Generalized-Ensemble Algorithms for Simulations of Complex Molecular Systems

Abstract

In molecular simulations of complex systems with many degrees of freedom, conventional Monte Carlo and molecular dynamics simulations in canonical ensemble or isobaric-isothermal ensemble suffer from a great difficulty, in which simulations tend to get trapped in states of energy local minima. A simulation in generalized ensemble performs a random walk in specified variables and overcomes this difficulty. In this chapter, we review the generalized-ensemble algorithms. Replica-exchange method, multicanonical algorithm, and their extensions are described. Some simulation results based on these generalized-ensemble algorithms are also presented.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

4.1 Introduction

In complex molecular systems such as biomolecular systems, conventional Monte Carlo (MC) and molecular dynamics (MD) simulations at low temperatures in the canonical ensemble and those at low temperatures or high pressures in the isobaric-isothermal ensemble tend to get trapped in states of energy local minima, giving results in error. In order to overcome this difficulty, a class of simulation methods, which are referred to as the generalized-ensemble algorithms, are often employed (for reviews, see e.g., Refs. [1–5]). In a generalized-ensemble simulation, each state is weighted by a non-Boltzmann probability weight factor so that a random walk in potential energy space may be realized. The random walk allows the simulation to overcome any energy barrier and to sample much wider conformational space than by conventional methods. The generalized-ensemble algorithm was introduced to the molecular simulation field almost 20 years ago [6].

One of the most well-known generalized-ensemble algorithms is perhaps replica-exchange method (REM) [7] (see Ref. [8] for the MD version). Multiple replicas of the system in the canonical ensemble at different temperatures are simulated simultaneously, and every few steps, a pair of replicas at neighboring temperatures is exchanged. This causes a random walk in temperature for each replica, and the simulation can avoid getting trapped in states of energy local minima.

REM was extended to multidimensions/multivariables so that not only temperature but also other parameter values of the system are exchanged, and the method is referred to as multidimensional replica-exchange method (MREM) [9]. Various special cases of MREM were then proposed [10–15] (MREM is also known as Hamiltonian replica-exchange method [10]).

Another widely used generalized-ensemble algorithm is multicanonical algorithm (MUCA) [16, 17] (for a textbook, see, e.g., Ref. [18]; see also Refs. [19, 20] for the MD version). The probability weight factor, which is referred to as the multicanonical weight factor, is defined to be inversely proportional to the density of states so that a flat distribution in potential energy may be obtained. The uniform distribution induces a free random walk in the potential energy space, and the multiple-minima problem is overcome.

MUCA was extended so that flat distributions in parameters other than potential energy and/or multidimensional parameter space may be realized [21–28].

We remark that general formulations for multidimensional/multivariable generalized-ensemble algorithms (including REM and MUCA) were recently worked out [29–31].

In this chapter, we describe both REM and MUCA. We then present several of newly developed generalized-ensemble algorithms that are multidimensional/multicomponent extensions of REM and MUCA. The first algorithm is an example of MREM and referred to as the van der Waals replica-exchange method (vWREM) [32], where different values of van der Waals radius are exchanged. The second one is the multioverlap algorithm (MUOV), which performs a random walk in the overlap space instead of the potential energy space [33–35]. Further extension of MUOV, which is referred to the multicanonical-multioverlap algorithm (MUCA-MUOV) [36–38] and realizes a random walk both in the potential energy space and the overlap space, is then given. The fourth method that we present here is the multibaric-multithermal algorithm (MUBATH), which realizes a random walk both in the potential energy space and in the volume space [39–45]. We remark that other generalized-ensemble algorithms for the isobaric-isothermal ensemble have also been developed [46–49]. Finally, examples of some simulation results based on these methods are presented.

4.2 Generalized-Ensemble Algorithms

4.2.1 Replica-Exchange Method

Let us consider a system of N atoms of mass m _k (k = 1, …, N) with their coordinate vectors and momentum vectors denoted by q ≡ {{ q} ₁, …, { q} _N } and p ≡ {{ p} ₁, …, { p} _N }, respectively. The Hamiltonian H(q, p) of the system is the sum of the kinetic energy K(p) and the potential energy E(q):

$$H(q,p) = K(p) + E(q),$$

(4.1)

where

$$K(p) ={ \sum \nolimits }_{k=1}^{N}\frac{{{\text{ <Emphasis Type="Bold">$p$}</Emphasis>}_{k}}^{2}} {2{m}_{k}} .$$

(4.2)

In the canonical ensemble at temperature T, each state x ≡ (q, p) with the Hamiltonian H(q, p) is weighted by the Boltzmann factor:

$${W}_{\mathrm{B}}(x;T) =\exp \left (-\beta H(q,p)\right ),$$

(4.3)

where the inverse temperature β is defined by β = 1 ∕ k _B T (k _B is the Boltzmann constant). The average kinetic energy at temperature T is then given by

$${ \left < K(p)\right >}_{T} ={ \left < {\sum \nolimits }_{k=1}^{N}\frac{{{\text{ <Emphasis Type="Bold">$p$}</Emphasis>}_{k}}^{2}} {2{m}_{k}} \right >}_{T} = \frac{3} {2}N{k}_{\mathrm{B}}T.$$

(4.4)

Because the coordinates q and momenta p are decoupled in Eq. 4.1, we can suppress the kinetic energy part and can write the Boltzmann factor as

$${W}_{\mathrm{B}}(x;T) \propto {W}_{\mathrm{B}}(E;T) =\exp (-\beta E).$$

(4.5)

The canonical probability distribution of potential energy P _NVT(E; T) is then given by the product of the density of states n(E) and the Boltzmann weight factor W _B(E; T):

$${P}_{\mathrm{NVT}}(E;T) \propto n(E){W}_{\mathrm{B}}(E;T).$$

(4.6)

Because n(E) is a rapidly increasing function and the Boltzmann factor decreases exponentially, the canonical ensemble yields a bell-shaped distribution of potential energy which has a maximum around the average potential energy at temperature T. The conventional MC or MD simulations at constant temperature are expected to yield P _NVT(E; T). A MC simulation based on the Metropolis algorithm [50] is performed with the following transition probability from a state x of potential energy E to a state x ^′ of potential energy E ^′:

$$w(x \rightarrow {x}^{{\prime}}) = \mathrm{min}\left (1, \frac{{W}_{\mathrm{B}}({E}^{{\prime}};T)} {{W}_{\mathrm{B}}(E;T)} \right ) = \mathrm{min}\left (1,\exp \left (-\beta \Delta E\right )\right ),$$

(4.7)

where

$$\Delta E = {E}^{{\prime}}- E.$$

(4.8)

A MD simulation, on the other hand, is based on the following Newton equations of motion:

$$\begin{array}{rcl} \dot{{\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}}& =& \frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}} {{m}_{k}} , \end{array}$$

(4.9)

$$\begin{array}{rcl}\dot{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}}& =& - \frac{\partial E} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}} = \text{ <Emphasis Type="Bold">F}</Emphasis>{\!}_{k}\!,\end{array}$$

(4.10)

where { F}​ _k is the force acting on the kth atom (k = 1, ⋯ , N). This set of equations actually yield the microcanonical ensemble, however, and we have to add a thermostat in order to obtain the canonical ensemble at temperature T. Here, we just follow Nosé’s prescription [51, 52], and we have

$$\begin{array}{rcl} \dot{{\text{ <Emphasis Type="Bold">q}</Emphasis>}}_{k}& =& \frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}} {{m}_{k}} , \end{array}$$

(4.11)

$$\begin{array}{rcl}\dot{{\text{ <Emphasis Type="Bold">p}</Emphasis>}}_{k}& =& - \frac{\partial E} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}} -\frac{\dot{s}} {s}{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k} = \text{ <Emphasis Type="Bold">F}</Emphasis>{\!}_{k} -\frac{\dot{s}} {s}\ {\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}, \end{array}$$

(4.12)

$$\begin{array}{rcl} \dot{s}& =& s\ \frac{{P}_{s}} {Q} , \end{array}$$

(4.13)

$$\begin{array}{rcl} \dot{{P}}_{s}& =& {\sum \nolimits }_{k=1}^{N}\frac{{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}}^{2}} {{m}_{k}} - 3N{k}_{\mathrm{B}}T = 3N{k}_{\mathrm{B}}\left (T(t) - T\right ),\end{array}$$

(4.14)

where s is Nosé’s scaling parameter, P _s is its conjugate momentum, Q is its mass, and the “instantaneous temperature” T(t) is defined by

$$T(t) = \frac{1} {3N{k}_{\mathrm{B}}}{ \sum \nolimits }_{k=1}^{N}\frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}{(t)}^{2}} {{m}_{k}} .$$

(4.15)

However, in practice, it is very difficult to obtain accurate canonical distributions of complex systems at low temperatures by conventional MC or MD simulation methods. This is because simulations at low temperatures tend to get trapped in one or a few of local-minimum-energy states. This difficulty is overcome by, for instance, the generalized-ensemble algorithms, which greatly enhance conformational sampling.

