EXPLORING PHASE SPACES OF BIOMOLECULES WITH MONTE CARLO METHODS

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1 Journal of Optoelectronics and Advanced Materials Vol. 7, o. 3, June 2005, p EXPLORIG PHASE SPACES OF BIOMOLECULES WITH MOTE CARLO METHODS A. Bu u * ational Institute for Research and Developent of Biological Sciences, Splaiul Independen ei 296, Buchurest, Roania Monte Carlo (MC) ethods have becoe an iportant tool in biophysics, structural olecular biology and rational drug design. They are used to predict quantities that either cannot be easured directly or when accurate experiental data are difficult to obtain. They are also helpful for the interpretation of experiental results since in a siulation the syste of interest can be studied in detail on the olecular level. (Received January 18, 2005; accepted May 26, 2005) Keywords: Monte Carlo ethod, Bioolecules conforations, Metropolis algorit 1. Introduction A central proble for olecular siulations is the sapling of conforational degree of freedo. The two ost widely used ethods for atoic-level odeling of fluids are the Monte Carlo (MC) ethods and the olecular dynaics (MD). Both procedures could use the sae olecular odels, classical force fields for the potential energy ters, and the ipleentation of boundary conditions. The principal differences are the ethods of sapling the configuration space available to the syste. The conventional for of olecular dynaics represents a realization of Boltzann's approach to statistical echanics, whereas the Monte Carlo ethod is rooted on the Gibbs' forulation of the proble. MC serves as a very robust algorith and can be applied to ore types of odels and potential functions. The advantages of the MD ethods are the efficiency [1,2,3] of searching in the phase space for high density systes and the built-in parallellizable nature. However, in soe situations and by eans of optiization of algoriths the MC ethods could be still ore efficient [3]. The Monte Carlo ethod was developed by von euann, Ula, and Metropolis at the end of the Second World War to study the diffusion of neutrons in fissionable aterial. The nae, which derives fro the faous Monaco casino, ephasizes the iportance of randoness, or chance, in the ethod and was coined by Metropolis in 1947 in the title of a paper describing the early work at Los Alaos [4]. It was when the MAIAC coputer in Los Angels becae operational in March 1952, Metropolis was interested in having as broad a spectru of probles as possible tried on the new achine, in order to evaluate its logical structure and to deonstrate the capabilities of the achine. Solving the classical statistical echanical -body proble via Monte Carlo technique was one of the first probles, done by Metropolis, in collaboration with the Tellers and the Rosenbluths, and led to the developent of what is now known as the Metropolis Monte Carlo ethod [5]. * Corresponding author: alina_butu@yahoo.co

2 1564 A. Bu u 2. Iportance of the sapling technique Monte Carlo (MC) ethods refer, in a very general sense, to any siulation of an arbitrary syste which uses a coputer algorith explicitly dependent on a series of (pseudo)rando nubers [6]. MC is particularly iportant in statistical physics, where systes have a large nuber of degrees of freedo and quantities of interest, such as theral averages, cannot be coputed exactly. Consider the integral of interest: x2 = ( ) (1) x1 I f x dx A naive way to calculate the above integral using the Monte Carlo approach would be the following quadrature forula: where i x2 x trial 1 I f ( ξi ) (2) trial ξ is the i-th chosen rando nuber fro a unifor distribution at the interval [, ] i= 1 x x. Iportance sapling techniques choose rando nubers fro a distribution ( x) 1 2, which allows the function evaluation to be concentrated in the regions of space that ake iportant contribution to the integral. If we rewrite the above integral as: ( x) ( x) x2 f I = ( x) dx x1 (3) where ( x) is an arbitrary positive weight function, then the integral I could be evaluated as the expectation value of f ( x) / ( x) with the probability density function ( x). Such a reforulation akes it possible to speed up the efficiency of the sapling. Fig. 1. Transforation of integrand in the iportance sapling technique. The solid line and dotted line in (a) represent the original integrand f(x) and weighting function (x), respectively. The solid line in (b) represents the transfored integrand f(x) / (x). The dashed lines in both parts represent the average values.

