Application of Biased Metropolis Algorithms: From protons to proteins

Abstract

We show that sampling with a biased Metropolis scheme is essentially equivalent to using the heatbath algorithm. However, the biased Metropolis method can also be applied when an efficient heatbath algorithm does not exist. This is first illustrated with an example from high energy physics (lattice gauge theory simulations). We then illustrate the Rugged Metropolis method, which is based on a similar biased updating scheme, but aims at very different applications. The goal of such applications is to locate the most likely configurations in a rugged free energy landscape, which is most relevant for simulations of biomolecules.

1. Introduction

Consider a random variable y which is sampled with a probability density function (PDF) P(y) on an interval [y₁, y₂]. The cumulative distribution function (CDF) is defined as

$$z=F(y)={\int }_{y_1}^{y}P(y^{\prime })\mathrm{d}{y}^{\prime }\text{and}P(y)=\frac{\mathrm{d}F(y)}{\mathrm{d}y},$$ (1)

where we assume that P(y) is properly normalized so that F(1) = 1 holds. Let us consider two popular local algorithms to achieve this sampling of y in a Markov chain Monte Carlo process.

1.1. Heatbath algorithm (HBA)

The HBA [12] generates y by converting a uniformly distributed random number 0 ≤ z < 1 into

$$y=F^{-1}(z).$$ (2)

We define the acceptance rate by the number of accepted changes divided by the total number of proposed moves. Thus the acceptance rate of the HBA is always 1 (a new value of y is generated on every step). In simulations the inversion of the CDF (1) may be unacceptably slow or the CDF itself may not be a priori known. Then one has to rely on other approaches.

1.2. Metropolis algorithm

In the conventional Metropolis scheme [21] (for historical accounts see Ref. [17] and for a textbook treatment [5]) y_new is generated uniformly in the range [y₁, y₂] (we refer to this as proposal) and then accepted with probability (accept/reject step)

$$p_{\text{Met}}=min\left\{1,\frac{P(y_{\text{new}})}{P(y_{\text{old}})}\right\}.$$ (3)

This process may have a low acceptance rate in the region of interest. Possible remedies are to decrease the proposal range, which makes the moves small, or propose a move multiple times (i.e., multi-hit) Metropolis, which needs a fixed number of hits. Both remedies are worse than an efficient HBA, which for many systems is the optimal solution in the considered class of local algorithms.

We also note that in certain cases faster decorrelation is achieved by using an overrelaxation algorithm [13,10,1] in which the proposed value is chosen as far as possible from the previous one. For such cases as U(1) and SU(2) gauge theories the overrelaxation is microcanonical, i.e., P(y_new) = P(y_old), thus it has to supplement Metropolis, HBA or BMA. In a simulation one normally tunes the ratio between overrelaxation and other algorithms for optimal performance. For instance, in a recent study of U(1) gauge theory at finite temperature [6] on large volumes one BMA sweep was supplemented by two overrelaxation sweeps. The performance of the overrelaxation algorithm mixed with HBA and BMA was also studied for the case of the fundamental-adjoint SU(2) lattice gauge theory [3].

1.3. Biased Metropolis Algorithm (BMA)

Hastings [19] identified proposal probabilities, which are more general than those of the conventional Metropolis scheme, but gave no guidance whether some probabilities may be preferable over others.

If one does not propose y_new uniformly, the name Biased Metropolis Algorithm (BMA) is often used. Some biased Metropolis simulations can be found in the literature where the bias is introduced in an ad hoc way [11,22,14,16,26]. However, it appears that the answer to the question, when to use biased Metropolis updating and when not, is far from clear.

The biased Metropolis scheme [4,9,2] we discuss in the following makes it possible to approximate heatbath probabilities. Like the conventional Metropolis scheme it can be constructed for more general situations than the HBA, but it achieves the performance which is typical for an efficient HBA.

Let us discretize y into n bins as

$$y_1=y^0<y^1<y^2<\dots <y^n=y_2$$ (4)

where lengths of the bins are

$$\mathit{\Delta }{y}^j=y^j-y^{j-1},\text{with}j=1,\dots ,n.$$ (5)

A BMA can then be summarized by the following steps:

• Propose a new value y_new by first randomly picking a bin j_new and then proposing y_new uniformly in the given bin. (r₁, r₂ are uniformly distributed):

  $$j_{\text{new}}=1+\text{Int}[n{r}_1]\text{and}{y}_{\text{new}}=y^{j_{\text{new}}-1}+\mathit{\Delta }{y}^{j_{\text{new}}}{r}_2,$$ (6)

  where Int[n r₁] denotes rounding to the largest integer ≤ n r₁.