The replica-exchange method (REM) [7] is one of effective generalized-ensemble algorithms. The system for REM consists of Mnoninteracting copies (or replicas) of the original system in the canonical ensemble at M different temperatures T _m (m = 1, …, M). We arrange the replicas so that there is always exactly one replica at each temperature. Then there exists a one-to-one correspondence between replicas and temperatures; the label i (i = 1, …, M) for replicas is a permutation of the label m (m = 1, …, M) for temperatures, and vice versa:

$$\left \{\begin{array}{rl} i& =\ i(m)\ \equiv \ f(m), \\ m& =\ m(i)\ \equiv \ {f}^{-1}(i), \end{array} \right .$$

(4.16)

where f(m) is a permutation function of m and f ^− 1(i) is its inverse.

Let $X = \left \{{x}_{1}^{[i(1)]},\ldots ,{x}_{M}^{[i(M)]}\right \} = \left \{{x}_{m(1)}^{[1]},\ldots ,{x}_{m(M)}^{[M]}\right \}$ stand for a “state” in this generalized ensemble. Each “substate” x _m ^[i] is specified by the coordinates q ^[i] and momenta p ^[i] of N atoms in replica i at temperature T _m :

$${x}_{m}^{[i]} \equiv {\left ({q}^{[i]},{p}^{[i]}\right )}_{ m}.$$

(4.17)

Because the replicas are noninteracting, the weight factor for the state X in this generalized ensemble is given by the product of Boltzmann factors for each replica (or at each temperature):

$${W}_{\mathrm{REM}}(X) = {\prod \nolimits }_{i=1}^{M}\exp \left \{-{\beta }_{ m(i)}H\left ({q}^{[i]},{p}^{[i]}\right )\right \} = {\prod \nolimits }_{m=1}^{M}\exp \left \{-{\beta }_{ m}H\left ({q}^{[i(m)]},{p}^{[i(m)]}\right )\right \},$$

(4.18)

where i(m) and m(i) are the permutation functions in Eq. 4.16.

We now consider exchanging a pair of replicas in this ensemble. Suppose we exchange replicas i and j which are at temperatures T _m and T _n , respectively:

$$X = \left \{\ldots ,{x}_{m}^{[i]},\ldots ,{x}_{ n}^{[j]},\ldots \right \}\rightarrow \ {X}^{{\prime}} = \left \{\ldots ,{x}_{ m}^{[j]{\prime}},\ldots ,{x}_{ n}^{[i]{\prime}},\ldots \right \}.$$

(4.19)

Here, i, j, m, and n are related by the permutation functions in Eq. 4.16, and the exchange of replicas introduces a new permutation function f ^′:

$$\left \{\begin{array}{rl} i& = f(m)\rightarrow j = {f}^{{\prime}}(m), \\ j & = f(n)\rightarrow i = {f}^{{\prime}}(n).\\ \end{array} \right .$$

(4.20)

The exchange of replicas can be written in more detail as

$$\left \{\begin{array}{rl} {x}_{m}^{[i]} \equiv {\left ({q}^{[i]},{p}^{[i]}\right )}_{m}&\rightarrow \ {x}_{m}^{[j]{\prime}}\equiv {\left ({q}^{[j]},{p}^{[j]{\prime}}\right )}_{m}, \\ {x}_{n}^{[j]} \equiv {\left ({q}^{[j]},{p}^{[j]}\right )}_{n}&\rightarrow \ {x}_{n}^{[i]{\prime}}\equiv {\left ({q}^{[i]},{p}^{[i]{\prime}}\right )}_{n},\end{array} \right .$$

(4.21)

where the definitions for p ^[i]′ and p ^[j]′ will be given below.

In the original implementation of the REM [7], Monte Carlo method was used, and only the coordinates q (and the potential energy function E(q)) had to be taken into account. In molecular dynamics method, on the other hand, we also have to deal with the momenta p. We proposed the following momentum assignment in Eq. 4.21 [8]:

$$\left \{\begin{array}{rl} {p}^{[i]{\prime}}& \equiv \sqrt{ \frac{{T}_{n } } {{T}_{m}}}\ {p}^{[i]}, \\ {p}^{[j]{\prime}}& \equiv \sqrt{\frac{{T}_{m } } {{T}_{n}}} \ {p}^{[j]}, \end{array} \right .$$

(4.22)

which we believe is the simplest and the most natural. This assignment means that we just rescale uniformly the velocities of all the atoms in the replicas by the square root of the ratio of the two temperatures so that the temperature condition in Eq. 4.4 may be satisfied immediately after replica exchange is accepted. We remark that general momentum rescaling formulae were derived for various thermostats in Ref. [53].

The transition probability of this replica-exchange process is given by the usual Metropolis criterion:

$$w(X \rightarrow {X}^{{\prime}}) \equiv w\left ({x}_{ m}^{[i]}\left \vert \ {x}_{ n}^{[j]}\right .\right ) = \mathrm{min}\left (1, \frac{{W}_{\mathrm{REM}}({X}^{{\prime}})} {{W}_{\mathrm{REM}}(X)} \right ) = \mathrm{min}\left (1,\exp \left (-\Delta \right )\right ),$$

(4.23)

where in the second expression (i.e., w(x _m ^[i] | x _n ^[j])), we explicitly wrote the pair of replicas (and temperatures) to be exchanged. From Eq. 4.22, the kinetic energy terms all cancel out in Eq. 4.23, and Δ becomes

$$\begin{array}{rcl} \Delta & =& {\beta }_{m}\left (E\left ({q}^{[j]}\right ) - E\left ({q}^{[i]}\right )\right ) - {\beta }_{ n}\left (E\left ({q}^{[j]}\right ) - E\left ({q}^{[i]}\right )\right ),\end{array}$$

(4.24)

$$\begin{array}{rcl} & =& \left ({\beta }_{m} - {\beta }_{n}\right )\left (E\left ({q}^{[j]}\right ) - E\left ({q}^{[i]}\right )\right ). \end{array}$$

(4.25)

Here, i, j, m, and n are related by the permutation functions in Eq. 4.16 before the replica exchange:

$$\left \{\begin{array}{ll} i & = f(m),\\ j&= f(n). \end{array} \right .$$

(4.26)

Note that after introducing the momentum rescaling in Eq. 4.22, we have the same Metropolis criterion for replica exchanges, i.e., Eqs. 4.23 and 4.25, for both MC and MD versions.

Without loss of generality, we can assume T ₁ < T ₂ < ⋯ < T _M . The lowest temperature T ₁ should be sufficiently low so that the simulation can explore the global-minimum-energy region, and the highest temperature T _M should be sufficiently high so that no trapping in an energy-local-minimum state occurs. A REM simulation is then realized by alternately performing the following two steps:

• 1. Each replica in canonical ensemble of the fixed temperature is simulated simultaneously and independently for a certain MC or MD steps.

• 2. A pair of replicas at neighboring temperatures, say x _m ^[i] and x _m + 1 ^[j], is exchanged with the probability $w\left ({x}_{m}^{[i]}\left \vert {x}_{m+1}^{[j]}\right .\right )$ in Eq. 4.23.

A random walk in “temperature space” is realized for each replica, which in turn induces a random walk in potential energy space. This alleviates the problem of getting trapped in states of energy local minima.

After a long production run of a REM simulation, the canonical expectation value of a physical quantity A at temperature T _m (m = 1, …, M) can be calculated by the usual arithmetic mean:

$${ \left < A\right >}_{{T}_{m}} = \frac{1} {{n}_{m}}{ \sum \nolimits }_{k=1}^{{n}_{m} }A\left ({x}_{m}(k)\right ),$$

(4.27)

where x _m (k) (k = 1, …, n _m ) are the configurations obtained at temperature T _m and n _m is the total number of measurements made at T = T _m . The expectation value at any intermediate temperature T ( = 1 ∕ k _Bβ) can also be obtained as follows:

$${ \left < A\right >}_{T} = \frac{{\sum \nolimits }_{E}\ A(E){P}_{\mathrm{NVT}}(E;T)} {{\sum \nolimits }_{E}\ {P}_{\mathrm{NVT}}(E;T)} = \frac{{\sum \nolimits }_{E}\ A(E)n(E)\exp (-\beta E)} {{\sum \nolimits }_{E}\ n(E)\exp (-\beta E)} .$$

(4.28)

Here, the explicit form of the physical quantity A should be known as a function of potential energy E. For instance, A(E) = E gives the average potential energy ${\left < E\right >}_{T}$ as a function of temperature, and $A(E) = {\beta }^{2}{(E -{\left < E\right >}_{T})}^{2}$ gives specific heat.

The density of states n(E) in Eq. 4.28 is given by the multiple-histogram reweighting techniques [54, 55] as follows (an extension of the multiple-histogram method is also referred to as the weighted histogram analysis method (WHAM) [55]). Let N _m (E) and n _m be respectively the potential energy histogram and the total number of samples obtained at temperature T _m  = 1 ∕ k _Bβ _m (m = 1, ⋯ , M). The best estimate of the density of states is then given by [54, 55]

$$n(E) = \frac{{\sum \nolimits }_{m=1}^{M}\ {g}_{ m}^{-1}\ {N}_{ m}(E)} {{\sum \nolimits }_{m=1}^{M}\ {g}_{ m}^{-1}\ {n}_{ m}\ \exp ({f}_{m} - {\beta }_{m}E)},$$

(4.29)

where we have for each m ( = 1, …, M)

$$\exp (-{f}_{m}) ={ \sum \nolimits }_{E}\ n(E)\ \exp (-{\beta }_{m}E).$$

(4.30)

Here, g _m  = 1 + 2τ _m , and τ _m is the integrated autocorrelation time at temperature T _m . For many systems, the quantity g _m can safely be set to be a constant in the reweighting formulae [55], and hereafter, we set g _m  = 1. Note that Eqs. 4.29 and 4.30 are solved self-consistently by iteration [54, 55] to obtain the density of states n(E) and the dimensionless Helmholtz free energy f _m .

