Introduction

Molecular dynamics (MD) and Monte Carlo (MC) simulations play a crucial role to obtaining properties of biomolecules from the atomic level. In many cases, however, MD and MC simulations sample conformations only in a few local-minimum free-energy states because of its time scale and degrees of freedom. This hampers accurate estimation of physical quantities such as free-energy differences. The specially designed hardware has extended upper limit of the time scale up to milliseconds (Shaw et al. 2010; Lindorff-Larsen et al. 2011). On the other hand, highly optimized software such as GENESIS (Jung et al. 2015; Kobayashi et al. 2017) and MODYLAS (Andoh et al. 2013) have extended upper limit of the system size. Yet, it remains difficult to sample sufficient conformations in the phase space.

Generalized-ensemble algorithms (Mitsutake et al. 2001) were developed to overcome the sampling difficulties. In a multicanonical ensemble (Berg and Neuhaus 1991, 1992b; Hansmann et al. 1996; Nakajima et al. 1997), the target system performs a free one-dimensional random walk in the potential-energy space; thus, the target system does not get trapped in local-minimum free-energy states. In a simulated tempering method (Lyubartsev et al. 1992; Marinari and Parisi 1992), temperature is treated as a dynamic variable so that a random walk in the temperature space can be realized. These methods introduce non-Boltzmann weight factors, which are not known a priori, and have to be determined before simulations. A replica-exchange (Hukushima and Nemoto 1996; Sugita and Okamoto 1999) and a replica-permutation (Itoh and Okumura 2012) methods are other generalized-ensemble algorithms, which are not necessary to determine the weight factor in advance. We prepare non-interacting copies (or replicas) of the target system at different temperatures and perform simulation in parallel. During the simulation, temperature exchange/permutation is conducted among the replicas. Each replica performs a random walk in the temperature space, which helps the target system to escape from a local-minimum free-energy state.

The generalized-ensemble algorithms were originally proposed in the canonical ensemble. In the canonical ensemble, the volume of the simulation box is constant and pressure is not controlled. In contrast, pressure as well as temperature is controlled in experiments; thus, physical quantities estimated from experiments follow isothermal–isobaric ensemble. Based on this point, the isothermal–isobaric ensemble simulation is important to compare the simulation results with experiments.

In this review article, we focus on a series of the generalized-ensemble algorithms in the isothermal–isobaric ensemble, namely multibaric–multithermal, pressure- and temperature-simulated tempering, replica-exhange, and replica-permutation methods. These methods achieve the efficient conformational sampling in the wider range of temperature and pressure. We also present applications of the generalized-ensemble algorithms in the isothermal–isobaric ensemble to the biomolecule systems: chignolin, AK16 peptide, C-peptide, and ubiquitin.

This review is organized as follows. First, we introduce the basics of the isothermal–isobaric ensemble and then we review the generalized-ensemble algorithms in the isothermal–isobaric ensemble. In the following section, the applications of the isothermal–isobaric generalized-ensemble algorithms to biomolecular systems are presented. The final section is devoted to conclusions.

Basics of isothermal–isobaric ensemble

In this section, we briefly review the fundamentals of thermodynamics and statistical mechanics for the isothermal–isobaric ensemble as described in many textbooks for example, McQuarrie and Simon (1997).

Isothermal–isobaric ensemble

In the isothermal–isobaric ensemble, the number of particles N, pressure P, and temperature T are treated as parameters. Let us consider N-particle system in a box with volume V. The system is specified by the coordinates q ≡{q1,⋯ ,qN}, momenta p ≡{p1,⋯ ,pN}, and volume of the box V. The potential energy of the system is given as a function of the coordinates and volume: U(q,V ). The probability distribution PNPT(U,V ;T,P) is given by the product of density of states n(U,V ) and distribution function WNPT(U,V ; T,P), as follows:

$$ \begin{array}{@{}rcl@{}} P_{\text{NPT}} (U, V; T, P) = n(U, V) W_{\text{NPT}} (U, V; T, P). \end{array} $$
(1)