• Locate the bin to which y_old belongs: find j_old which satisfies the condition

  $$y^{j_{\text{old}}-1}\le y_{\text{old}}\le y^{j_{\text{old}}}.$$ (7)
• Accept y_new with probability:

  $$p_{\text{BMA}}=min\left\{1,\frac{P(y_{\text{new}})}{P(y_{\text{old}})}\frac{\mathrm{\Delta }{y}^{j_{\text{new}}}}{\mathrm{\Delta }{y}^{j_{\text{old}}}}\right\}.$$ (8)

  p_BMA in (8) differs from p_Met in (3) by the bias Δy^j_new/Δy^j_old.

The scheme outlined in (6)–(8) satisfies the same balance or detailed balance conditions (defined, e.g., in Ref. [5]) as the original Metropolis algorithm. The bias influences only the acceptance rate. Choosing, for example, equidistant partitioning for y (Δy^j = Δy^k for any j, k) would turn the bias into 1 and get us back to the original Metropolis algorithm.

So far the partitioning y^j has not been introduced explicitly. A particular choice that achieves equidistant partitioning on the CDF axis is:

$$\frac{j}{n}=F(y^j)\text{or}{y}^j=F^{-1}\left(\frac{j}{n}\right).$$ (9)

Let us pick a bin initially labeled j and take the limit n → ∞ so that this bin collapses into a point labeled z. This corresponds to the limit:

$$n\to \infty ,j\to \infty \text{so}\text{that}\frac{j}{n}\to z.$$ (10)

Also, as the CDF axis is partitioned into n bins of the size Δz = 1/n, we have Δz → 0 for n → ∞. In this limit

$$\frac{\mathit{\Delta }{y}^j}{\mathrm{\Delta }z}=\frac{1}{\mathrm{\Delta }z}\left(F^{-1}\left(\frac{j}{n}\right)-F^{-1}\left(\frac{j-1}{n}\right)\right)\to \frac{d{F}^{-1}(z)}{dz}=\frac{1}{P(y)}$$ (11)

holds. Then the probability of the accept/reject step (8) is

$$\frac{P(y_{\text{new}})}{P(y_{\text{old}})}\frac{\mathit{\Delta }{y}^{j_{\text{new}}}}{\mathrm{\Delta }{y}^{j_{\text{old}}}}=\frac{P(y_{\text{new}})}{P(y_{\text{old}})}\frac{\mathit{\Delta }{y}^{j_{\text{new}}}/\mathrm{\Delta }z}{\mathrm{\Delta }{y}^{j_{\text{old}}}/\mathrm{\Delta }z}\to \frac{P(y_{\text{new}})}{P(y_{\text{old}})}\frac{1/P(y_{\text{new}})}{1/P(y_{\text{old}})}=1.$$ (12)

So, in the limit of an infinitely small discretization step this BMA approaches the HBA and the acceptance rate converges to 1. Therefore we call a BMA with a partitioning similar to (9) Biased Metropolis–heatbath algorithm (BMHA).

2. Application to lattice gauge theories

The fundamental interactions of Nature known nowadays are the gravitational, electromagnetic, weak and strong interactions. The last three are gauge field theories. For example, the Lagrangian of electrodynamics is invariant under local gauge transformations that belong to the U(1) gauge group.

Description on the quantum level requires switching from the classical to the quantum point of view: all fields in the Lagrangian of the theory are promoted from functions to operators satisfying certain (anti)commutation relations. Then a physical observable of interest is evaluated as an action of some operator on the vacuum state of the theory. Along these lines observables can be represented as path integrals, i.e., integrals over all possible values of the fields that live on a four-dimensional space-time. These integrals can be evaluated using perturbation theory when they can be expanded in series of parameters that are “small” enough to ensure convergence. Quantum Electrodynamics (QED) provides a good example of a theory where many physical observables are calculated order by order in perturbation theory and match experiments with high accuracy. For instance the magnetic moment of the electron is known to seven significant digits [25].