Moreover, the ensemble averages of any physical quantity A (including those that cannot be expressed as functions of potential energy) at any temperature T ( = 1 ∕ k _Bβ) can now be obtained from the “trajectory” of configurations of the production run. Namely, we first obtain f _m (m = 1, ⋯ , M) by solving Eqs. 4.29 and 4.30 self-consistently, and then we have [56]

$${ \left < A\right >}_{T} = \frac{{\sum \nolimits }_{m=1}^{M}{ \sum \nolimits }_{k=1}^{{n}_{m} }A({x}_{m}(k)) \frac{1} {{\sum \nolimits }_{\mathcal{l}=1}^{M}{n}_{ \mathcal{l}}\exp \left [{f}_{\mathcal{l}} - {\beta }_{\mathcal{l}}E({x}_{m}(k))\right ]}\exp \left [-\beta E({x}_{m}(k))\right ]} {{\sum \nolimits }_{m=1}^{M}{ \sum \nolimits }_{k=1}^{{n}_{m} } \frac{1} {{\sum \nolimits }_{\mathcal{l}=1}^{M}{n}_{ \mathcal{l}}\exp \left [{f}_{\mathcal{l}} - {\beta }_{\mathcal{l}}E({x}_{m}(k))\right ]}\exp \left [-\beta E({x}_{m}(k))\right ]} ,$$

(4.31)

where x _m (k) (k = 1, ⋯ , n _m ) are the configurations obtained at temperature T _m .

4.2.2 Extensions of the Replica-Exchange Method

4.2.2.1 Multidimensional Replica-Exchange Method

We now describe the multidimensional replica-exchange method (MREM) [9]. The crucial observation that led to this algorithm is as follows: As long as we have Mnoninteracting replicas of the original system, the Hamiltonian H(q, p) of the system does not have to be identical among the replicas, and it can depend on a parameter with different parameter values for different replicas.

Let us consider a generalized potential energy function E _λ(x), which depends on L parameters λ = (λ⁽¹⁾, …, λ^(L)), of a system in state x. The system for MREM consists of M noninteracting replicas of the original system in the “canonical ensemble” with M( = M ₀ ×M ₁ ×⋯ ×M _L ) different parameter sets Λ _m (m = 1, …, M), where ${\Lambda }_{m} \equiv ({T}_{{m}_{0}},{\lambda }_{m}) \equiv ({T}_{{m}_{0}},{\lambda }_{{m}_{1}}^{(1)},\ldots ,{\lambda }_{{m}_{L}}^{(L)})$ with m ₀ = 1, …, M ₀, m _ℓ  = 1, …, M _ℓ (ℓ = 1, …, L). Because the replicas are noninteracting, the weight factor is given by the product of Boltzmann-like factors for each replica:

$${W}_{\mathrm{MREM}} \equiv {\prod \nolimits }_{{m}_{0}=1}^{{M}_{0} }{ \prod \nolimits }_{{m}_{1}=1}^{{M}_{1} }\cdots {\prod \nolimits }_{{m}_{L}=1}^{{M}_{L} }\exp \left (-{\beta }_{{m}_{0}}{E}_{{\lambda }_{m}}\right ).$$

(4.32)

Without loss of generality, we can order the parameters so that ${T}_{1}<\,{T}_{2}<\,\cdots < {T}_{{M}_{0}}$ and ${\lambda }_{1}^{(\mathcal{l})} < {\lambda }_{2}^{(\mathcal{l})} < \cdots < {\lambda }_{{M}_{\mathcal{l}}}^{(\mathcal{l})}$ (for each ℓ = 1, ⋯ , L). A MREM simulation is realized by alternately performing the following two steps:

• 1. For each replica, a “canonical” MC or MD simulation at the fixed parameter values is carried out simultaneously and independently for a certain steps.

• 2. We exchange a pair of replicas i and j which are at the parameter sets Λ _m and Λ _m + 1, respectively. The transition probability for this replica-exchange process is given by

  $$w({\Lambda }_{m} \leftrightarrow { \Lambda }_{m+1}) = \mathrm{min}\left (1,\exp (-\Delta )\right ),$$

  (4.33)

  where we have

  $$\Delta = \left ({\beta }_{{m}_{0}} - {\beta }_{{m}_{0}+1}\right )\left ({E}_{{\lambda }_{m}}\left ({q}^{[j]}\right ) - {E}_{{ \lambda }_{m}}\left ({q}^{[i]}\right )\right ),$$

  (4.34)

  for T-exchange, and

  $$\begin{array}{rcl} \Delta = {\beta }_{{m}_{0}}\left [\left ({E}_{{\lambda }_{{m}_{ \mathcal{l}}}}({q}^{[j]}) - {E}_{{ \lambda }_{{m}_{\mathcal{l}}}}({q}^{[i]})\right ) -\left ({E}_{{ \lambda }_{{m}_{\mathcal{l}}+1}}({q}^{[j]}) - {E}_{{ \lambda }_{{m}_{\mathcal{l}}+1}}({q}^{[i]})\right )\right ],& & \\ & & \end{array}$$

  (4.35)

  for λ^(ℓ)-exchange (for one of ℓ = 1, ⋯ , L). Here, q ^[i] and q ^[j] stand for configuration variables for replicas i and j, respectively, before the replica exchange.

4.2.2.2 van der Waals Replica-Exchange Method

We now describe a special example of MREM, which we refer to as the van der Waals Replica-Exchange Method (vWREM) [32].

We consider a system consisting of solute molecule(s) in explicit solvent. We can write the total potential energy as follows:

$$\begin{array}{rcl}{ E}_{\lambda }(q) = {E}_{\mathrm{p}}({q}_{\mathrm{p}}) + {E}_{\mathrm{ps}}({q}_{\mathrm{p}},{q}_{\mathrm{s}}) + {E}_{\mathrm{s}}({q}_{\mathrm{s}}),& &\end{array}$$

(4.36)

where E _p is the potential energy for the atoms in the solute only, E _ps is the interaction term between solute atoms and solvent atoms, and E _s is the potential energy for the atoms of the solvent molecules only. Here, q = { q _p, q _s}, where q _p and q _s are the coordinate vectors of the solute atoms and the solvent atoms, respectively, and denoted by ${q}_{\mathrm{p}} \equiv \left \{{\mbox{ <Emphasis Type="Bold">q</Emphasis>}}_{1},\ldots ,{\mbox{ <Emphasis Type="Bold">q</Emphasis>}}_{{N}_{\mathrm{p}}}\right \}$ and ${q}_{\mathrm{s}} \equiv \left \{{\mbox{ <Emphasis Type="Bold">q</Emphasis>}}_{{N}_{\mathrm{p}}+1},\ldots ,{\mbox{ <Emphasis Type="Bold">q</Emphasis>}}_{N}\right \}$. (N _p is the total number of atoms in the solute.)

We are more concerned with effective sampling of the conformational space of the solute itself than that of the solvent molecules. The steric hindrance of the solute conformations are governed by the van der Waals radii of each atom in the solute. Namely, when the van der Waals radii are large, the solute molecule is bulky, and we have more steric hindrance among the solute atoms by the Lennard-Jones interactions, and when it is small, the solute molecule can move more freely. We thus introduce a parameter λ that scales the van der Waals radius of each atom in the solute by

$${\sigma }_{k\mathcal{l}}\rightarrow \lambda {\sigma }_{k\mathcal{l}}$$

(4.37)

and write the Lennard-Jones energy term within E _p in Eq. 4.36 as follows:

$$\begin{array}{rcl}{ V }_{\lambda }\left ({q}_{\mathrm{p}}\right ) ={ \sum \nolimits }_{k=1}^{{N}_{\mathrm{p}}-1}{ \sum \nolimits }_{\mathcal{l}=k+1}^{{N}_{\mathrm{p}} }4{\epsilon }_{k\mathcal{l}}\left \{{\left (\frac{\lambda {\sigma }_{k\mathcal{l}}} {{r}_{k\mathcal{l}}} \right )}^{12} -{\left (\frac{\lambda {\sigma }_{k\mathcal{l}}} {{r}_{k\mathcal{l}}} \right )}^{6}\right \}\!,& &\end{array}$$

(4.38)

where r _kℓ is the distance between atoms k and ℓ in the solute and ε _kℓ and σ _kℓ are the corresponding Lennard-Jones parameters. The original potential energy is recovered when λ = 1, and the steric hindrance of solute conformations is reduced when λ < 1. We remark that this is the only λ-dependent term in E _λ in Eq. 4.36.

We prepare M values of λ, λ _m (m = 1, …, M). Without loss of generality, we can assume that the parameter values are ordered as λ₁ < ⋯ < λ _M . Here, we consider the case in which temperature is fixed to be T ₀ = 1 ∕ k _Bβ₀. The vWREM is realized by alternately performing the following two steps:

• 1. For each replica, a canonical MC or MD simulation at the corresponding parameter value λ _m is carried out simultaneously and independently for a certain steps with the corresponding Boltzmann factor of Eq. 4.3 for each replica.