The distribution function in the isothermal–isobaric ensemble is given by

$$ \begin{array}{@{}rcl@{}} W_{\text{NPT}} (U, V; T, P) &=& \exp { - \beta(U + PV)}\\ &=& \exp {- \beta \mathcal{H} }, \end{array} $$
(2)

where β = 1/(kBT) is the inverse temperature (kB is the Boltzmann constant) and \(\mathcal {H}\) is the enthalpy. Scaled coordinates are used in the isothermal–isobaric molecular simulation (McDonald 1972). When an isotropic box is used, the coordinates are scaled as s = V− 1/3q. In this case, the distribution function is transformed as

$$ \begin{array}{@{}rcl@{}} \exp[-\beta\{U(q, V) + PV\}] dq &=& \exp[-\beta\{U(s, V) + PV\}] V^{N} ds \\ &=& \exp[-\beta \{U(s, V) + PV - Nk_{B}T \log V \}] ds . \end{array} $$
(3)

Thermodynamic quantities

The difference in the Gibbs free-energy ΔG between states A and B is calculated as follows:

$$ {\Delta} G = G_{B} - G_{A} =-RT \log \left( \frac{f_{B}}{f_{A}} \right) , $$
(4)

where R is the gas constant, fA and fB are fractions of the states A and B, respectively. In the case of evaluating protein stability, for example, fA and fB are probabilities of the folded and unfolded states, respectively. The differences in the partial molar enthalpy ΔH, the heat capacity ΔCP, and the partial molar volume ΔV are calculated as follows:

$$ \begin{array}{@{}rcl@{}} {\Delta} H {\kern5pt} &=& - R \left[ \frac{\partial \log (f_{B}/f_{A})}{\partial(1/T)} \right]_{P} = \left[ \frac{\partial \left( {\Delta} G / T\right)} {\partial \left( 1/T \right)} \right]_{P} , \end{array} $$
(5)
$$ \begin{array}{@{}rcl@{}} {\Delta} C_{P} &=& - T \left( \frac{\partial^{2} {\Delta} G}{\partial T^{2}}\right)_{P} , \end{array} $$
(6)
$$ \begin{array}{@{}rcl@{}} {\Delta} V {\kern6pt} &=& - RT \left[\frac{\partial \log (f_{B}/f_{A})}{\partial P} \right]_{T} = \left[\frac{\partial {\Delta} G}{\partial P}\right]_{T} . \end{array} $$
(7)

The differences in the internal energy ΔU and the entropy ΔS are calculated as follows:

$$ \begin{array}{@{}rcl@{}} {\Delta} U &=& {\Delta} H - P {\Delta} V , \end{array} $$
(8)
$$ \begin{array}{@{}rcl@{}} {\Delta} S &=& \frac{{\Delta} H - {\Delta} G}{T} . \end{array} $$
(9)

The temperature and pressure dependence of ΔG can be fitted by the following equation proposed by Hawley (1971):

$$ \begin{array}{@{}rcl@{}} {\Delta} G(T,P) &= &{\Delta} G_{0} - {\Delta} S_{0} (T - T_{0}) -{\Delta} C_{P} \left[T \left\{\ln \left( \frac{T}{T_{0}} \right) -1 \right\} + T_{0} \right] \\ &&+ {\Delta} V_{0}(P-P_{0}) + \frac{{\Delta} \beta}{2}(P-P_{0})^{2} + {\Delta} \alpha (P-P_{0})(T-T_{0}) . \end{array} $$
(10)

Here, T0 and P0 are the reference temperature and pressure. ΔG0, ΔS0, and ΔV0 are the differences in the Gibbs free-energy, entropy, and partial molar volume between states A and B, respectively, at T0 and P0. Δα and Δβ are the differences in the thermal expansivity factor and compressibility factor between states A and B, respectively; these factors are related to, but are not identical to the thermal expansion coefficient and compressibility in thermodynamics, respectively. By substituting (10) into (5) and (7), the temperature and pressure dependence of ΔH and ΔV are obtained:

$$ \begin{array}{@{}rcl@{}} {\Delta} H (T,P) &=& {\Delta} G_{0} + T_{0} {\Delta} S_{0} + (T - T_{0}){\Delta} C_{p} \\ &+& {\Delta} V_{0} (P - P_{0}) + \frac{{\Delta} \beta}{2} (P - P_{0})^{2} - {\Delta} \alpha (P - P_{0})T_{0} , \end{array} $$
(11)
$$ \begin{array}{@{}rcl@{}} {\Delta} V (T,P) &=& {\Delta} V_{0} + {\Delta} \beta (P - P_{0}) + {\Delta} \alpha (T - T_{0}) . \end{array} $$
(12)

Generally, from Le Chatelier’s principle, the volume of a substance becomes smaller by applying pressure. In other words, the probability of the smaller volume state increases as pressure increases.