The theory of strong interaction is Quantum Chromodynamics (QCD). The strong force is responsible for binding fundamental constituents of matter, quarks, into protons, neutrons and other particles observed experimentally, and, in turn, protons and neutrons into atomic nuclei. The gauge group of QCD is SU(3). As this group is non-Abelian the theory possesses a richer structure and introduces more difficulties than QED. One of them is a non-perturbative regime where, as the name implies the theory cannot be expanded into a series. To overcome this difficulty lattice gauge theory was introduced by Wilson [28]. In principle QCD allows for calculations of low energy properties, as for instance the mass of the proton, by Markov chain Monte Carlo simulations, which are suitable for calculating path integrals in Euclidean space (connected by a Wick rotation to the physical Minkowski space). However, in practice such calculation require tremendous computational resources, so that (besides Moore’s law at work) major progress on the algorithmic front has still to be made before ultimate answers may be computed.

In the following we illustrate lattice gauge theory calculations on a simple example: The U(1) gauge group. In Ref. [2] we have applied the same method to the SU(2) gauge group and it can also be extended to SU(3). We should emphasize that this deals only with the pure gauge part of the theory, whereas the notorious difficulties of including fermions in these calculations remain at the moment untouched by biased Metropolis calculations.

2.1. U(1) pure gauge theory

For the U(1) gauge group the “matrices” are complex numbers on the unit circle, which can be parameterized by an angle ϕ ∈ [0, 2π). After defining the theory on the links of a four-dimensional lattice the PDF

$$P_{\alpha }(\varphi )=N_{\alpha }{e}^{\alpha cos(\varphi )}$$ (13)

has to be sampled, where α is a parameter associated to the interaction of the link being updated with its environment. The corresponding CDF is

$$F_{\alpha }(\varphi )=N_{\alpha }{\int }_0^{\varphi }d{\varphi }^{\prime }{e}^{\alpha cos({\varphi }^{\prime })}\text{with}{F}_{\alpha }(2\pi )=1.$$ (14)

For U(1) HBAs of type (2) were introduced in Refs. [27,20]. As $F_{\alpha }^{-1}(z)$ is approximated one needs a repeat until accepted (RUA) step to generate the correct distribution, although the acceptance rate is always 1. ¹F_α(ϕ) depends on the parameter α, which incorporates the effect of interaction with the neighbors, and is a function of ϕ, the variable being updated. In the following we consider U(1) gauge theory at a coupling close to the critical point for which one finds 0 ≤ α ≤ 6. For this case F_α(ϕ) is plotted on Fig. 1. Contour lines on the surface represent levels where F_α(ϕ) increases from 0 to 1 by a chosen constant value (in this case 1/8). Lines in the α − ϕ plane are projections of these contours and constitute a level map similar to those used to encode height on maps in geography. To construct a BMHA we need a discretized version of this level map.

Fig. 1.

Cumulative distribution function F_α(ϕ) with the level map in the α − ϕ plane.

Let us discretize the parameter α into m = 2^n₁ = 16 (n₁ = 4) bins. For simplicity we choose equidistant partitioning. Other discretizations are possible too. Then in each αⁱ bin we discretize ϕ using the condition (9) with n = 2^n₂ = 16 (n₂ = 4). In this way we achieve a discretized version of the level map at the bottom of Fig. 1, which is shown in Fig. 2.

Fig. 2.

m × n partitioning of Δϕ^i,j for U(1) at the coupling constant value discussed in the text.

Two two-dimensional arrays are needed: one for storing ϕ^i,j (levels themselves) and another for Δϕ^i,j = ϕ^i,j − ϕ^i,j−1 (distances between levels). Let us assume that for a link being updated α falls into the 11th bin, so i = 11. Finding i is achieved with an operation of the form: Int[m α/α_max] with α_max = 6. For a given αⁱ it is straightforward to apply BMA step (6).

The cross-section of the F_α(ϕ) surface by the α = α¹¹ plane is shown in Fig. 3. To determine the bin label j_old which belongs to the (known) value ϕ_old (BMA step (7)) one may use the n₂-step recursion j → j + 2^i₂sign(ϕ − ϕ^i,j), i₂ → i₂ − 1. Once j_old is known it gives the length of the bin: Δϕ^i,j_old and the final accept/reject step (8) can be applied:

$$p_{\text{BMA}}=min\left\{1,\frac{exp(\alpha cos{\varphi }_{\text{new}})}{exp(\alpha cos{\varphi }_{\text{old}})}\frac{\mathrm{\Delta }{\varphi }_0^{i,j_{\text{new}}}}{\mathrm{\Delta }{\varphi }_0^{i,j_{\text{old}}}}\right\}.$$ (15)

Fig. 3.