• 2. We exchange a pair of replicas i and j which are at the neighboring parameter values λ _m and λ _m + 1, respectively. The transition probability for this replica-exchange process is given by Eq. 4.33, where Δ in Eq. 4.35 now reads

  $$\begin{array}{rcl} \Delta = {\beta }_{0}\left [\left ({V }_{{\lambda }_{m}}\left ({q}_{\mathrm{p}}^{[j]}\right ) - {V }_{{ \lambda }_{m}}\left ({q}_{\mathrm{p}}^{[i]}\right )\right ) -\left ({V }_{{ \lambda }_{m+1}}\left ({q}_{\mathrm{p}}^{[j]}\right ) - {V }_{{ \lambda }_{m+1}}\left ({q}_{\mathrm{p}}^{[i]}\right )\right )\right ].& & \\ & & \end{array}$$

  (4.39)

  Here, V _λ is the Lennard-Jones potential energy in Eq. 4.38 among the solute atoms only.

Note that because the λ dependence of E _λ exists only in V _λ, the rest of the terms have been canceled out in Eq. 4.35.

We see that Eq. 4.39 includes only the coordinates q _p of the atoms in the solute only and is independent of the coordinates q _s of solvent molecules. Because N _p ≪ N usually holds, the difficulty in the usual REM that the number of required replicas increases with the number of degrees of freedom is much alleviated in this formalism.

We remark that in order to further enhance the conformational sampling, we can perform a two-dimensional REM in both temperature and λ, using Eqs. 4.34 and 4.35.

4.2.2.3 Reweighting Techniques

The results from MREM simulations with different parameter values can be analyzed by the reweighting techniques [54, 55]. Suppose that we have carried out a MREM simulation at a constant temperature T ₀ with M replicas corresponding to M parameter values λ _m (m = 1, …, M).

For appropriate reaction coordinates ξ₁ and ξ₂, the canonical probability distribution P _T, λ(ξ₁, ξ₂) with any parameter value λ at any temperature T can be calculated from

$$\begin{array}{rcl}{ P}_{T,\lambda }({\xi }_{1},{\xi }_{2}) ={ \sum \nolimits }_{{E}_{{\lambda }_{ 1}},\ldots ,{E}_{{\lambda }_{M}}}\frac{{\sum \nolimits }_{m=1}^{M}{N}_{ m}({E}_{{\lambda }_{1}},\ldots ,{E}_{{\lambda }_{M}};{\xi }_{1},{\xi }_{2}){e}^{-\beta {E}_{\lambda } }} {{\sum \nolimits }_{m=1}^{M}{n}_{ m}{e}^{{f}_{{T}_{0},{\lambda }_{m}}-{\beta }_{0}{E}_{{\lambda }_{m}} }} ,& &\end{array}$$

(4.40)

and

$$\begin{array}{rcl}{ e}^{-{f}_{{T}_{0},{\lambda }_{m}} } ={ \sum \nolimits }_{{\xi }_{1},{\xi }_{2}}{P}_{{T}_{0},{\lambda }_{m}}({\xi }_{1},{\xi }_{2}).& &\end{array}$$

(4.41)

Here, ${N}_{m}({E}_{{\lambda }_{1}},\ldots ,{E}_{{\lambda }_{M}};{\xi }_{1},{\xi }_{2})$ is the histogram of the M-dimensional energy distributions at the parameter value λ _m and the reaction coordinate values (ξ₁, ξ₂), which was obtained by the MREM simulation, and n _m is the total number of samples obtained at the parameter value λ _m . Note that this probability distribution is not normalized. Equations 4.40 and 4.41 are solved self-consistently by iteration. Note also that these equations can be easily generalized to any reaction coordinates (ξ₁, ξ₂, …).

From the probability distribution P _T, λ(ξ₁, ξ₂) in Eq. 4.40, the expectation value of a physical quantity A with any parameter value λ at any temperature T is given by

$$\begin{array}{rcl}{ \left < A\right >}_{T,\lambda } = \frac{{\sum \nolimits }_{{\xi }_{1},{\xi }_{2}}A({\xi }_{1},{\xi }_{2}){P}_{T,\lambda }({\xi }_{1},{\xi }_{2})} {{\sum \nolimits }_{{\xi }_{1},{\xi }_{2}}{P}_{T,\lambda }({\xi }_{1},{\xi }_{2})} .& &\end{array}$$

(4.42)

We can also calculate the free energy (or the potential of mean force) as a function of the reaction coordinates ξ₁ and ξ₂ with any parameter value λ at any temperature T from

$${F}_{T,\lambda }({\xi }_{1},{\xi }_{2}) = -{k}_{\mathrm{B}}T\mathrm{ln}{P}_{T,\lambda }({\xi }_{1},{\xi }_{2}).$$

(4.43)

By utilizing these equations, therefore, we can obtain various physical quantities from the MREM simulations with the original and non-original parameter values. We remark that although we wrote anyT in Eqs. 4.40, 4.42, and 4.43 above, the valid T value is limited in the vicinity of T ₀. We also need the T-exchange process in Eq. 4.34 in order to have accurate average quantities for a wide range of T values.

4.2.3 Multicanonical Algorithm

The next generalized-ensemble algorithm that we present is the multicanonical algorithm (MUCA) [16, 17]. In the multicanonical ensemble, each state is weighted by a non-Boltzmann weight factor W _MUCA(E) (which we refer to as the multicanonical weight factor) so that a uniform potential energy distribution P _MUCA(E) may be obtained:

$${P}_{\mathrm{MUCA}}(E) \propto n(E){W}_{\mathrm{MUCA}}(E) \equiv \mathrm{constant}.$$

(4.44)

The flat distribution implies that a free one-dimensional random walk in the potential energy space is realized in this ensemble. This allows the simulation to escape from any local-minimum-energy states and to sample the configurational space much more widely than the conventional canonical MC or MD methods.

The definition in Eq. 4.44 implies that the multicanonical weight factor is inversely proportional to the density of states, and we can write it as follows:

$${W}_{\mathrm{MUCA}}(E) \equiv \exp \left [-{\beta }_{0}{E}_{\mathrm{MUCA}}(E;{T}_{0})\right ] = \frac{1} {n(E)},$$

(4.45)

where we have chosen an arbitrary reference temperature, T ₀ = 1 ∕ k _Bβ₀, and the “multicanonical potential energy” is defined by

$${E}_{\mathrm{MUCA}}(E;{T}_{0}) \equiv {k}_{\mathrm{B}}{T}_{0}\ln n(E) = {T}_{0}S(E).$$

(4.46)

Here, S(E) is the entropy in the microcanonical ensemble. Because the density of states of the system is usually unknown, the multicanonical weight factor has to be determined numerically by iterations of short preliminary runs [16, 17].

A multicanonical MC simulation is performed, for instance, with the usual Metropolis criterion [50]: The transition probability of state x with potential energy E to state x ^′ with potential energy E ^′ is given by

$$\begin{array}{rcl} w(x \rightarrow {x}^{{\prime}})& =& \mathrm{min}\left (1, \frac{{W}_{\mathrm{MUCA}}({E}^{{\prime}})} {{W}_{\mathrm{MUCA}}(E)} \right ) = \mathrm{min}\left (1, \frac{n(E)} {n({E}^{{\prime}})}\right ) \\ & =& \mathrm{min}\left (1,\exp \left (-{\beta }_{0}\Delta {E}_{\mathrm{MUCA}}\right )\right ), \end{array}$$

(4.47)

where

$$\Delta {E}_{\mathrm{MUCA}} = {E}_{\mathrm{MUCA}}({E}^{{\prime}};{T}_{ 0}) - {E}_{\mathrm{MUCA}}(E;{T}_{0}).$$

(4.48)

The MD algorithm in the multicanonical ensemble also naturally follows from Eq. 4.45, in which the regular constant temperature MD simulation (with T = T ₀) is performed by replacing E by E _MUCA in Eq. 4.12 [19, 20]:

$$\dot{{\text{ <Emphasis Type="Bold">p}</Emphasis>}}_{k} = -\frac{\partial {E}_{\mathrm{MUCA}}(E;{T}_{0})} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}} -\frac{\dot{s}} {s}\ {\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k} = \frac{\partial {E}_{\mathrm{MUCA}}(E;{T}_{0})} {\partial E} \ {\text{ <Emphasis Type="Bold">F}</Emphasis>}_{k} -\frac{\dot{s}} {s}\ {\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}.$$

(4.49)

Let N _MUCA(E) be the histogram of potential energy distribution P _MUCA(E) obtained by the production run. The best estimate of the density of states can then be given by the single-histogram reweighting techniques [57] as follows (see the proportionality relation in Eq. 4.44):

$$n(E) = \frac{{N}_{\mathrm{MUCA}}(E)} {{W}_{\mathrm{MUCA}}(E)}.$$

(4.50)

By substituting this quantity into Eq. 4.28, one can calculate ensemble averages of physical quantity A(E) as a function of temperature. Moreover, the ensemble averages of any physical quantity A (including those that cannot be expressed as functions of potential energy) at any temperature T ( = 1 ∕ k _Bβ) can also be obtained as long as one stores the “trajectory” of configurations from the production run. Namely, we have

$${ \left < A\right >}_{T} = \frac{{\sum \nolimits }_{k=1}^{{n}_{s} }A({x}_{k}){W}_{\mathrm{MUCA}}^{-1}(E({x}_{ k}))\exp \left [-\beta E({x}_{k})\right ]} {{\sum \nolimits }_{k=1}^{{n}_{s} }{W}_{\mathrm{MUCA}}^{-1}(E({x}_{ k}))\exp \left [-\beta E({x}_{k})\right ]} ,$$

(4.51)

where x _k is the configuration at the kth MC (or MD) step and n _s is the total number of configurations stored.