Generalized-ensemble algorithms in isothermal–isobaric ensemble

In this section, we first introduce the general view of Markov chain Monte Carlo algorithms (Berg 2004). Then, we review formulations on a series of generalized-ensemble algorithms in the isothermal–isobaric ensemble: multibaric–multithermal, pressure- and temperature-simulated tempering, replica-exchange, and replica-permutation methods.

Markov chain Monte Carlo algorithm

Let us consider the Markov chain Monte Carlo (MCMC) algorithm. We assume here that the number of candidates for a state transition is n. Each state i has a weight wi and transits to state j with a transition probability P(i → j). To converge the unique distribution, Markov chain should satisfy ergodicity (that is, the transitions are regular and non-periodic). In addition, the following equations are imposed to obtain the required equilibrium distribution (Manousiouthakis and Deem 1999) such as isothermal–isobaric ensemble:

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{n} P(i \to j) &=& 1 , \end{array} $$
(13)
$$ \begin{array}{@{}rcl@{}} \sum\limits_{i=1}^{n} w_{i} P(i \to j) &=& w_{j} . \end{array} $$
(14)

Equation 13 is the normalization condition and Eq. 14 is the balance condition. These equations are summarized by using a stochastic flow from state i to state j defined by v(i → j) = wiP(i → j), as follows:

$$ \sum\limits_{i=1}^{n} v(i \to j) = \sum\limits_{i=1}^{n} v(j \to i). $$
(15)

The Metropolis algorithm (Metropolis et al. 1953) (its generalization is called Metropolis–Hastings algorithm (Hastings 1970)) and heat bath (Gibbs sampler) algorithm (Geman and Geman 1984) are the two most practical implementations of the MCMC algorithms. These algorithms only satisfy the detailed balance condition, which is the sufficient condition for Eq. 15:

$$ v(i \to j) = v(j \to i). $$
(16)

In the Metropolis algorithm, the amount of stochastic flow from state i to state j is given as follows (see Fig. 1a):

$$ \begin{array}{@{}rcl@{}} v(i \to j) = \frac{1}{n-1} \min \left( w_{i},w_{j} \right), i \neq j, \end{array} $$
(17)

where the coefficient 1/(n − 1) arises from the random selection of state j from n − 1 candidates except state i. In the heat bath algorithm, the amount of stochastic flow from state i to state j is given as follows (see Fig. 1b):

$$ \begin{array}{@{}rcl@{}} v(i \to j) = \frac{w_{i}w_{j}}{{\sum}_{k=1}^{n} w_{k}}, \forall i, j. \end{array} $$
(18)
Fig. 1
figure 1

Schematic figures of the a Metropolis, b heat bath, and c Suwa–Todo algorithms

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Recently, Suwa and Todo proposed a new type of MCMC algorithm (Suwa and Todo 2010), which satisfies (15) without imposing the detailed balance condition (16). This method succeeds in minimizing the rejection ratio, and in many cases, the rejection ratio becomes zero. The amount of stochastic flow is given as follows (see Fig. 1c):

$$ v(i \to j) = \max [0, \min[{\Delta}_{i j}, w_{i}+w_{j}-{\Delta}_{i j}, w_{i},w_{j} ] ], $$
(19)

where

$$ \begin{array}{@{}rcl@{}} {\Delta}_{i j} &\equiv& S_{i} -S_{j - 1} + w_{k} , \\ S_{i} &\equiv& \left\{\begin{array}{llll} \displaystyle \sum\limits_{j=k}^{i} w_{j} , &&\text{for}\ i \geq k \\ \displaystyle \sum\limits_{j=k}^{n} w_{j} + \sum\limits_{j=1}^{i} w_{j}, &&\text{for}\ i < k, \end{array}\right. \\ S_{0} &\equiv& S_{n} , \end{array} $$

and wk is a maximum weight among n states. Note that the Suwa–Todo algorithm is identical to the Metropolis algorithm when n = 2.

Multibaric–multithermal method

A multibaric–multithermal (MBT) method (Okumura and Okamoto 2004a, b, c, 2006) was proposed as an extension of the multicanonical method. In the MBT method, the uniform distribution in the potential energy and volume spaces is realized so that the system can perform a random walk in the potential energy and volume spaces, as shown in Fig. 2a. Hence, we can obtain the isothermal–isobaric ensemble in a wide range of temperature and pressure values from one MBT simulation.