3 Exploring phase spaces of bioolecules with Monte Carlo ethods 1565 If we choose a (x), which behaves approxiately as f (x) does (i.e., (x) is large where f (x) is large and sall where f (x)is sall), then the integrand in eq.(3.), f(x) / (x), can be ade very sooth, see Fig. 1 with a consequent reduction in the standard deviation of the Monte Carlo estiate: i= 1 σ 1 i ( ( ) ) 2 f ξ f (4) i= 1 ( ζ ) ( ) f f ζ i 1 i 2, σ (5) ζ is the i-th chosen rando nuber according to the probability distribution function ( x) where i. This is a result fro the central liit theore, and the values of σ and σ ' defined above are the deviation of the integral of the function f fro that of the Monte Carlo evaluation. Coon integrals we will encounter in statistical echanics are as follows: = p q p q (6) (, ) (, ) 3 3 A d pd q A where ( p, q ) = ( p1, p2,..., p3, q1, q2,..., q3 ), is the phase point in the 6-diensional phase space and the choice of ( p, depends on the enseble of interest. For the icrocanonical enseble, the ( p, is: ( p, = ( ( p, ) δ H ε (7) where ε is the total energy of the syste, and for the canonical enseble, the ( p, reads: (, ) (, ) = e β H p q p q (8) For the isobaric-isotheral enseble, the ( p, reads: ( p, = e β ( H ( p, + P V) (9) 3. Markovian chain and the aster equation It is clear now that we should generate a stochastic process which is distributed according to p, q in order to calculate integrals like eq.(6). the desired distribution ( ) For siplicity we discuss the stochastic process with discrete tie step. Suppose that the p, q; t = p, q, where t = t0 + τ t, τ = 0,1,2.... Then we have probability density function at tie ( ) ( ) the following aster equation for a Markovian stochastic process:

4 1566 A. Bu u ( p q ) ( p q ) r + 1 n, n = r n, n + (, ) ( ) (, ) ( ) r p q P n p q P n dp dq r n n (10) where P( n) is the probability that the syste ( rando walker ) will ake a transition fro state to state n. We ai at reaching the desired distribution, ( p, q ), for the enseble of interest after long enough tie, i.e., In this liit we have: ( ) = ( ) li p, q p, q (11) r r (, ) ( ) = (, ) ( ) p q P n dp dq p q P n dp dq (12) n n n n A sufficient but not necessary detailed balancing condition (also called icroscopic reversibility ) could satisfy above equation with great siplicity: or (, ) P( n) = (, ) p ( n ) p q p q (13) n n ( n) ( ) P = P n ( pn, qn ) ( p, q ) (14) 4. Metropolis algorith Usually the transition probability P ( n) consists of two parts: P( n) = T ( n) A( n) (15) where T ( n) is the probability of aking a trial fro state to state n, and A( n) probability of accepting that step. If state n can be reached fro state in a single step, then is the T ( n) = T ( n ) (16) so that the equilibriu distribution of the rando walkers satisfies ( pn, qn ) ( p, q ) ( ) ( ) ( ) ( ) ( n) ( ) T n A n = T n A n A = A n (17) It is easy to prove that the following choice for A( n) Metropolis et al. [5] ( ) in 1, (, ) / (, ) which was first proposed by A n = pn qn p q (18)