4.2.4 Extensions of Multicanonical Algorithm

4.2.4.1 Multioverlap Algorithm and Multicanonical-Multioverlap Algorithm

While MUCA yields a flat distribution in potential energy and performs a random walk in potential energy space, we can, in principle, choose any other variable and induce a random walk in that variable. One such example is the multioverlap algorithm (MUOV) [33–35]. Here, we choose a protein system and define the overlap in the space of dihedral angles by [58]

$$O = 1 - d,$$

(4.52)

where d is the dihedral-angle distance given by

$$\begin{array}{rcl} d = \frac{1} {n\pi }{\sum \nolimits }_{i}{d}_{a}({\theta }_{i},{\theta }_{i}^{0}).& &\end{array}$$

(4.53)

θ _i is the dihedral angle i, and θ _i ⁰ is the dihedral angle i of the reference conformation. The distance d _a (θ _i , θ _i ⁰) between two dihedral angles is defined by

$$\begin{array}{rcl}{ d}_{a}({\theta }_{i},{\theta }_{i}^{0}) = \mathrm{min}(\vert {\theta }_{ i} - {\theta }_{i}^{0}\vert ,2\pi -\vert {\theta }_{ i} - {\theta }_{i}^{0}\vert ).& &\end{array}$$

(4.54)

The dihedral-angle distance d in Eq. 4.53 takes a value in the range 0 ≤ d ≤ 1. If d = 0, all dihedral angles are coincident with those of the reference conformation. The dihedral-angle distance is thus an indicator of how similar the conformation is to the reference conformation. As one can see in Eq. 4.52, the dihedral-angle distance d is equivalent to the overlap O. We will deal with the dihedral-angle distance instead of the overlap hereafter.

In the multioverlap ensemble at a constant temperature T ₀, the probability distribution is given by the following non-Boltzmann weight factor, which we refer to as the multioverlap weight factor:

$$\begin{array}{rcl}{ W}_{\mathrm{muov}}(d,E;{T}_{0}) = {e}^{-{\beta }_{0}{E}_{\mathrm{muov}} },& &\end{array}$$

(4.55)

where E _muov is the “multioverlap potential energy” defined by

$$\begin{array}{rcl}{ E}_{\mathrm{muov}}(d,E;{T}_{0}) = E - {k}_{\mathrm{B}}{T}_{0}f(d;{T}_{0}).& &\end{array}$$

(4.56)

The function f(d; T ₀) is the dimensionless free energy at dihedral-angle distance d.

The generalization to the multidimensional dihedral-angle distance space is straightforward, and the multioverlap weight factor is given by

$$\begin{array}{rcl}{ W}_{\mathrm{muov}}({d}_{1},\ldots ,{d}_{L},E;{T}_{0}) = {e}^{-{\beta }_{0}{E}_{\mathrm{muov}} } \equiv {e}^{-{\beta }_{0}E+f({d}_{1},\ldots ,{d}_{L};{T}_{0})},& &\end{array}$$

(4.57)

where L is the number of the reference conformations and d _i is the dihedral-angle distance, with respect to reference conformation i (i = 1, …, L). The function f(d ₁, …, d _L ; T ₀) is the dimensionless free energy with the fixed value of dihedral-angle distances d ₁, ⋯ , d _L . The dimensionless free energy f(d ₁, …, d _L ; T ₀) is defined so that the probability distribution of dihedral-angle distances P _muov(d ₁, …, d _L ; T ₀) is flat:

$$\begin{array}{rcl}{ P}_{\mathrm{muov}}({d}_{1},\ldots ,{d}_{L};{T}_{0})& =& \int \nolimits \nolimits dE\ {P}_{\mathrm{muov}}({d}_{1},\ldots ,{d}_{L},E;{T}_{0}) \\ & \propto & \int \nolimits \nolimits dE\ n({d}_{1},\ldots ,{d}_{L},E){W}_{\mathrm{muov}}({d}_{1},\ldots ,{d}_{L},E;{T}_{0}) \\ & =& \int \nolimits \nolimits dE\ n({d}_{1},\ldots ,{d}_{L},E){e}^{-{\beta }_{0}E+f({d}_{1},\cdots \,,{d}_{L};{T}_{0})} \\ & \equiv & \mathrm{constant}, \end{array}$$

(4.58)

where P _muov(d ₁, …, d _L , E; T ₀) is the probability distribution of potential energy and dihedral-angle distances, and n(d ₁, …, d _L , E) is its density of states.

The MD algorithm in the multioverlap ensemble also naturally follows from Eq. 4.57, in which the regular constant temperature MD simulation (with T = T ₀) is performed by replacing E by E _muov in Eq. 4.12 [35, 36]:

$$\begin{array}{rcl} \dot{{\text{ <Emphasis Type="Bold">p}</Emphasis>}}_{k}& =& -\frac{\partial {E}_{\mathrm{muov}}} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}} ({d}_{1},\ldots ,{d}_{L},E;{T}_{0}) -\frac{\dot{s}} {s}\ {\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k} \\ & =&{ \text{ <Emphasis Type="Bold">F}</Emphasis>}_{k} + {k}_{\mathrm{B}}{T}_{0} \frac{\partial f} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}}({d}_{1},\ldots ,{d}_{L};{T}_{0}) -\frac{\dot{s}} {s}\ {\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}.\end{array}$$

(4.59)

The multioverlap weight factor, or the dimensionless free energy, is not a priori known and has to be determined by the usual iterations of short simulations [2, 18]. Suppose that we have determined an appropriate dimensionless free energy f(d ₁, …, d _L ; T ₀) at temperature T ₀ and that we have made a production run at this temperature. The results of the multioverlap production run can then be analyzed by the reweighting techniques [57]. Namely, the expectation value of a physical quantity A at any temperature T is given by

$$\begin{array}{rcl}{ \left < A\right >}_{T}& =& \frac{{\sum \nolimits }_{{d}_{1},\cdots \,,{d}_{L},E}A({d}_{1},\cdots \,,{d}_{L},E){N}_{\mathrm{muov}}({d}_{1},\cdots \,,{d}_{L},E){{W}_{\mathrm{muov}}({d}_{1},\cdots \,,{d}_{L},E;{T}_{0})}^{-1}{e}^{-\beta E}} {{\sum \nolimits }_{{d}_{1},\cdots \,,{d}_{L},E}{N}_{\mathrm{muov}}({d}_{1},\cdots \,,{d}_{L},E){{W}_{\mathrm{muov}}({d}_{1},\cdots \,,{d}_{L},E;{T}_{0})}^{-1}{e}^{-\beta E}} \\ & =& \frac{{\sum \nolimits }_{{d}_{1},\cdots \,,{d}_{L},E}A({d}_{1},\cdots \,,{d}_{L},E){N}_{\mathrm{muov}}({d}_{1},\cdots \,,{d}_{L},E){e}^{-(\beta -{\beta }_{0})E-f({d}_{1},\cdots \,,{d}_{L};{T}_{0})}} {{\sum \nolimits }_{{d}_{1},\cdots \,,{d}_{L},E}{N}_{\mathrm{muov}}({d}_{1},\cdots \,,{d}_{L},E){e}^{-(\beta -{\beta }_{0})E-f({d}_{1},\cdots \,,{d}_{L};{T}_{0})}} , \end{array}$$

(4.60)

where N _muov(d ₁, …, d _L , E) is the histogram of the probability distribution P _muov(d ₁, …, d _L , E; T ₀) of potential energy and dihedral-angle distances that was obtained by the multioverlap production run.

The multioverlap algorithm can further be combined with the multicanonical algorithm as follows (this method is referred to as the multicanonical-multioverlap algorithm (MUCA-MUOV)) [36]. In analogy with the multicanonical ensemble in Eq. 4.44 or the multioverlap ensemble in Eq. 4.58, by employing the non-Boltzmann weight factor W _mcmo(d ₁, …, d _L , E), which we refer to as the multicanonical-multioverlap weight factor, a uniform probability distribution with respect to the potential energy and dihedral-angle distances is obtained:

$${P}_{\mathrm{mcmo}}({d}_{1},\ldots ,{d}_{L},E) \propto n({d}_{1},\ldots ,{d}_{L},E){W}_{\mathrm{mcmo}}({d}_{1},\ldots ,{d}_{L},E) \equiv \mathrm{constant}.$$

(4.61)

In this method, we obtain a random walk not only in the dihedral-angle distance space but also in the potential energy space.