Fig. 2
figure 2

Schematic figures of generalized-ensemble algorithms in isothermal–isobaric ensemble: a multibaric–multithermal, b pressure- and temperature-simulated tempering, c replica-exchange, and d replica-permutation methods

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In the MBT method, a non-Boltzmann weight factor

$$ W_{\text{MBT}} \{U(q,V), V\} \equiv \exp [ -\beta_{0} \mathcal{H}_{\text{MBT}} \{U(q,V), V\}] $$
(20)

is used instead of the usual distribution function in the isothermal–isobaric ensemble given in Eq. 2. Here, β0 = 1/(kBT0) is a reference temperature and \(\mathcal {H}_{\text {MBT}}\) is the multibaric–multithermal enthalpy. \(\mathcal {H}_{\text {MBT}}\) is determined so as to realize the uniform probability distribution of the potential energy and volume:

$$ \begin{array}{@{}rcl@{}} P_{\text{MBT}} (U, V) &=& n(U, V) W_{\text{MBT}} (U, V) \\ &=& n(U, V) \exp[-\beta_{0} \mathcal{H}_{\text{MBT}} \{U(q,V), V\}] \\ &=& \text{const} . \end{array} $$

To perform the MBT simulation, the non-Boltzmann weight factor (or multibaric–multithermal enthalpy) must be determined in advance. An iterative procedure of trial simulations (Berg and Celik 1992a; Lee 1993; Okamoto and Hansmann 1995; Mitsutake et al. 2001) or Wang–Landau method (Wang and Landau 2001a, 2001b) is used to obtain the weight factor (also see Refs. Okumura and Okamoto 2008, 2012). By using the obtained weight factor, we perform one long multibaric–multithermal simulation. For MC simulation, the transition probability is calculated using multibaric–multithermal enthalpy \(\mathcal {H}_{\text {MBT}}\) instead of the enthalpy \(\mathcal {H}\) (Okumura and Okamoto 2004b, 2004c). For MD simulation, the enthalpy term \(\mathcal {H} = U + PV\) in the equations of motion is substituted by multibaric–multithermal enthalpy \(\mathcal {H}_{\text {MBT}}\) (Okumura and Okamoto 2004a, 2006).

Pressure- and temperature-simulated tempering method

Let us introduce a simulated tempering (ST) method associated with pressure and temperature (PTST) (Mori and Okamoto 2010). The PTST realizes a uniform distribution in the temperature and pressure spaces by treating those as dynamic variables. Temperature and pressure as well as the coordinates and momenta are changed during the PTST simulation.

The weight factor for the PTST is given by

$$ \begin{array}{@{}rcl@{}} W_{\text{PTST}} (U, V; T, P) \equiv \exp \{ -\beta (U +PV) + g(T,P) \}. \end{array} $$
(21)

Here, g(T,P) is introduced to realize the uniform distribution in the temperature and pressure spaces. g(T,P) is determined so as to satisfy the following equation:

$$ \begin{array}{@{}rcl@{}} P_{\text{PTST}} (T, P) = {\int}_{0}^{\infty}\! dV\! {\int}_{V} dq W_{\text{PTST}} \{ U(q,V), V; T, P \} = \text{const.} ,\\ \end{array} $$
(22)

and thus, g(T,P) is written as

$$ \begin{array}{@{}rcl@{}} g(T,P) = - \ln \left[ {\int}_{0}^{\infty} dV {\int}_{V} dq \exp [ -\beta \{ U(q,V) + PV \} ] \right]. \end{array} $$
(23)

We find that g(T,P) corresponds to a dimensionless Gibbs free-energy omitting a constant. In practice, the temperature and pressure are discretized. Let us consider using M0 temperatures and M1 pressures. Here, we introduce indices m0 and m1 that indicate one of the M0 temperatures and one of M1 pressures, respectively. The weight factor and the dimensionless Gibbs free-energy \(g_{m} = g(T_{m_{0}}, P_{m_{1}})\) at \(T_{m_{0}}\) and \(P_{m_{1}}\) are given as follows:

$$ \begin{array}{@{}rcl@{}} &&W_{\text{PTST}} (U, V; T_{m_{0}}, P_{m_{1}}) = \exp \left\{ - \beta_{m_{0}} (U + P_{m_{1}} V) + g_{m} \right\} , \end{array} $$
(24)
$$ \begin{array}{@{}rcl@{}} &&g(T_{m_{0}}, P_{m_{1}}) \!= \!- \ln \left[ {\int}_{0}^{\infty} dV {\int}_{V} dq \exp [ -\beta_{m_{0}} \{ U(q,V) \!+ P_{m_{1}}V \} ] \right] . \end{array} $$
(25)