Discretization of the cumulative distribution function F_α¹¹(ϕ) for U(1) corresponding to the 11th bin of Fig. 2.

2.1.1. Performance

In our simulations we used a finer discretization than in the figures, m = 32 and n = 128. Table 1 illustrates the performance of the U(1) BMHA for a long run on a 4 × 16³ lattice. At the used coupling the system exhibits critical slowing down, because of its proximity to the U(1) phase transition. We used 16,384 sweeps for reaching equilibrium and, subsequently, 32 × 20,480 sweeps for measurements. Simulations were performed on 2 GHz Athlon PCs with the -O2 option of the (freely available) g77 Fortran compiler.

Table 1.

Efficiency of three algorithms for U(1) lattice gauge theory on a 4 × 16³ lattice at a coupling constant value close to the phase transition value.

U(1)	HBA [20]	Metropolis [21]	BMHA
CPU time [s]	131,111	84,951	107,985
Acceptance rate	1 (1.093 proposals)	0.286	0.972
〈 cosϕ□ 〉	0.59113 (8)	0.59103 (16)	0.59106 (12)
τint	127 (7)	341 (26)	142 (10)

Our comparison is with the Hattori–Nakajima HBA [20] and with the conventional Metropolis algorithm [21]. A direct measure for the performance of an algorithm is the integrated autocorrelation time τ_int. Values of τ_int are given in the Table 1 for the Wilson plaquette, 〈 cos ϕ□ 〉 (a reference physical observable whose expectation value we use to check consistency of the algorithms). Error bars are given in parenthesis and apply to the last digits. They are calculated with respect to 32 bins (jackknife bins in case of τ_int) using the data analysis software of [5].

In this example the CDF is known. We have shown that sampling with the BMHA is essentially equivalent to using the HBA, but can be numerically faster, as shown here for U(1). SU(2) lattice gauge theory with the fundamental-adjoint action is a case for which more substantial gains are achieved by using a BMHA [3]. In the next part of the article we show how a similar biasing procedure can be used when the CDF is not known (making a HBA impossible) and how it can be extended to a multi-variable case.

3. Application to biophysics

Simulations of biomolecules remain one of the major challenges in computational science today. Rugged free energy landscapes are typical for such systems and conventional Metropolis updating suffers from low acceptance rates at the temperatures of interest.

We consider biomolecule models for which the energy E is a function of a number of dynamical variables v_i, i = 1, …, n. The fluctuations in the Gibbs canonical ensemble are described by a probability density function ρ(v₁, …, v_n; T) = const exp(−β E(v₁, …, v_n)), where T is the temperature, β = 1/(kT), and E is the energy of the system. To be consistent with the notation of [4,9] we now use ρ(v₁, …, v_n; T) instead of P(y) introduced in previous one-variable example. Proposing a new variable (with the other variables fixed) from the PDF constitutes a HBA. However, an implementation of a HBA is only possible when the CDF of the PDF can be controlled. In particular this requires the normalization constant in front of the exp(−β E(v₁, …, v_n)) Boltzmann factor. In practice this is often not the case. Then the following strategy provides a useful approximation.

For a range of temperatures

$$T_1>T_2>\dots >T_r>\dots >T_{f-1}>T_f$$ (16)

the simulation at the highest temperature, T₁, is performed with the usual Metropolis algorithm and the results are used to construct an estimator

$$\overline{\rho }(v_1,\dots ,v_n;T_1)$$

which is used to bias the simulation at T₂. Recursively, the estimated PDF

$$\overline{\rho }(v_1,\dots ,v_n;T_{r-1})$$

is expected to be a useful approximation of ρ(v₁, …, v_n; T_r). Formally this means that BMA acceptance step (8) at temperature T_r is of the form

$$P_{\text{RM}}=min\left\{1,\frac{exp\left(-\beta E^{\prime }\right)}{exp(-\beta E)}\frac{\overline{\rho }(v_1,\dots ,v_n;T_{r-1})}{\overline{\rho }(v_1^{\prime },\dots ,v_n^{\prime };T_{r-1})}\right\}$$ (17)

where β = 1/(kT). For this type of BMA where the bias is constructed by using information from a higher temperature the name Rugged Metropolis (RM) was given in Ref. [4].