4.2.4.2 Multibaric-Multithermal Algorithm

Besides the canonical ensemble, molecular simulations in the isobaric-isothermal ensemble are also commonly used. This is because most experiments are carried out under the constant pressure and constant temperature conditions. The canonical probability distribution P _B(E; T ₀) in Eq. 4.6 is here replaced by the isobaric-isothermal distribution P _NPT(E, V ; T ₀, P ₀) for potential energy E and volume V :

$${P}_{\mathrm{NPT}}(E,V ;{T}_{0},{P}_{0}) \equiv n(E,V ){\mathrm{e}}^{-{\beta }_{0}\mathcal{H}}.$$

(4.62)

Here, the density of states n(E, V ) is given as a function of both E and V , and $\mathcal{H}$ is the “enthalpy” (without the kinetic energy contributions):

$$\mathcal{H} = E + {P}_{0}V,$$

(4.63)

where P ₀ is the pressure at which simulations are performed. This weight factor produces an isobaric-isothermal ensemble at constant temperature (T ₀) and constant pressure (P ₀). This ensemble has bell-shaped distributions in both E and V .

As for the MD methods in this ensemble, we just present the Nosé-Andersen algorithm [51, 52, 59]. The equations of motion in Eqs. 4.11–4.14 are now generalized as follows:

$$\begin{array}{rcl} \dot{{\text{ <Emphasis Type="Bold">q}</Emphasis>}}_{k}& =& \frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}} {{m}_{k}} + \frac{\dot{V }} {3V }\ {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}, <EquationNumber>4.64</EquationNumber> \\ \dot{{\text{ <Emphasis Type="Bold">p}</Emphasis>}}_{k}& =& - \frac{\partial \mathcal{H}} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{k}} -\left (\frac{\dot{s}} {s} + \frac{\dot{V }} {3V }\right ){\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k} ={ \text{ <Emphasis Type="Bold">F}</Emphasis>}_{k} -\left (\frac{\dot{s}} {s} + \frac{\dot{V }} {3V }\right ){\text{ <Emphasis Type="Bold">p}</Emphasis>}_{k}, <EquationNumber>4.65</EquationNumber> \\ \dot{s}& =& s\ \frac{{P}_{s}} {Q} , <EquationNumber>4.66</EquationNumber> \\ \dot{{P}}_{s}& =& {\sum \nolimits }_{i=1}^{N}\frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{i}^{2}} {{m}_{i}} - 3N{k}_{\mathrm{B}}{T}_{0} = 3N{k}_{\mathrm{B}}\left (T(t) - {T}_{0}\right ), <EquationNumber>4.67</EquationNumber> \\ \dot{V }& =& s\frac{{P}_{V }} {M} , <EquationNumber>4.68</EquationNumber> \\ \dot{{P}}_{V }& =& s\left \{ \frac{1} {3V }\left ({\sum \nolimits }_{i=1}^{N}\frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{i}^{2}} {{m}_{i}} -{\sum \nolimits }_{i=1}^{N}{\text{ <Emphasis Type="Bold">q}</Emphasis>}_{ i} \cdot \frac{\partial \mathcal{H}} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{i}}\right ) -\frac{\partial \mathcal{H}} {\partial V }\right \} = s\left [P(t) - {P}_{0}\right ],\end{array}$$

(4.69)

where M is the artificial mass associated with the volume, P _V is the conjugate momentum for the volume, and the “instantaneous pressure” P(t) is defined by

$$\begin{array}{rcl} P(t)& =& \frac{1} {3V }\left ({\sum \nolimits }_{i=1}^{N}\frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{i}{(t)}^{2}} {{m}_{i}} -{\sum \nolimits }_{i=1}^{N}{\text{ <Emphasis Type="Bold">q}</Emphasis>}_{ i}(t) \cdot \frac{\partial \mathcal{H}} {\partial {\text{ <Emphasis Type="Bold">q}</Emphasis>}_{i}}(t)\right ) \\ & =& \frac{1} {3V }\left ({\sum \nolimits }_{i=1}^{N}\frac{{\text{ <Emphasis Type="Bold">p}</Emphasis>}_{i}{(t)}^{2}} {{m}_{i}} +{ \sum \nolimits }_{i=1}^{N}{\text{ <Emphasis Type="Bold">q}</Emphasis>}_{ i}(t) \cdot {\text{ <Emphasis Type="Bold">F}</Emphasis>}_{i}(t)\right )\!.\end{array}$$

(4.70)

We now introduce the idea of the multicanonical technique into the isobaric-isothermal ensemble method and refer to this generalized-ensemble algorithm as the multibaric-multithermal algorithm (MUBATH) [39, 40, 42, 43]. The molecular simulations in this generalized ensemble perform random walks both in the potential energy space and in the volume space.

In the multibaric-multithermal ensemble, each state is sampled by the multibaric-multithermal weight factor ${W}_{\mathrm{mbt}}(E,V ) \equiv \exp \{-{\beta }_{0}{\mathcal{H}}_{\mathrm{mbt}}(E,V )\}$ (${\mathcal{H}}_{\mathrm{mbt}}$ is referred to as the multibaric-multithermal enthalpy) so that a uniform distribution in both potential energy and volume may be obtained:

$${P}_{\mathrm{mbt}}(E,V ) \propto n(E,V ){W}_{\mathrm{mbt}}(E,V ) = n(E,V )\exp \{ - {\beta }_{0}{\mathcal{H}}_{\mathrm{mbt}}(E,V )\} \equiv \mathrm{constant}.$$

(4.71)

In order to perform the multibaric-multithermal MD simulation, we just solve the above equations of motion (Eqs. 4.64–4.69) for the regular isobaric-isothermal ensemble (with T = T ₀ and P = P ₀), where the enthalpy $\mathcal{H}$ is replaced by the multibaric-multithermal enthalpy ${\mathcal{H}}_{\mathrm{mbt}}$ in Eqs. 4.65 and 4.69 [42].

The multibaric-multithermal weight factor is, however, not a priori known and has to be determined by the usual iterations of short simulations [2, 18]. After an optimal weight factor W _mbt(E, V ) is obtained, a long production simulation is performed for data collection. We employ the reweighting techniques [57] for the results of the production run to calculate the isobaric-isothermal-ensemble averages. The probability distribution P _NPT(E, V ; T, P) of potential energy and volume in the isobaric-isothermal ensemble at the desired temperature T and pressure P is given by

$${P}_{\mathrm{NPT}}(E,V ;T,P) = \frac{{N}_{\mathrm{mbt}}(E,V )\ {{W}_{\mathrm{mbt}}(E,V )}^{-1}\ {\mathrm{e}}^{-\beta (E+PV )}} {{\sum \nolimits }_{E,V }\ {N}_{\mathrm{mbt}}(E,V )\ {{W}_{\mathrm{mbt}}(E,V )}^{-1}\ {\mathrm{e}}^{-\beta (E+PV )}},$$

(4.72)

where N _mbt(E, V ) is the histogram of the probability distribution P _mbt(E, V ) of potential energy and volume that was obtained by the multibaric-multithermal production run. The expectation value of a physical quantity A at T and P is then obtained from

$${ \left < A\right >}_{T,P} = {\sum \nolimits }_{E,V }\ A(E,V )\ {P}_{\mathrm{NPT}}(E,V ;T,P).$$

(4.73)

4.3 Examples of Simulation Results

We now present several examples of the simulation results by the generalized-ensemble algorithms described in the previous section.

The first example is a vWREM simulation of a small peptide [32]. In order to demonstrate the effectiveness of vWREM, in which we exchange pairs of the van der Waals radius parameter values, we applied the vWREM MD algorithm, which we refer to as the vWREMD, to the system of an alanine dipeptide in explicit water solvent and compared the results with those obtained by the replica-exchange MD (REMD) simulation [8] and conventional canonical MD simulations. The N-terminus and the C-terminus were blocked by the acetyl group and the N-methyl group, respectively. The number of water molecules was 67. The force field that we adopted was the AMBER parm96 parameter set [60], and the model for the water molecules was the TIP3P rigid-body model [61]. The vWREMD, REMD, and canonical MD simulations were carried out with the symplectic integrator with rigid-body water molecules, in which the temperature was controlled by the Nosé-Poincaré thermostat [44, 45, 62–65]. The system was put in a cubic unit cell with the side length of 13.4 $\r{A}$, and we imposed the periodic boundary conditions.

In the vWREMD simulation, we needed only four replicas (M = 4). That is, we employed four different parameter values λ _m (m = 1, …, 4), and their values were λ₁ = 0. 85, λ₂ = 0. 9, λ₃ = 0. 95, and λ₄ = 1. 0. The original potential energy corresponds to the scale factor λ₄ = 1. 0. The temperature of the system T ₀ was set to be 300 K for all the replicas in the vWREMD simulation. We also employed four replicas for the REMD simulation to compare the sampling efficiency with those of the vWREMD simulation, and the four different temperatures were 300 K, 315 K, 335 K, and 360 K, and these temperatures were determined so that exchanges between pairs of replicas were accepted sufficiently. Moreover, we carried out four canonical MD simulations at 300 K, and the difference among these four simulations was initial velocities. We employed the original parameter value λ = 1. 0 for the REMD and canonical MD simulations. The initial conformations were the same for all the simulations, and the initial backbone dihedral angles ϕ and ψ of the alanine dipeptide were set (ϕ, ψ) = (180^∘, 180^∘), as shown in Fig. 4.1. The total time of the MD simulations was 2.5 ns per replica for the vWREMD and REMD simulations and 2.5 ns for each canonical simulation, including equilibration for 0.1 ns.

Fig. 4.1

The common initial conformation of the alanine dipeptide for the vWREMD, REMD, and canonical MD simulations. Reprinted from Ref. [32] with kind permission of © The American Institute of Physics (2010)

Figure 4.2 shows the time series of the backbone dihedral angles ϕ for the vWREMD, REMD, and the conventional canonical MD simulations. From the figure, we see that the samplings in the ϕ space in the vWREMD simulation were the most effective, then those in the REMD simulation, and the least effective in the conventional MD simulation.