The weight factor (i.e., the dimensionless Gibbs free-energy) should be determined before performing the PTST simulation. The weight factor is determined by an iterative procedure of short trial simulations (Irbäck and Potthast 1995; Hansmann and Okamoto 1997; Mitsutake et al. 2001) or short replica-exchange simulation combining with reweighting techniques (Kumar et al. 1992; Shirts and Chodera 2008). On-the-fly weight determination scheme during the ST simulation was also proposed (Nguyen et al. 2013).

The PTST simulation is performed as follows (schematic figure is shown in Fig. 2b): (i) isothermal–isobaric MD or MC simulation at temperature \(T_{m_{0}}\) and \(P_{m_{1}}\) is performed for certain steps, (ii) calculate the transition probability from \((T_{m_{0}}, P_{m_{1}})\) to another set of temperature and pressure value \((T_{n_{0}}, P_{n_{1}})\) in M0 temperatures and M1 pressures, and (iii) update the temperature and pressure following the obtained transition probability. Usually, the transition probability is calculated by the Metropolis algorithm. Instead of the Metropolis algorithm, the Gibbs sampler (heat bath algorithm) or Suwa–Todo algorithm is also applicable to enhance acceptance ratio (Chodera and Shirts 2011; Mori and Okumura 2015).

Isothermal–isobaric replica-exchange method

In an isothermal–isobaric replica-exchange method (TP-REM) (Okabe et al. 2001; Mori and Okamoto 2010), simulations at different thermodynamic states, characterized by a set of temperature and pressure \(\lambda _{m}=[T_{m_{0}}, P_{m_{1}}]\), are performed in parallel. M0 different temperatures and M1 different pressures are assigned to M(= M0 × M1) non-interacting replicas (copies of the system). A state in the TP-REM is expressed by a combination of the thermodynamic state indices and the replica indices: \( X_{\alpha } = \left [x_{1}^{[i(1)]},x_{2}^{[i(2)]},\dots ,x_{M}^{[i(M)]}\right ] = \left [x_{m(1)}^{[1]},x_{m(2)}^{[2]},\dots ,x_{m(M)}^{[M]}\right ] \), where i(m) is a permutation function from the thermodynamic state index to replica index and m(i) is the inverse.

The weight factor for the state Xα is given by the product of the weight factor of each replica i (or each thermodynamic state m):

$$ \begin{array}{@{}rcl@{}} w_{\text{R}}(X_{\alpha}) &=& \prod\limits_{i=1}^{M} \exp \left[-\beta_{m(i)} \left\{ H(x_{m(i)}^{[i]}) + P_{m(i)}V(x_{m(i)}^{[i]}) \right\} \right] \\ &=& \prod\limits_{m=1}^{M} \exp \left[-\beta_{m} \left\{H(x_{m}^{[i(m)]}) + P_{m}V(x_{m}^{[i(m)]}) \right\} \right] . \end{array} $$
(26)

In the TP-REM, exchange trials for a pair of parameters

$$ \begin{array}{@{}rcl@{}} X_{\alpha} = [\dots, x_{m}^{[i]},\dots, x_{n}^{[j]}, \dots] \to X_{\gamma} = [\dots, x_{n}^{[i]},\dots, x_{m}^{[j]}, \dots] \end{array} $$
(27)

are performed during the simulation (Fig. 2c). Typically, neighboring parameters are selected as a pair of replica exchange. The temperature exchange and pressure exchange trials are performed alternately or randomly. The Metropolis algorithm is employed to calculate the amount of stochastic flow v(Xα → Xγ) and transition probability P(Xα → Xγ). Note that the weight factors wi and wj in Eq. 17 are replaced by wR(Xα) and wR(Xα), respectively.

