for A. several SeT! calculations were performed in forward and reverse directions with different total simulation times. The free energy results and the hys­tereses obtained are listed in Table II. The first thing to note is the slow and poor convergence of the final free energy

values as a function of simulation time. Second, the hystere­sis does not monotonically decrease with increasing total simulation time. The cause of this problem becomes appar­ent when the free energy change for a single thermodynamic integration is plotted as a function of the simulation time, together with the calculated derivatives. In Fig. 1 these prop­erties are shown for the thermodynamic integration in 25 ps. Figure 1 (a) gives the free energy result for the forward di­rection as the full line, and the curve for the reverse direction as the broken line. Figures 1 (b) and 1 (c) give the derivative (JG / JA.) for the forward and reverse direction, respectively. The free energy curves are smoother and more similar in the latter part of the simulation, where the fluctuations in the derivative are small. The first half of the thermodynamic integration shows large fluctuations in the derivative and larger differences between the forward and reverse free ener­gy curves. It is important to observe that, at certain values of

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1182 T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration

TABLE III. SeT! free energy difference results for the mutation Ne ..... Xe in aqueous solution. using the functional form ,1= (t IT )".

Time I::.G- I::.G- Hysteresis ps kJmol- ' kJ mol-I kJ mol-I

10 5.6 4.6 10.2 25 -0.3 -0.1 -0.4 50 0.2 4.0 4.1

100 - 1.4 0.8 -0.6 250 - 1.9 1.5 - 0.4 350 - 1.6 0.5 -1.1 500 -0.8 0.8 0.0 625 - 1.5 1.1 -0.3 750 -2.2 1.8 -0.3

A., the free energy curves for the forward and reverse direc­tions, deviate more than the hysteresis, the difference at the end of the simulations. Between 8 and 9 ps in Fig. 1 (a) there is a difference between the two curves of about 2 kJ mol - 1 ,

while the final hysteresis is only of the order of 0.5 kJ mol - I .

AG j kJ mol-1

6.

4.

2.

o.

-2.

o.

(8GjaX) j kJ mol-1

400.

300.

200.

100.

o.

o.

-100.

-200.

-300.

-400. O.

/

- - --

250. 500.

250. 500.

Yet at every time step the Hamiltonian is exactly the same for the forward and reverse simulations. Therefore, the two curves should be exactly the same, not only at beginning and end, but over the whole range. The fact that the difference does not increase monotonically, and the fact that the final hysteresis can be found to be much smaller than the differ­ence between the two curves at some other point during the simulation, illustrate the limited value of the hysteresis as a measure of the error.

For the longest thermodynamic integration (750 ps) using the same time dependence of A., the corresponding curves are plotted in Fig. 2. Forward and reverse free energy changes are much smoother and resemble each other more compared to the 25 ps simulation. Still, as expected, fluctu­ations in the derivative behave in the same way, and, conse­quently, the largest deviation between forward and reverse free energy result is found in the first part of the calculation, where high fluctuations in the derivative are found. For this seemingly simple process 750 ps still appears to give an ap-

a

FIG. 3. Single configuration thermody­namic integration free energy Ne ..... Xe.

750. with A = (t IT)" in 750 ps. (a) I::.G vs time, forward (solid) and reverse (dashed); (b) (oG laA) for forward SeTI; (c) (oG laA) for reverse sen.

b

c

750.

Time j pH

J. Chem. Phys .• Vol. 95, No.2, 15 July 1991

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T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration 1183

TABLE IV. MCTI free energy difference results for the mutation Ne-Xe in aqueous solution.

Equilibration Minimum Maximum Total time steps steps steps ps

100 100 100 20.000 100 100 500 31.188 100 100 1000 46.544 250 250 500 59.364 250 250 1000 85.048 100 100 5000 194.572 25 100 10000 363.880

100 250 17500 669.450 100 250 25000 776.529

preciable uncertainty in the final free energy result. The hys­teresis is of the order of 30% of the calculated free energy difference.

6.G kJ mol-I

1.4 -0.4 -2.0 -2.7 -1.5 - 1.3 - 2.3 - 1.5 -2.3

Error. kJ mol-I

0.9 0.2 0.2 0.2 0.1 0.2 0.2 0.2 0.2

A(t)=(~r (33)

The series of thermodynamic integrations was repeated using a different time dependence of A,

This redistributes the stepsize in A, making the LUi smaller in the range with large fluctuations in the derivative, and

AG / kJ mol-1

5.0 r----------------.----------------.---------------~~

2.5

0.0

-2.5

O. (8G/8X) / kJ mol-1

75.