For the following illustration we use the all-atom energy function Empirical Conformational Energy Program for Peptides/2 (ECEPPs) [24] (and references given therein) as implemented in the Simple Molecular Mechanics for Proteins (SMMPs) [15] program package. Our dynamical variables v_i are the dihedral angles, each chosen to be in the range −π ≤ v_i < π, so that the volume of the configuration space is K = (2π)ⁿ. Details of the energy functions are expected to be irrelevant for the algorithmic questions addressed here. Our test case is the small brain peptide Met-Enkephalin (Tyr-Gly-Gly-Phe-Met), which features 24 dihedral angels as dynamical variables (we use the conventions of Ref. [15]). Besides the ϕ, ψ angles, we keep also the ω angles unconstrained, which are usually restricted to [π − π/9, π + π/9]. This allows us to illustrate the RM idea for a particularly simple case.

3.1. The RM₁ approximation

To get things started, we need to construct an estimator ρ̄(v₁, …, v_n; T_r) from the numerical data of the RM simulation at temperature T_r. Although this is neither simple nor straightforward, a variety of approaches offer themselves to define and refine the desired estimators.

In Ref. [4] the approximation

$$\overline{\rho }(v_1,\dots ,v_n;T_r)=\prod _{i=1}^n{\overline{p}}_i^1(v_i;T_r)$$ (18)

was investigated, where ${\overline{\rho }}_i^1(v_i;T_r)$ are estimators of reduced one-variable PDFs defined by

$${\rho }_i^1(v_i;T)={\int }_{-\pi }^{+\pi }\prod _{j\ne i}d{v}_j\rho (v_1,\dots ,v_n;T).$$ (19)

The resulting algorithm, called RM₁, constitutes the simplest RM scheme possible.

The cumulative distribution functions are defined by

$$F_i(v)-{\int }_{-\pi }^v\mathrm{d}{v}^{\prime }{\rho }_i^1(v^{\prime }).$$ (20)

The estimate of F₁₀, the cumulative distribution function for the dihedral angle Gly-3 ϕ(v₁₀), from the vacuum simulations at our highest temperature, T₁ = 400 K, is shown in Fig. 4. For our plots in this part of the paper we use degrees, while we use radians in our theoretical discussions and in the computer programs. Fig. 4 is obtained by sorting all n_dat values of v₁₀ in our time series in ascending order and increasing the values of F₁₀ by 1/n_dat whenever a measured value of v₁₀ is encountered. Using a heapsort approach, the sorting is done in n_dat log₂(n_dat) steps (see, e.g., Ref. [5]).

Fig. 4.

Estimate of the cumulative distribution function for the Met-Enkephalin dihedral angle v₁₀ (Gly-3 ϕ) at 400 K.

Fig. 5 shows the cumulative distribution function for v₉ (Gly-2 ω) at 400 K, which is the angle of lowest acceptance rate in the conventional Metropolis updating. This distribution function corresponds to a histogram narrowly peaked around ±π, which is explained by the specific electronic hybridization of the CO–N peptide bond. From the grid shown in Fig. 5 it is seen that the RM₁ updating concentrates the proposal for this angle in the range slightly above −π and slightly below +π. Thus the procedure has a similar effect as the often used restriction to the range [π − π/9, π + π/9], which is also the default implementation in SMMP.

Fig. 5.

Estimate of the cumulative distribution function for the Met-Enkephalin dihedral angle v₉ (Gly-2 ω) at 400 K.