Isothermal–isobaric replica-permutation method

An isothermal–isobaric replica-permutation method (TP-RPM) (Yamauchi and Okumura 2017, 2019) is an improved alternative to the TP-REM. In the TP-RPM, the parameter permutation among more than two replicas is performed during the simulation (Fig. 2d):

$$ \begin{array}{@{}rcl@{}} X_{\alpha} = \left[ x_{m(1)}^{[1]}, \cdots, x_{m(M)}^{[M]}\right] \to X_{\gamma} = \left[ x_{n(1)}^{[1]}, \cdots, x_{n(M)}^{[M]}\right] , \end{array} $$
(28)

where m and n are permutation functions. All possible permutations between replicas and parameters are considered; thus, γ takes values from 1 to M!. However, it becomes impractical to consider all possible permutations as the number of replicas increases, especially exceeding ten replicas. To avoid this problem, the temperature permutations and the pressure permutations are performed alternately. If the number of permutation candidates is still large, the sets of temperatures or pressures are divided into several subsets. Usually, subsets are set so that the number of replicas in each subset is the same and the temperature or pressure labels in each subset are sequential and cyclic. Parameter permutation is performed among the replicas in each subset.

The weight factor for state Xα is identical to the REM as in Eq. 26. The Suwa–Todo algorithm is employed to calculate the amount of stochastic flow v(Xα → Xγ) and transition probability P(Xα → Xγ). Of course, instead of the Suwa–Todo algorithm, the Metropolis or heat bath algorithm is also able to be employed. However, sampling efficiency, such as transition ratio of the parameters, is lower than the Suwa–Todo algorithm (Yamauchi and Okumura 2017). The comparison of the transition ratio of temperature and pressure values between TP-RPM with the Suwa–Todo, Metropolis, heat bath algorithms, and TP-REM is presented in Figure 3. Here, the transition ratio at a parameter label (i.e., a set of temperature and pressure labels) is defined as a probability with which the replica at that parameter label is transferred to other parameter labels. These results clearly show that the TP-RPM with the Suwa–Todo algorithm achieves the highest transition ratio, indicating that it is the most efficient among the four methods.

Fig. 3
figure 3

Comparison of the transition ratio between TP-RPM with the Suwa–Todo algorithm and a TP-REM, b TP-RPM with the heat bath algorithm, and c TP-RPM with the Metropolis algorithm

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Estimation of physical quantities: reweighting techniques

After performing the generalized-ensemble simulation, we can obtain statistical averages of physical quantities by using reweighting techniques.

For the MBT method, a single-histogram reweighting technique (Ferrenberg and Swendsen 1988, 1989a, Okumura and Okamoto 2004c) is employed to obtain the density of states and probability distribution at any temperature and pressure values.

For the PTST, TP-REM, and TP-RPM, the arithmetic mean of the trajectory data from the one equilibrium state gives the isothermal–isobaric average at the set temperature and pressure values for the simulation. On the other hand, a weighted histogram analysis method (WHAM) (Kumar et al. 1992; Chodera et al. 2007), which is also called multiple-histogram reweighting techniques (Ferrenberg and Swendsen 1989b), and a multistate Bennett acceptance ratio (MBAR) method (Shirts and Chodera 2008) are widely applied to calculate the isothermal–isobaric averages of the physical quantities. These methods enable us to evaluate the physical quantities more precisely because of considering all trajectory data from multiple equilibrium states. The WHAM gives the density of states, whereas the MBAR cannot directly calculate. However, the histogram construction of the WHAM makes bias due to the binning, and is a time-consuming process. The MBAR does not require making the histogram; thus, the MBAR provides the weight for each trajectory and the direct estimation of statistical errors.

A generic transition-based reweighting analysis method (TRAM) (Wu et al. 2016) has been developed, which is a general formalization of the WHAM and MBAR. The TRAM takes advantage of the Markov states model. The key assumption of the TRAM is that the data are sampled from the local equilibrium distributions, while the WHAM and MBAR assume those from the global equilibrium distribution of each ensemble. The TRAM provides not only statistical average of the physical quantities, but also the transition probability from one state to the other states. Thus, we can obtain information about kinetics of the target system.

Application of isothermal–isobaric generalized methods to biomolecules

Pressure dependence for β-hairpin formation peptide: chignolin

Chignolin is an artificially designed peptide comprised of ten amino acid residues, and its amino acid sequence is GYDPETGTWG (Honda et al. 2004). Chignolin folds into a β-hairpin structure. Molecular dynamics simulation studies revealed that chignolin is able to form two types of β-hairpin structure: one is a folded structure and the other is a misfolded structure (see Fig. 4a and b) (van der Spoel and Seibert 2006; Satoh et al. 2006; Suenaga et al. 2007; Harada and Kitao 2011; Kührová et al. 2012).