50.

25.

o.

E(AG) / kJ mol-1

0.20

0.15 I-

0.10

0.05

0.00 O.

250.

I

I

250.

J I

500. 750.

b

I I

<:

-

-

I T 500. Time / ps 750.

J. Chem. Phys., Vol. 95, No.2, 15 July 1991

FIG. 4. Multiconfiguration thermody­namic integration free energy Ne-Xe, in 776ps. (a) 6.Gvstime; (b) (aGlaA); (c) Statistical error vs time.

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1184 T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration

AG / kJ mol-t

r--------------.--------------r-------------~------------~

40.

20.

o.

0.00

(IJG/IJA) / kJ mol-t

250.

o.

-250.

E(AG) / kJ mol-1

1.5

1.0

0.5

0.0 0.00

0.25 0.50

0.25 0.50

larger in the range with small fluctuations. The free energy results and hystereses obtained for a series of calculations with different total simulation times are given in Table III. The forward and reverse results appear to converge more rapidly with increasing total simulation time, compared to the first series of calculations. The hysteresis, although smaller than in the first series at comparable simulation lengths, still does not monotonically decrease with increas­ing total simulation time. Again, this lack of trend in the magnitude of the hysteresis with increasing total simulation time and the fact that negative values for the hysteresis are found, which contradicts the argument that systematic er­rors will not cancel between forward and reverse thermody­namic integrations, lead to the observation that hysteresis is a poor measure of the quality of this type of calculation.

The accumulated free energy change for the 750 ps sim­ulation is shown in Fig. 3, together with the derivatives for the forward and reverse simulation. In the large fluctuation

a

0.75

c

0.75

FIG. 5. Free energy difference for the forced rotation around the X dihedral an­gle in ifJ and", restrained valine dipeptide in aqueous solution. (a) Free energy change as a function of A after 2000 (dot­ted line), 3000 (dash-<lot line), 4000 (dashed line), and 5000 (solid line) con-

1.00 figurations in each ensemble for A. (b)

1.00

Free energy derivative as a function of A after 2000 (dotted line), 3000 (dash-<lot line), 4000 (dashed line), and 5000 (solid line) configurations in each ensemble for A. (c) Accumulated statistical error in free energy curve as a function of A after 2000 (dotted line), 3000 (dash-<lot line), 4000 (dashed line), and 5000 (solid line) configurations in each ensemble for A.

part of the derivative, where the steps in A, are very small, as well as in the last part of the curve where steps in A, are relatively large but the fluctuations in the derivative are small, the free energy curves have a very similar slope. It is in the region in between where steps in A, are becoming larger, that the fluctuations of the derivative are still large enough to cause the free energy difference curves to show a noticeable difference in slope. As a result the hysteresis of the total run is still found to be appreciable, considering the length of the simulation, 750 ps.

The results from the two series of SCTI simulations de­scribed above show the difficulty of efficiently distributing molecular dynamics time steps over the range of values of A,. Table IV lists the results of a series ofMCTI simulations. For each simulation the number of steps in A, was 100, equally spaced from 0 to 1. This table lists the number of equilibra­tion steps for each value of A" the minimum and maximum number of steps for the evaluation of the ensemble average of

J. Chem. Phys., Vol. 95, No.2, 15 July 1991

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T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration 1185

11' / kJ mol-1

50. r------------.------------~----------_,------------~

40.

30.

20.

10.

o. -11' o

AG ...... / J.cJ mol-1

-2.

-3.

-4.

-5.

-6.

o

the derivative, the total simulation time, including the time used for the equilibrations, and the accumulated Gibbs free energy differences using Eq. (7), as well as the accumulated statistical error evaluated using Eq. (16). The statistical er­ror threshold was set to be 1.0 kJ mol - I in the ensemble average of the derivative. The free energy difference seems to converge much faster than in the SCT! simulations de­scribed above. The variation in final free energy differences is of the same order as the statistical errors found.

In Fig. 4 the accumulated free energy as a function of simulation time is given for the longest of the MCT! simula­tions in Table IV, together with the derivative in Fig. 4(b). The derivative shows hardly any fluctuations, leading to a very smooth free energy change. The free energy shows a very interesting feature. In the last 50 ps of the simulation the fluctuations in the derivative are so small that the minimum number of steps were already sufficient to bring the statisti­cal error below the preset threshold. The peculiar shape of the free energy curve illustrates that distribution of the simu­lation steps according to the fluctuations in the derivative gives a more efficient and more accurate result than equal distribution of free energy change over the molecular dy­namics simulation.