After the empirical CDFs are constructed for each angle v_i, they are discretized using the condition (9). Here we denote differences (5) needed for the bias as

$$\mathit{\Delta }{v}_{i,j}=v_{i,j}-v_{i,j-1}\text{with}{v}_{i,0}=-\pi .$$ (21)

The RM₁ updating of each dihedral angle v_i follows the BMA procedure (6)–(8). The accept/reject step in the v_i,j notation is

$$p_{{\text{RM}}_1}=min\left\{1,\frac{exp(-\beta E^{\prime })}{exp(-\beta E)}\frac{\mathit{\Delta }{v}_{i,j_{\text{new}}}}{\mathrm{\Delta }{v}_{i,j_{\text{old}}}}\right\}.$$ (22)

3.2. The RM₂ approximation

In Ref. [9] the RM₁ scheme of Eq. (22) was generalized to the simultaneous updating of two dihedral angles. For i₁ ≠ i₂ the reduced two-variable PDFs are defined by

$${\rho }_{i_1,i_2}^2(v_{i_1},v_{i_2};T)={\int }_{-\pi }^{+\pi }\prod _{j\ne i_1,i_2}d{v}_j\rho (v_j,\dots ,v_n;T).$$ (23)

The one-variable cumulative distribution functions F_i1 and the discretization v_i1,_j, j = 0, …, n are already given by Eqs. (20) and (21). We define conditional CDFs by

$$F_{i_1,i_2;j}(v)={\int }_{-\pi }^{v}d{v}_{i_2}{\int }_{v_{i_1,j-1}}^{v_{i_1,j}}d{v}_{i_1}{\rho }_{i_1,i_2}^2(v_{i_1},v_{i_2})$$ (24)

for which the normalization F_i1,i2;j(π) = 1/n holds. To extend the RM₁ updating to two variables we define for each integer k = 1, …, n the value F_i1,i2;j,k = k/n². Next we define v_i1,i2;j,k through F_i1,i2;j,k = F_i1,i2;j(v_i1,i2;j,k) and also the differences

$$\mathit{\Delta }{v}_{i_1,i_2;j,k}=v_{i_1,i_2;j,k}-v_{i_1,i_2;j,k-1}\text{with}{v}_{i_1,i_2;j,0}=-\pi .$$ (25)

The RM₂ procedure for the simultaneous update of (v_i1, v_i2) is then specified as follows:

• Propose a new value v_i1,new using two uniform random numbers r₁, r₂ (BMA step (6) for the angle i₁):

  $$j_{\text{new}}=1+\text{Int}[n{r}_1]\text{and}{v}_{i_1,\text{new}}=v_{i_1,j_{\text{new}}-1}+\mathit{\Delta }{v}_{i_1,j_{\text{new}}}{r}_2.$$ (26)
• Propose a new value v_{i2, new} using two uniform random numbers r₃, r₄ (BMA step (6) for the angle i₂):

  $$k_{\text{new}}=1+\text{Int}[n{r}_3]\text{and}{v}_{i_2,\text{new}}=v_{i_1,i_2;j_{\text{new}},k_{\text{new}}-1}+\mathrm{\Delta }{v}_{i_1,i_2;j_{\text{new}},k_{\text{new}}}{r}_4.$$ (27)

• Find the bin index j_old for the present angle v_{i1, old} through v_{i1, jold−1} ≤ v_{i1, old} ≤ v_{i1, jold}, just like for RM₁ updating (BMA step (7) for v_i1).

• Find the bin index k_old for the present angle v_{i2, old} through v_{i1,i2;jold,kold−1} ≤ v_{i2, old} ≤ v_{i1,i2;jold,kold} (again step (7) but for v_i2).

• Accept (v_{i1 ,new,} v_i2,new) with the probability

  $$p_{{\text{RM}}_2}=min\left\{1,\frac{exp(-\beta E^{\prime })}{exp(-\beta E)}\frac{\mathit{\Delta }{v}_{i_1,j_{\text{new}}}}{\mathrm{\Delta }{v}_{i_1,j_{\text{old}}}}\frac{\mathrm{\Delta }{v}_{i_1,i_2;j_{\text{new}},k_{\text{new}}}}{\mathrm{\Delta }{v}_{i_1,i_2;j_{\text{old}},k_{\text{old}}}}\right\}.$$ (28)

  As for RM₁, estimates of the conditional CDFs and the intervals Δ v_{i1, i2;j,k} are obtained from the conventional Metropolis simulation at 400 K. In the following we focus on the pairs (v₇, v₈), (v₁₀, v₁₁) and (v₁₅, v₁₆). These angles correspond to the largest integrated autocorrelation times of the RM₁ procedure and are expected to be strongly correlated with one another because they are pairs of dihedral angles around a C_α atom.