Fig. 4
figure 4

Typical conformations of the a folded and b misfolded chignolin. c The pressure dependence of fraction for folded, misfolded, and unfolded chignolin

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To investigate the folding–unfolding mechanism of chignolin, the multibaric–multithermal molecular dynamics simulation was performed (Okumura 2012). Folding and unfolding processes were discussed by calculating free-energy landscapes. The unfolding process begins from only the C terminus or both C and N termini of the β-hairpin structures. The intermediate structures, for example, 310-helix and α-helix, can be formed under the unfolding pathway. The folding is the reverse process. Chignolin is denatured by not only temperature but also pressure. The pressure denaturation occurs because water molecules approach the hydrophobic residues with increasing pressure, and the water molecules break the hydrophobic contacts among side chains.

The isothermal–isobaric replica-permutation MD simulation was also performed (Yamauchi and Okumura 2017), revealing that the fraction of the misfolded structure increases while that of folded structure decreases with increasing pressure (see Fig. 4c). The partial molar volume difference was estimated as ΔV = Vmisfold − Vfold = − 4.6 ± 0.2 cm3/mol at 300 K, indicating that the misfolded structure has smaller partial volume. The molecular mechanism of the different stability between folded and misfolded structures under high pressure arises from different orientation of the Tyr2 and Trp9 side chains (Fig. 4a and b). For the folded structure, as pressure increases, water molecules approach not only hydrophobic core, but also intra-hydrogen bonds from the opposite side of the hydrophobic core. Therefore, the water molecules break the intra-hydrogen bonds as well as the hydrophobic core. On the other hand, in the misfolded structure, the hydrogen bonds that are important to form the misfolded structure are covered with the Try2 and Trp9 side chains. The side chains protect the hydrogen bonds from the approaching water molecules. Therefore, the misfolded structure becomes more stable with compressing its structure under high-pressure condition.

Pressure dependence for helix formation peptide: AK16

AK16 peptide is comprised of 16 amino acid residues, and its amino acid sequence is YGAAKAAAAKAAAAKA. AK16 peptide forms α-helix conformation at room temperature and atmospheric pressure (Chakrabartty and Baldwin 1995). FT-IR spectroscopic studies investigated pressure dependence of the alanine-rich peptide including AK16 peptide, revealing that the fraction of α-helix conformation increases with increasing pressure (Takekiyo et al. 2005; Imamura and Kato 2009).

To understand the molecular mechanism for the pressure-induced conformation changes of AK16 peptide, the ST simulations associated with pressure were performed (Mori and Okumura 2013, 2014). The pressure dependence of the fraction for helical conformation and the partial molar volume difference between folded and unfolded structures, ΔV = Vunfold − Vfold, were calculated, as presented in Fig. 5a, b. In the high-pressure condition, especially more than 0.6 GPa, the fraction for the helical conformation increases and the ΔV takes a positive value. These results are consistent with experimental results (Takekiyo et al. 2005). The typical conformations of the AK16 peptide at various pressures are illustrated in Fig. 5c. As pressure increases, the diameter of the helix structure and the distance between hydrogen bonds that form the helix structure become short. These results indicate that the helical structure at high pressure is more compressed from the longitudinal and transversal directions than that at atmospheric pressure. This is also the reason why the ΔV takes positive value at high pressure.

Fig. 5
figure 5

Pressure dependence of the a fraction for the folded AK16 peptide and b partial molar volume difference. c Typical conformations of the AK16 peptide at P = 0.1 MPa, 0.6 GPa, and 1.4 GPa

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Pressure dependence for helix formation peptide: C peptide

C-peptide analogs are polypeptides that exist at the N terminus of ribonuclease A. Here, we only discuss the C-peptide analog comprised of thirteen amino acid residues (AETAAAKFLRAHA). Hereafter, we refer to this C-peptide analog as simply C-peptide. The C-peptide forms α-helix conformation with stabilization of a salt bridge (ionic interaction) between Glu2 and Arg10 (Scholtz and Baldwin 1992).