Fig. 4.2

Time series of the dihedral angle ϕ during the vWREMD simulation (left), REMD simulation (center), and canonical MD simulation (right)

The second example is a multioverlap MD simulation of the system of a pentapeptide, Met-enkephalin, in vacuum [34]. The amino-acid sequence is Tyr-Gly-Gly-Phe-Met. The N-terminus and the C-terminus were blocked with the acetyl group and the N-methyl group, respectively. The force field that we adopted is the CHARMM param 22 parameter set [66]. Our multioverlap MD simulations were performed by implementing the method in the CHARMM macromolecular mechanics program [67].

We considered two energy-local-minimum states of Met-enkephalin as reference conformations. In Fig. 4.3, we show these two reference conformations. We then set L = 2 in Eq. 4.57, and the dimensionless free energy is expressed as f(d ₁, d ₂; T ₀). The multioverlap MD simulation was carried out at T ₀ = 300 K with a time step of 0.5 fs.

Fig. 4.3

(a) Reference conformation 1 and (b) reference conformation 2. The side chains are suppressed, and only backbone structures are shown. The dotted lines denote the hydrogen bonds. The N-terminus and the C-terminus are on the right-hand side and on the left-hand side, respectively

Figure 4.4 shows the time series of the dihedral-angle distances with respect to each of the two reference conformations. While Fig. 4.4a, b shows the results of the conventional canonical MD simulation at T ₀ = 300 K, Fig. 4.4c, d shows the results of the multioverlap MD simulation at the same temperature. When d ₁ = 0, the values of dihedral angles of backbone completely coincide with those of reference conformation 1 and d ₂ = 0. 122. Conversely, when d ₂ = 0, d ₁ = 0. 122. When d ₁ (d ₂) is near zero, the conformation is similar to reference conformation 1 (2). Therefore, Fig. 4.4 implies that the multioverlap MD simulation performed a random walk in the dihedral-angle distance space between reference conformation 1 and reference conformation 2, whereas the usual canonical MD simulation got trapped in a local-minimum state near conformation 2.

Fig. 4.4

The time series of the dihedral-angle distances d ₁ and d ₂. (a) and (b) are from the conventional canonical MD simulation, and (c) and (d) are the results from the multioverlap MD simulation at T ₀ = 300 K

The free energy F(d ₁, d ₂; T) (or the potential of mean force) at temperature T is defined by

$$\begin{array}{rcl} F({d}_{1},{d}_{2};T) = -{k}_{\mathrm{B}}T\ \mathrm{ln}{P}_{\mathrm{B}}({d}_{1},{d}_{2};T),& &\end{array}$$

(4.74)

where P _B(d ₁, d ₂; T) is the reweighted canonical probability distribution of d ₁ and d ₂ at T and given by (see Eq. 4.60)

$$\begin{array}{rcl}{ P}_{\mathrm{B}}({d}_{1},{d}_{2};T) = \frac{{\sum \nolimits }_{E}{N}_{\mathrm{muov}}({d}_{1},{d}_{2},E){e}^{-(\beta -{\beta }_{0})E-f({d}_{1},{d}_{2};{T}_{0})}} {{\sum \nolimits }_{{d}_{1},{d}_{2},E}{N}_{\mathrm{muov}}({d}_{1},{d}_{2},E){e}^{-(\beta -{\beta }_{0})E-f({d}_{1},{d}_{2};{T}_{0})}}.& &\end{array}$$

(4.75)

In Fig. 4.5, we illustrate the free-energy landscapes with respect to the dihedral-angle distances that were calculated from the results of the conventional canonical MD simulation and those of the multioverlap MD simulation. While in Fig. 4.5a only one local-minimum state exists near reference conformation 2, in Fig. 4.5b, we find a local-minimum state A and a local-minimum state B near reference conformation 1 and reference conformation 2, respectively. This result again implies that the canonical MD simulation got trapped in the latter local-minimum state. The local-minimum state B near reference conformation 2 corresponds to the global-minimum state at 300 K. The local-minimum state A near reference conformation 1 is another local-minimum state at 300 K. The free-energy difference between the global-minimum state (B) and the local-minimum state (A) is about 3 kcal∕mol.

Fig. 4.5

The free-energy landscape obtained from (a) the conventional canonical MD simulation and (b) the multioverlap MD simulation at T ₀ = 300 K. Contour lines are drawn every 1 kcal/mol. The labels A and B locate the local-minimum states. The label C stands for the saddle point, or the transition state, between these two local-minimum states

The saddle point C in Fig. 4.5b corresponds to the transition state between the global-minimum state (B) and the local-minimum state (A). The free-energy difference between B and C is about 5 kcal/mol and that between A and C is 2 kcal/mol. Because k _B T ≈ 0. 6 kcal∕mol at T = 300 K, these barrier heights are rather high. This is why the conventional canonical MD simulation got trapped in the vicinity of the global-minimum state B.

Our next simulation is the multicanonical-multioverlap MD simulation of Alzheimer’s amyloid-β (Aβ) peptide fragment [37]. The amino-acid sequence was Ace-GAIIGLMVGGVVIA-Nme. In multicanonical-multioverlap simulations, we must have a reference conformation. We adopted the conformation that was obtained from the corresponding part in the conformation whose PDB ID code is 2BEG. Here, we took into account only the backbone dihedral angles ϕ (the rotation angles around the N–C_α bonds) and ψ (the rotation angles around the C_α–C bonds) of the residues 30–41 of Aβ(29–42) as the reference dihedral angles in our simulations. The force field that we adopted is the CHARMM 22 parameter set [66]. We employed the GB/SA model [68–70] as an implicit solvent model. We also introduced the harmonic constraint k(r − r ₀)² ∕ 2 when the distance between the center of mass of two Aβ(29–42) molecules exceeded 20 $\r{A}$ in order to avoid the states in which two molecules are too much spatially separated. Here, r is the distance between the center of mass of two molecules, and k is a force constant whose value is 200 kcal/(mol $\r{{A}}^{2}$), and the value of r ₀ is set 20 $\r{A}$.

In Fig. 4.6, we show conformations of Aβ(29–42) monomer in the case when the distance between the center of mass of two peptides is more than 15 $\r{A}$ at 300 K. We identified three major metastable states. These states correspond to low concentrations of Aβ(29–42) peptides or to their monomeric states. Conformation 1 in Fig. 4.6 is a β-helix-like structure, conformation 2 is an α-helix (or sometimes π-helix) structure, and conformation 3 is an intramolecular antiparallel β-sheet (β-hairpin) structure. When the Aβ(29–42) peptide is in a monomeric state, therefore, it seems that the conformations of Aβ(29–42) peptides have the same structure as those in Fig. 4.6.

Fig. 4.6

Typical conformations of the Aβ(29–42) molecule at 300 K when two molecules are spatially separated. Reprinted from Ref. [37] with kind permission of © The American Chemical Association (2008)

We show the free-energy landscape of the dimer system at 300 K in Fig. 4.7a. The free-energy landscape was obtained from the results of the multicanonical-multioverlap MD simulation by the reweighting techniques. The abscissa is the number of backbone C_α intermolecular contacts, and we regard a pair of C_α atoms as being in contact if the distance between the two atoms is within 6.5 $\r{A}$. d _α and d _β in the label of the ordinate are dihedral-angle distances, which we introduced to set the reaction coordinates of the free-energy data analysis. When the value of d _α (d _β) is close to 0, the structures of Aβ(29–42) molecules are helical (extended strand). From the free-energy landscape in Fig. 4.7a, we identified seven local-minimum states. In Fig. 4.7b, we show typical conformations of the Aβ(29–42) in each local-minimum state.

Fig. 4.7

(a) Free-energy landscape of an Aβ(29–42) dimer system at 300 K. The ordinate is an indicator of structure for helix and strand. When the value of the ordinate is close to 1, conformations of Aβ(29–42) become helical. Conversely, if the value is close to 0, the conformations have extended forms. The abscissa is the number of backbone C_α intermolecular contacts. Contour lines are drawn every 1 kcal/mol. (b) Typical structures in the corresponding local-minimum states in (a). The arrows indicate possible pathways of the early stages of amyloidogenesis. Reprinted from Ref. [37] with kind permission of © The American Chemical Association (2008)

From Figs. 4.6 and 4.7, we deduce the dimerization (oligomerization) process, which corresponds to a seeding process in amyloidogenesis, for Aβ(29–42) peptides as follows: Stage 1: When the Aβ(29–42) peptides are in the monomeric state, the peptides are mainly in one of the three conformational states in Fig. 4.6. Stage 2: Aβ(29–42) peptides come close to each other and create dimers (or oligomers) as a result of hydrophobic effects. If the structures are intramolecular antiparallel β-sheet structures before dimerization, such as conformation 3 in Fig. 4.6, the conformation after dimerization will correspond to conformation 2 in the local-minimum state E in Fig. 4.7b. If the structures are like conformation 1 or 2 in Fig. 4.6, on the other hand, the Aβ(29–42) dimer will have structures like those of the conformations in A or B in Fig. 4.7b. Stage 3: If the conformations in stage 2 are in states A or B in Fig. 4.7b, then the peptides have helical conformations with extended parts like those in C. If the conformations in stage 2 are already in E in Fig. 4.7b, on the other hand, this corresponds to Stage 4 below. Stage 4: The extended parts will create intermolecular β-ladders such as those in D or E. Stage 5: The intramolecular secondary structures are broken, and the peptides will have a fully extended form such as those in F. Stage 6: The Aβ(29–42) dimer has intermolecular parallel or antiparallel β-sheet structure like those in G. These pathways are summarized in Fig. 4.7b (see the arrows). In the early process of amyloidogenesis, these intermolecular parallel or antiparallel β-sheet structure can be a seed of amyloid fibrils.