5 Exploring phase spaces of bioolecules with Monte Carlo ethods 1567 satisfies the above stochastic equation. In the canonical enseble, we have: where E = ( p, q ) ( p, q ) H H. n n β H ( p, q ) n n e A( n) = in1, β H ( p, ) q e (19) = in 1, e β E It should be noted that eq.(19) is not the only solution for the stochastic equation eq.(18). In systes containing large potential energy barriers, the Monte Carlo ethods ay not be able to search the whole configuration space, leading to the so-called quasi-ergodicity. This can be avoided by juping over the barriers by coupling the conventional Metropolis sapling to the Boltzann distribution generated by another rando walker at higher teperature [7]. On the other hand, if kinetic echaniss or continuous dynaics are still of interest, the efficient oves based on the inforations of the local properties of potential functions, e.g., forces, torques and Hessian atrices, have been proposed in the force bias ethods [8, 9], sart Monte Carlo [10], or energy biased ethod [11]. In a MC siulation a segent of a olecule oves stochastically through rando oves of a sall nuber of onoers at a tie. In general MC serves as a very robust algorith with welldefined ensebles. Unfortunately the MC ethods suffers fro two disadvantages. First, it is very difficult to ipleent an efficient algorith in parallel coputer architectures. Second, in dense systes, the otions are ostly reduced to cooperative fluctuations. However, the MC ethod has been very successful and will reain very iportant at low and oderate densities. 5. Monte Carlo ethods of siulating polyer systes In the siple sapling algorith the polyer chains are each tie grown fro scratch. This is a good ethod for Rando Walks (but Rando Walks are trivial anyway). For Self Avoiding Walks this would ean that each tie a self-intersection occurs the chain should be grown anew fro the beginning. All the effort put in growing the chain is lost. For large the probability that a walk is grown without self-intersection is very low. Most of the attepts to grow a Self Avoiding Walks fail. This is why siple sapling is ineffective for large. This proble can be partially reedied by choosing at each step only fro the unoccupied lattice sites. However, this introduces a bias which should be copensated for. This is done by giving a weight to the conforation. If si is the nuber of unoccupied neighbors (i.e. the nuber of possible steps) at step i the conforation should be weighted by: w = s (20) This ethod is also known as the Rosenbluth ethod [12]. A drawback of the ethod is that one spends a lot of tie generating conforations which have a low weight. It is desirable if one could generate conforations with the correct probability so no reweighting is necessary. This is called Iportance sapling [13]: one avoids to saple conforations which have a low weight. The Metropolis ethod [5] is a frequently used ethod to achieve iportance sapling. It consists of repeatedly trying to change the conforation a little bit (oving a few atos or so) and then deciding whether to accept the new conforation or to retain the previous one. Such a prospective change in the conforation is called a ove. The acceptance criterion should be i i

6 1568 A. Bu u chosen in such a way that the conforations are sapled with the desired probability (usually the Boltzann distribution). The ethod of generating a new conforation (on the basis of the previous one) is stochastic and should satisfy detailed balance : the probability of generating state j when in state i should be the sae as the probability of generating state i when in state j. Furtherore ergodicity is required: one should be able to reach each icroscopic state by one or ore of these sapling steps. If the sapling procedure satisfies detailed balance and ergodicity the conforations are sapled with unifor probability. A popular acceptance criterion for achieving Boltzann statistics is [13]: where P(accept) = in(1, exp( E)) (21) E = E(new) E(previous) (22) A convenient way to ipleent this in a coputer progra is: 1. copute the change in energy E 2. if E < 0 accept the new conforation always. 3. if E > 0 copute a rando nuber r fro a unifor distribution on the interval [0, 1) and copare it to exp( E). If r < exp( E) accept the new conforation. Otherwise retain the previous conforation. It is essential that, when a ove is rejected, the previous conforation is taken into account once ore in the statistics (for instance in the averages of physical quantities). ot doing so would result in biased statistics and incorrect enseble averages. ote that a ove which leads to excluded volue overlap should be handled as if E = and should always be rejected. The Metropolis procedure described above generates a sequence of conforations, soe identical to the previous one, soe slightly altered. Enseble averages can be calculated siply by averaging over the generated conforations. o weights are needed. One of the drawbacks of the ethod is that subsequent conforations are very uch alike or identical (if the ove was rejected). It takes a nuber of steps to obtain a conforation which is uncorrelated. This nuber is called the correlation tie [14]. Soething siilar happens when one starts a siulation with an initial conforation that is designed by hand (for exaple a copletely stretched chain). Usually such a conforation is artificial and a nuber of steps is needed to reach a conforation that is uncorrelated to the initial conforation. This is called equilibration". The conforations that were generated during equilibration should not be included in the statistics; they ust be discarded. Fig. 2. Monte Carlo oves: kink jup and crankshaft oves. Fig. 3. Monte Carlo oves: end ove.