To reveal the pressure-induced conformation changes of the C-peptide, the ST simulations associated with pressure were performed (Mori and Okumura 2014). Figure 6a presents pressure dependence of the fraction for the helical conformation. Below 0.6 GPa, the fraction of the helical conformation of the C-peptide decreases as pressure increases. In contrast, above 0.6 GPa, the fraction of the helical conformation increases. Figure 6b presents partial molar volume difference between folded and unfolded structures: ΔV = Vunfold − Vfold. The partial molar volume difference takes a positive value at high-pressure conditions. The typical conformations of the C-peptide at various pressures are illustrated in Fig. 6c. It is noteworthy that the helical conformation of the C-peptide at high pressure does not form the salt bridge. This is a different point from the helical conformation at atmospheric pressure. The salt bridge contributes to the stabilization of the helical conformation at room temperature and atmospheric pressure. As the pressure increases up to 0.6 GPa, the salt bridge is broken by the hydration of water molecules. This is the reason why the fraction for helical conformation takes minimum value around 0.6 GPa. As further pressure is applied, the volume of the helical conformation becomes smaller than the helical conformation at the atmospheric pressure and the unfolded structure; thus, the helical conformation is stable under high-pressure conditions. Indeed, the diameter of the helix structure and distance between hydrogen bonds that form the helix structure gradually become short with increasing pressure. These results reflect compression of the helical conformation under the high-pressure condition.

Fig. 6
figure 6

Pressure dependence of the a fraction for the folded C-peptide and b partial molar volume difference. c Typical conformations of the C-peptide at P = 0.1 MPa, 0.6 GPa, and 1.4 GPa

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Ubiquitin

Ubiquitin (UB), consisting of 76 amino acid residues, is involved in many cellular phenomena such as proteasomal degradation of damaged proteins. Its structure, functions, and protein folding have been studied thoroughly both in experimental and theoretical aspects. NMR experiments detected conformational change of the UB induced by pressure perturbation (Kitahara et al. 2001, 2003, 2005, Nisius and Grzesiek 2012). The pressure-simulated tempering (PST) simulations also demonstrated the pressure-induced conformational changes of the UB (Mori and Okamoto 2017). In particular, larger fluctuations were found between 7–9 and 30–40 amino acid residues during the simulations. As presented in Fig. 7a, the peak position of the distribution for the distance between Leu8 and Glu34 is shifted with increasing pressure from 0.1 MPa to 1.0 GPa, which is consistent with the constant pressure MD simulation (Imai and Sugita 2010). The conformations at 0.1 GPa and 1.0 GPa are illustrated in Fig. 7b, c. The water molecules penetrate into the inside of the UB in the high-pressure condition, which induces denaturation. The trajectory data from the PST simulations were also employed to 15N NMR chemical shift estimation (La Penna et al. 2016). This technique is useful to integrate experiment and simulation results.

Fig. 7
figure 7

a The probability distribution of the distance between Leu8 and Glu34. Typical conformation of ubiquitin at bP = 0.1 MPa and cP = 1.0 GPa

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Conclusions

In this review, we outlined the generalized-ensemble algorithms in the isothermal–isobaric ensemble and its applications to the biomolecule system. The generalized-ensemble simulation in the isothermal–isobaric ensemble enables us to sample conformations of the target system at various temperature and pressure values. Furthermore, a random walk in the parameter spaces enhances the sampling efficiency in the conformational space; thus, physical quantities are estimated more precisely.

We also reviewed the application of the generalized-ensemble algorithms in the isothermal–isobaric ensemble to the following biomolecular systems: chignolin, AK16 peptide, C-peptide, and ubiquitin. The physical quantities, such as ΔV and 15N chemical shift, were estimated from the simulations. We can directly compare these quantities from simulations with those from experiments. Furthermore, the generalized-ensemble simulation in the isothermal–isobaric ensemble enables us not only to elucidate the mechanism behind the conformational change, but also to predict the stable conformation under high-pressure conditions. We believe that integrating experimental and theoretical results will lead to deep understanding of life system.

Although we only focused on the conformational change induced by pressure pertubation in this review, the generalized-ensemble methods in the isothermal–isobaric ensemble are also powerful tools to investigate the molecular mechanism of the conformational change induced by cold temperature (Privalov 1989). To access higher pressure and lower temperature region, more efficient generalized-ensemble algorithms are necessary. The post analysis methods such as reweighting become key techniques for not only estimating physical quantities more accurately but also extracting more information from the generalized-ensemble simulations. Furthermore, it should be emphasized that using accurate force field, especially water model, is crucial for quantitative estimation of physical quantities at various temperature and pressure values (Yang et al. 2014; Best et al. 2014).