We now present the results of a multibaric-multithermal MD simulation [42]. We considered a Lennard-Jones 12–6 potential system. The length and the energy are scaled in units of the Lennard-Jones diameter σ and the depth of the potential ε, respectively. We use an asterisk ( ∗ ) for quantities reduced by σ and ε.

We used 500 particles (N = 500) in a cubic unit cell with periodic boundary conditions. We started the multibaric-multithermal weight factor determination from a regular isobaric-isothermal simulation at T ₀ ^∗ = 2. 0 and P ₀ ^∗ = 3. 0 (the multibaric-multithermal production run was also performed at this set of temperature and pressure values). These temperature and pressure values are respectively higher than the critical temperature T _c ^∗ and the critical pressure P _c ^∗ [71, 72]. Recent reliable data are T _c ^∗ = 1. 3207(4) and P _c ^∗ = 0. 1288(5) [72]. The cutoff radius r _c ^∗ was taken to be r _c ^∗ = 4. 0. A cutoff correction was added for the pressure and the potential energy.

In order to carry out the multibaric-multithermal MD simulation in Eqs. 4.64–4.69 with the replacement of $\mathcal{H}$ by ${\mathcal{H}}_{\mathrm{mbt}}$, we employed the Nosé-Poincaré formalism [44, 45, 62–65]. This gives the same equations of motion as the Nosé thermostat and provides a symplectic integrator. Therefore, it has an advantage that the secular deviation of the Hamiltonian is suppressed. We have recently shown that this integrator is also very effective for rigid-body molecules [64]. We performed a long production run of 10⁶ MD steps.

In Fig. 4.8a, we show the probability distribution P _NPT(E ^∗ ∕ N, V ^∗ ∕ N) from the isobaric-isothermal simulation that was carried out first. It is a bell-shaped distribution. As the iteration of the multibaric-multithermal weight factor determination proceeds, P _mbt(E ^∗ ∕ N, V ^∗ ∕ N) will become flat and broad gradually. Figure 4.8b depicts the probability distribution P _mbt(E ^∗ ∕ N, V ^∗ ∕ N) from the multibaric-multithermal simulation that was finally performed. It shows a flat distribution, and the multibaric-multithermal MD simulation indeed sampled the conformational space in wider ranges of E ^∗ ∕ N and V ^∗ ∕ N than the conventional isobaric-isothermal MD simulation.

Fig. 4.8

(a) The probability distribution P _NPT(E ^∗ ∕ N, V ^∗ ∕ N) in the isobaric-isothermal MD simulation at (T ₀ ^∗, P ₀ ^∗) = (2. 0, 3. 0) and (b) the probability distribution P _mbt(E ^∗ ∕ N, V ^∗ ∕ N) in the multibaric-multithermal MD simulation. Reprinted from Ref. [42] with kind permission of © Elsevier (2004)

The time series of E ^∗ ∕ N from two conventional isobaric-isothermal MD simulations at (T ₀ ^∗, P ₀ ^∗) = (1.6, 3.0) and (2.4, 3.0) are given in Fig. 4.9a. The potential energy fluctuates in narrow ranges of E ^∗ ∕ N =  − 4. 0 ∼ − 3. 5 at the higher temperature of T ₀ ^∗ = 2. 4 and in the ranges of E ^∗ ∕ N =  − 5. 1 ∼ − 4. 7 and at the lower temperature of T ₀ ^∗ = 1. 6. On the other hand, Fig. 4.9b shows that the multibaric-multithermal MD simulation realizes a random walk in the potential energy space and covers a wide energy range.

Fig. 4.9 Fig. 4.10

The time series of E ^∗ ∕ N from (a) the conventional isobaric-isothermal MD simulations at (T ₀ ^∗, P ₀ ^∗) = (2. 4, 3. 0) and at (T ₀ ^∗, P ₀ ^∗) = (1. 6, 3. 0) and (b) the multibaric-multithermal MD simulation. Reprinted from Ref. [42] with kind permission of © Elsevier (2004)

The time series of V ^∗ ∕ N from (a) the conventional isobaric-isothermal MD simulations at (T ₀ ^∗, P ₀ ^∗) = (2. 0, 2. 2) and at (T ₀ ^∗, P ₀ ^∗) = (2. 0, 3. 8) and (b) the multibaric-multithermal MD simulation. Reprinted from Ref. [42] with kind permission of © Elsevier (2004)

A similar situation is observed in V ^∗ ∕ N. In Fig. 4.10a the time series of two conventional isobaric-isothermal MD simulations at (T ₀ ^∗, P ₀ ^∗) = (2.0, 2.2) and (2.0, 3.8), is shown. The volume fluctuations are only in the range of V ^∗ ∕ N = 1. 3 ∼ 1. 4 and V ^∗ ∕ N = 1. 5 ∼ 1. 6 at P ₀ ^∗ = 3. 8 and at P ₀ ^∗ = 2. 2, respectively. On the other hand, the multibaric-multithermal MD simulation performs a random walk that covers even a wider volume range, as shown in Fig. 4.10b.

We applied the MUBATH MD algorithm to a system consisting of one alanine dipeptide molecule and 63 water molecules. We used enough water molecules so that the alanine dipeptide molecule was always held perfectly within the simulation box. We used both AMBER parm99 [73] and AMBER parm96 [60] force fields for the alanine dipeptide molecule and the TIP3P [61] rigid-body model for the water molecules. We employed a cubic unit cell with periodic boundary conditions. The electrostatic potential was calculated by the Ewald method. We calculated the van der Waals interaction, which is given by the Lennard-Jones 12–6 term, for all pairs of the atoms within the minimum image convention instead of introducing the spherical potential cutoff. The time step was taken to be Δt = 0. 5 fs.

Figure 4.11 shows $\mathcal{P}(\phi ,\psi )$ obtained from the MUBATH MD simulations by the reweighting techniques at T = 298 K and P = 0.1 MPa. In the case of longer peptides or proteins, the α_R state corresponds to an α-helix structure, and the P_II and C₅ states correspond to a β-strand structure. It is known that, in general, the AMBER parm99 force field tends to form an α-helix structure, and the AMBER parm96 force field tends to form a β-sheet structure [74]. The distributions $\mathcal{P}(\phi ,\psi )$ in Fig. 4.11 are consistent with this feature.

Fig. 4.11

The probability distributions $\mathcal{P}(\phi ,\psi )$ of the backbone dihedral angles ϕ and ψ at T = 298 K and P = 0. 1 MPa, which were obtained by the reweighting techniques from the results of the multibaric-multithermal MD simulation (a) in the AMBER parm99 force field and (b) in the AMBER parm96 force field. (a ^′) and (b ^′) are the contour map of (a) and that of (b), respectively. Reprinted from Ref. [45] with kind permission of © The American Chemical Society (2008)

Figure 4.12 shows the population ratio of each state and the P_II state as a function of P at the constant temperature of T = 298 K. A pressure increase at constant temperature generally causes a decrease in the volume. The decreases in the population ratio of some state and the P_II state mean that the volume of that state is larger than that of the P_II state. The difference in partial molar volume ΔV of the C₅ state from that of the P_II state, for example, is calculated from the derivative of $\log ({W}_{{\mathrm{C}}_{5}}/{W}_{\mathrm{{P}_{II}}})$ with respect to P by

$$\Delta V = -RT{\left [\frac{\partial \log ({W}_{{\mathrm{C}}_{5}}/{W}_{\mathrm{{P}_{II}}})} {\partial P} \right ]}_{T}.$$

(4.76)

The difference between the partial molar volume of the other states and that of the P_II state was also obtained in the same way. The values of ΔV are shown in Table 4.1. Note that all the experimental data lie in between the corresponding simulation results with the two force fields.

Fig. 4.12

The population ratios $W/{W}_{\mathrm{{P}_{II}}}$ against the P_II state as functions of pressure P at constant temperature of T = 298 K, which was obtained by the reweighting techniques from the results of the multibaric-multithermal MD simulation. Reprinted from Ref. [45] with kind permission of © The American Chemical Society (2008)

Table 4.1 Differences ΔV / (cm³mol^− 1) in partial molar volume of the C₅, α_R, α_P, α_L, and C₇ ^ax states from that of the P_II state calculated by the MUBATH MD simulations. Raman experimental data are taken from Ref. [75]

4.4 Conclusions

In this chapter, we described two powerful generalized-ensemble algorithms, namely, replica-exchange method (REM) and multicanonical algorithm (MUCA), which are effective for molecular simulations. We also introduced multidimensional/multivariable extensions of the two methods, namely, MREM, vWREM, MUOV, MUCA-MUOV, and MUBATH. These generalized-ensemble algorithms are particularly useful for biomolecular simulations.