At the long time scale of the simulations reported here, an additional cause for lack of convergence may be the long time fluctuations in an aqueous environment. 1.30

11'/2 x

x

FIG. 6. Potential of mean force for rota­tion of dihedral angle X in valine dipeptide (a) PMF as obtained from a MCT! free energy calculation (dashed curve), fitted to a sum of six sinusoidal functions (dot­ted curve), and as obtained from the dis­tribution of the angle X from a simulation using the negative of the fitted function as an umbrella potential. (b) Correction to the MCT! free energy curve to obtain the PMF.

V. POTENTIAL OF MEAN FORCE IN VALINE AND THREONINE DIPEPTIDE

Multiconfiguration thermodynamic integrations have been carried out to determine the X potential of mean force for the ¢ and t/J restrained valine dipeptide and threonine dipeptide in aqueous solution. This serves to illustrate the use of this type offree energy evaluation in a case in which a A-dependent restraining potential with a large force constant is employed to obtain a potential of mean force. The back­bone dihedral angles ¢ and t/J were restrained, first to avoid the necessity of considering the multiple stable isomeric states that would otherwise be possible, and second, to mod­el the degree of freedom in X as it would occur in a protein in which the backbone is less flexible. The angles ¢ and t/J were restrained at - 2.617 99 and 2.61799 rad, respectively, with sinusoidal restraining potentials with force constants 175 kJ mol- I . Simulations were performed on the dipeptide in 198 SPC/E water molecules in a cubic periodic box, in the isothermal isobaric ensemble. The restraining potential on the X dihedral angle is a sinusoidal potential with a force constant of 175 kJ mol-I. The reference angle of this poten­tial is changed during the MCT! from a to 21T, so that the angle X is changed from 1T to - 1T.

The complete change was made in 101 equally spaced steps in A. At each value of A, 500 equilibration steps were taken, followed by 5000 steps of data acquisition. To illus-

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1186 T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration

trate the convergence of the free energy curve with increas­ing number of configurations generated for the evaluation of the ensemble averages, and the ease with which additional configurations can be generated for an already completed simulation, the data acquisition was performed in four sepa­rate calculations. For the valine case, the free energy change, the ensemble average of the derivative and the accumulated statistical error are shown in Figs. 5(a), 5(b), and 5(c), respectively, as a function of A after 2000 (dotted line), 3000 (dash-dot line), 4000 (dashed line), and 5000 data acquisi­tion steps (solid line). The first thing to note is the difference between the initial and final values of the free energy curve, which should be zero. The fact that the final value tends to approach zero with increasing number of configurations in each ensemble, and the fact that there is a converging change of the entire free energy curve with increasing size of the ensembles, points towards a systematic error that becomes less when more configurations are generated for each value of A. The most probable cause is that the equilibration with 500 steps for each ensemble is insufficiently long to fully equilibrate the ensemble after the change in A. The ensemble derivatives for the four cases in Fig. 5 (b) show no noticeable differences. An important result is the extremely slow con­vergence of the statistical error with increasing ensemble size, as can be seen from Fig. 5 (c). It clearly shows the slow convergence of the statistical error with increasing number of data. A correlation analysis was performed in calculating

the statistical error. The errors were found to be more than five times larger than the standard errors. Although the eval­uation of the statistical error in MCTI using Eq. (16) may represent a conservative estimate, the importance of a corre­lation analysis is obvious. A second observation is that the statistical error is monotonically increasing as the integra­tion proceeds. Unlike in the example in the previous section, it appears not to be necessary to let the statistical error deter­mine the number of configurations in each ensemble. The statistical error from the fairly long simulation of 500 ps is 1.4 kJ mol-I.

The potential of mean force for rotation of dihedral an­gle X, evaluated using Eq. (30) with a force constant of 826.21 kJ mol - I for the restraint potentials, is shown by the dashed curve in Fig. 6(a). The correction to the MCn free energy to obtain this PMF is shown in Fig. 6(b). These curves were evaluated using data obtained from a simulation with 5000 steps per A. The correction applied is small com­pared to the free energy values it is applied to. Consequently, there is only a small difference between the free energy curve and the evaluated potential of mean force. The small correc­tion needed also indicates that, for each value of A, the reac­tion coordinate X only deviates slightly from the minimum in the restraining potential.