The bias of the acceptance probability given in Eq. (28) is governed by the areas

$$\mathrm{\Delta }{A}_{i_1,i_2;j,k}=\mathrm{\Delta }{v}_{i_1,j}\mathrm{\Delta }{v}_{i_1,i_2;j,k}.$$

For i₁ = 7 and i₂ = 8 our 400 K estimates of these areas are depicted in Fig. 6. For the RM₂ procedure these areas take the role which the intervals on the abscissa of Fig. 4 play for RM₁ updating. The small and the large areas are proposed with equal probabilities, so the a priori probability for our two angles is high in a small area and low in a large area. In Fig. 6 the largest area is 503.4 times the smallest area. Areas of high probability correspond to allowed regions in the Ramachandran map of a Gly residue [23].

Fig. 6.

Areas of equal probabilities (sorting v₇ then v₈).

Note that the order of the angles matters. The difference between Figs. 6 and 7 is that we plot in Fig. 6 the areas A_7,8;j,k and in Fig. 7 the areas A_8,7;j,k while the labeling of the axes is identical. This means that for Fig. 6 sorting is first done on the angle v₇ (regardless of the value of v₈) and then done on v₈ for which the corresponding value of v₇ is within a particular bin Δv₇, but for Fig. 7 it is first done one v₈ and then on v₇. In Fig. 7 the largest area is 396.4 times the smallest area.

Fig. 7.

Areas of equal probabilities (sorting v₈ then v₇).

Figs. 8 and 9 give plots for the (v₁₀, v₁₁) and (v₁₅, v₁₆) pairs in which the angle with the smaller subscript is sorted first. The ratio of the largest area over the smallest area is 650.9 for (v₁₀, v₁₁) and 2565.8 for (v₁₅, v₁₆). The large number in the latter case is related to the fact that (v₁₅, v₁₆) is the pair of ϕ, ψ angles around the C_α atom of Phe-4, for which positive ϕ values are disallowed [23].

Fig. 8.

Areas of equal probabilities (sorting v₁₀ then v₁₁).

Fig. 9.

Areas of equal probabilities (sorting v₁₅ then v₁₆).

3.3. Performance

The RM₂ scheme which we have tested adds updates for the three pairs (v₇, v₈), (v₁₀, v₁₁) and (v₁₅, v₁₆) after one-angle updates for all the 24 angles with the RM₁ scheme. For each pair both orders of sorting are used, so that we add altogether six new updates.

For the angles used in the figures the performance of the RM₁ and RM₂ schemes is illustrated in Table 2. Integrated autocorrelation times (computed along the lines of [5]) are compiled. The units are chosen, so that the computer time needed with the different algorithms to achieve the same accuracy is directly proportional to the integrated autocorrelation times of the table. At 300 K we read off that the improvement over the conventional Metropolis algorithm is typically a factor of two for the RM₁ and a factor of four for the RM₂ approach. It stays about the same at lower temperatures [9].

Table 2.

Integrated autocorrelation times for dihedral angle movements in units of 32 sweeps for Metropolis and RM₁ and in units of 26 sweeps for RM₂.

var	400 K (Metro)	300 K (Metro)	300 K (RM1)	300 K (RM2)
v7	5.83 (29)	103 (14)	52.9 (4.3)	24.3 (1.3)
v8	7.36 (22)	125 (12)	74.2 (6.9)	35.0 (2.7)
v9	4.39 (13)	32.0 (2.2)	14.2 (1.0)	8.84 (48)
v10	9.08 (88)	124 (12)	80.6 (6.9)	34.3 (2.8)
v11	5.39 (45)	105 (08)	72.4 (5.5)	31.3 (1.9)
v15	6.72 (28)	105 (12)	45.6 (2.7)	27.5 (4.5)
v16	9.28 (28)	133 (09)	75.2 (5.2)	33.9 (2.1)
E	4.89 (21)	50.7 (5.0)	26.0 (1.4)	14.2 (0.7)

4. Conclusions

High energy physics and biophysics are certainly far apart in their scientific objectives. Nevertheless quite similar computational techniques allow for efficient Metropolis simulations in either field. Cross-fertilization may go in both directions. For instance, generalized ensemble techniques propagated from lattice gauge theory [8] over statistical physics [7] into biophysics [18]. It appears that biased Metropolis techniques propagate in the opposite direction. It remains to be seen whether they will indeed gain widespread acceptance.