7 Exploring phase spaces of bioolecules with Monte Carlo ethods 1569 There is a lot of freedo in choosing the Monte Carlo oves, provided they satisfy detailed balance and ergodicity. This freedo of choice should be used to optiize the efficiency of the sapling: the phase space (the space of icroscopic states) should be traversed as fast as possible. We will now give a few exaples of the Monte Carlo algoriths used in polyer siulation. For polyers on a siple cubic lattice several Monte Carlo oves have been suggested. The kink jup and crankshaft oves are illustrated in Fig. 2. The kink jup is a single ato ove while the crankshaft ove oves two atos at a tie. For a linear polyer also oves that reposition a loose end should be included in the sapling algorith (Fig. 3). The cobination of kink jups, crankshaft oves and end oves is called the Verdier Stockayer algorith. Fig. 4. Monte Carlo oves: reptation ove. A reptation ove [15] consists of reoving a onoer fro one chain end and randoly adding a onoer to the other end. (Fig. 4). It is also called the slithering snake ove for obvious reasons. The growth direction can be chosen anew at every attepted step, but it can be shown that it is also correct to let the polyer grow consistently in one direction reversing the growth direction only when a ove is rejected. Another ore involved lattice odel for polyers is the Bond Fluctuation Model [16]. Monoers occupy a square of 4 lattice sites in 2 diensions or a cube of 8 sites in 3 diensions. There is a set of allowed bond vectors between the onoers. This set has been carefully chosen such that no crossing bonds will for during the siulation (provided that there were no crossing bonds in the initial conforation). Only one type of Monte Carlo ove is used: the chosen onoer is oved by one lattice constant. The advantage of the Bond Fluctuation Model is that the chains have ore flexibility: the nuber of allowed bond lengths and bond angles is larger than the siple cubic lattice odel where only bonds of length 1 and bond angles of 90 and 180 are allowed. The Bond Fluctuation Model can be viewed as an interediate between lattice odels and off-lattice odels. Fig. 5. Trapped conforation.

8 1570 A. Bu u There are a nuber of off-lattice odels that are used in polyer siulations.the odels that will be described below are again coarse grained odels. In the bead and spring odel spherical beads represent the onoers. They are not allowed to overlap (excluded volue interaction). The beads are connected by springs: there is a quadratic or FEE potential between bonded onoers. A Monte Carlo ove consists of randoly choosing a onoer and oving it in soe convenient way that satisfies detailed balance. In the freely jointed odel [17] the length of the bonds is held fixed. This can be achieved by starting in a conforation with the correct bond lengths and using Monte Carlo oves that do not change the bond lengths. The tangent spheres odel consists of ipenetrable spheres for onoers and fixed bond lengths which are chosen such that the bonded beads touch one another. 6. Molecular Monte Carlo Monte Carlo siulations of olecules are perfored in a siilar anner to those of atoic systes. However, there are different degrees of freedo to consider and so additional Monte Carlo oves are required. For rigid olecules it is necessary to consider orientational as well as translational degrees of freedo. These are usually cobined into a single trial ove consisting of a centre of ass translation and a rotation around the centre-of-ass. A translational ove is perfored by randoly displacing the centre of ass of the olecule. An orientational ove is then perfored by randoly changing the orientation of the olecule. In order to perfor rotational oves, it is necessary to have a set of coordinates to describe the olecular orientation. One such set are the Euler angles [18]. These are described in ters of a sequence of rotations of a set of Cartesian axes about the origin. The first is through an angle φ about the z axis. This is followed by a rotation ofθ about the new x axis and then a final rotation of ψ about the new z axis. The change in orientation can then be achieved through rando changes in these angles. For flexible olecules, changes to the internal coordinates occur along side translational and rotational oves. Again we need a set of coordinates to describe the olecular conforations. This is coonly done in ters of internal coordinates, the bond lengths, bond angles, and dihedral angles in each olecule. As for rigid olecules, these are usually cobined into a single trial ove. A Monte Carlo ove for a flexible olecule could then consist of a centre of ass displaceent, and rotation of the olecule about its centre of ass, and then a change in an internal dihedral angle, bond angle and bond length. 7. Hybrid ethods It is tepting to cobine the nice features of both Monte Carlo ethod and olecular dynaics ethods to reach higher efficiency for searching in the phase space. A hybrid Monte Carlo ethod was proposed [19] in the field of quantu chroodynaics which contains ferion degree of freedo, and was later applied to condensed-atter systes [20, 21, 22]. However, this algorith does not reach higher efficiency although it was claied to be ore robust. For the lipid bilayer siulation, a new equilibration procedure for the atoic level siulation was proposed recently by Chiu et al. [23], which is also a hybrid algorith with configurational bias Monte Carlo oves searching in the configuration space. 8. Conclusion The iportance of the siulation Monte Carlo ethods in the field of structural olecular biology, as well as the peculiarities of different procedures are pointed out.

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