To make a comparison with the potential of mean force as evaluated using Eq. (25) the PMF as obtained from the MCn was fitted to the function

" / kJ mol-t

35. ~-----------r------------~-----------'------------'

30.

25.

20.

15.

10.

5.

o. -71"

AG •• rr / kJ mol-1

-2.

-3.

-4.

-5.

-6. -71"

".. a

o 71"/2

b

o

J. Chem. Phys., Vol. 95, No.2, 15 July 1991

71"

FIG. 7. Potential of mean force for rota­tion of dihedral angle X t in threonine di­peptide (a) PMF as obtained from a MCTI free energy calculation (dashed curve), fitted to a sum of six sinusoidal functions (dotted curve), and as obtained from the distribution of the angle X from a simulation using the negative of the fitted function as an umbrella potential. (b) Correction to the MCTI free energy curve to obtain the PMF.

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T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration 1187

TABLE V. Parameters obtained for the fit of the potential of mean force around X in valine dipeptide with the function given in the text.

C, 8, n, kJ mol- I kJ mol- I

1 7.73337 - 0.78235 2 10.36670 1.26350 3 9.43613 2.98581 4 1.25923 - 0.866 43 5 0.59568 1.540 57 6 0.24668 2.87143

6

<1>(x) = L C; [1 + cos(n;x - 0;)]. (34) ;= I

The parameters n;,e;, and 0; for this fit are given in Tables V and VI, and the function is shown by the dotted curve in Fig. 6 (a). The distribution p* of the angle X was then obtained from a series of molecular dynamics simulations using an umbrella potential U(X) = - <1>(X)' With this umbrella potential the rotation around X is expected to be almost free. Each of the final configurations at each step in A. in the MeT! was used as a starting configuration for a lOps molec­ular dynamics simulation. Using the distribution p* ob­tained from these simulations in Eq. (25) resulted in the PMF given as the solid curve in Fig. 6 (a). The two curves for the PMF are, considering the statistical error, in very good agreement.

These calculations were repeated for the threonine di­peptide, under the same conditions, again using a force con­stant of 826.21 kJ mol- I for the restraint potentials. Figure 7 (a) shows the potential of mean force from a MeT! simula­tion (dashed curve), the fit of this PMF to the function given by Eq. (34) (dotted curve), and the PMF evaluated by use in Eq. (2S) of the angle distribution p* as obtained from a series of molecular dynamics simulations in which UCr) = - <P(l') was used as an umbrella potential (solid curve). Figure 7 (b) gives the correction that was applied to the MeT! free energy curve to obtain the PMF. Again good agreement is found between the two potentials of mean force.

VI. DISCUSSION

The multiconfiguration thermodynamic integration technique presented here and the multiple step perturbation method are based on the same basic idea that the most reli-

TABLE VI. Parameters obtained for the fit of the potential of mean force around XI in threonine dipeptide with the function given in the text.

C, 0, n, kJ mol- I kJmol- 1

1 1.799 10 0.55666 2 7.50140 1.08722 3 9.29020 3.05060 4 0.504 76 - 0.17836 5 0.35520 0.89194 6 0.26550 4.077 23

able way of obtaining properties is by averaging over ensem­bles of configurations. The basic problem in perturbation method calculations is that the reference ensemble has to be representative for a system described by the perturbed Ham­iltonian as well. This problem does not exist in multiconfi­guration thermodynamic integration since the derivative of the Hamiltonian should be evaluated from the generated en­semble.

The MCTI technique makes a more reliable assessment of the statistical error possible compared to the commonly used single configuration thermodynamic integration in which the error is usually obtained from the hysteresis. This is important because the reliability of the hysteresis as a mea­sure of error is questionable. The MeT! also allows adding to the quality of previous simulations, without loss of pre­viously obtained results. Moreover, it is possible, at the same time, to increase the quality of free energy evaluations in limited parts of the complete change of the system based upon the contribution to the statistical error. This is not pos­sible with SCT! simulations. MeT! simulations therefore make more efficient use of computing resources, while pro­viding more soundly based error estimates.

ACKNOWLEDGMENTS

This work has been supported in part by the National Science Foundation, the Robert A. Welch Foundation, the Texas Advanced Research Program, the Houston Ad­vanced Research Center, and HNS Supercomputers. J. A. M. is the recipient of the George H. Hitchings Award from the Burroughs Wellcome Fund.

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1188 T. P. Straatsma and J. A. McCammon: Multiconfiguration thermodynamic integration

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