Water's Hydrogen Bonds in the Hydrophobic Effect: A Simple Model

J. Phys. Chem. B 2005, 109, 23611-23617

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Water’s Hydrogen Bonds in the Hydrophobic Effect: A Simple Model Huafeng Xu and Ken A. Dill* Department of Pharmaceutical Chemistry and Graduate Group of Biophysics, UniVersity of California, San Francisco, San Francisco, California 94143 ReceiVed: May 21, 2005; In Final Form: October 17, 2005

We propose a simple analytical model to account for water’s hydrogen bonds in the hydrophobic effect. It is based on computing a mean-field partition function for a water molecule in the first solvation shell around a solute molecule. The model treats the orientational restrictions from hydrogen bonding, and utilizes quantities that can be obtained from bulk water simulations. We illustrate the principles in a 2-dimensional MercedesBenz-like model. Our model gives good predictions for the heat capacity of hydrophobic solvation, reproduces the solvation energies and entropies at different temperatures with only one fitting parameter, and accounts for the solute size dependence of the hydrophobic effect. Our model supports the view that water’s hydrogen bonding propensity determines the temperature dependence of the hydrophobic effect. It explains the puzzling experimental observation that dissolving a nonpolar solute in hot water has positive entropy.

Introduction The “hydrophobic effect” refers to the poor solubility of nonpolar solutes in water. Thermodynamically, the hydrophobic effect is characterized by the positive free energy (∆G > 0) of transferring a nonpolar solute into water. A very interesting aspect of the hydrophobic effect is its temperature dependence: the entropy (∆S) and the enthalpy (∆H) of the transfer process both increase quickly with temperature T. This is reflected in the large heat capacity change, ∆Cp ) ∂∆H/∂T ) T∂∆S/∂T, that occurs in hydrophobic solvation. Although ∆G itself varies little with temperature, due to almost perfect enthalpy-entropy compensation,1 the large ∆Cp is a ubiquitous thermodynamic characteristic in the hydrophobic effect. Since this same thermodynamic signature is observed in many biophysical processes such as protein folding,2-7 proteinprotein binding,8 and protein-DNA binding,9 it has been taken as evidence that the hydrophobic effect is a key driving force in those processes. In this work we develop a simple analytical model of the hydrophobic effect that captures this large heat capacity. Modern modeling of aqueous solvation and the hydrophobic effect usually involves either serious computational expense, as in detailed all-atom computer simulations, or substantial simplification of the physics, whereby (a) water is represented as a continuum, (b) hydrogen bonding is not treated explicitly, and therefore (c) effects of temperature and solute shape are not handled. Our aim here is a middle ground. We seek a model that contains water’s hydrogen bonding physics yet is simple enough to be treated analytically and with computational efficiency. There has been significant progress in modeling the hydrophobic effect.10-18 First, there have been studies involving computer simulations and atomically detailed force fields.15,19-22 Also, there are analytical treatments of simple models that aim to explain the temperature dependence of the hydrophobic effect in terms of the properties of water.4,23-27 Some of these analytical approaches are based on the microscopic properties * E-mail: [email protected].

of water, such as the oxygen-oxygen radial distribution function that is measured experimentally, to construct a perturbative approximation of water structure near the nonpolar solute.4,23 Other models use computer simulations to generate the water structure near nonpolar solute and relate this structural information to the thermodynamics of solvation.25,28 A simplified twostate model, which partitions water hydrogen bonds into broken or formed states, has also been proposed to explain the heat capacity in hydrophobic solvation, where the energies and entropies of the two states are treated as empirical parameters to fit the experimental heat capacity curve.27 A key component of the hydrophobic effect is thought to be water’s hydrogen bonding to neighboring waters.14,26,29,30 With one important exception,24 previous analytical models do not aim to determine how the hydrophobic effect arises from water’s hydrogen bonding. Ashbaugh et al.24 consider explicitly the perturbation by the nonpolar solute to water-water hydrogen bonds. Our aim here is to develop an even simpler analytical treatment of the hydrophobic effect that starts from a model of water-water hydrogen bonding. Theory Our starting point is the 2-dimensional Mercedes-Benz (MB) model of water31 (see Figure 1). This simple model has previously been shown in Monte Carlo simulations to exhibit all the characteristic properties of water and hydrophobic solvation.32-35 Each water molecule is a circular Lennard-Jones disk with three hydrogen bonding arms, each situated at an angle 2π/3 from the other two, as in the Mercedes-Benz logo. Two water molecules form a hydrogen bond when: (a) the two water molecules are sufficiently close to each other, within the ideal hydrogen bonding distance lhb, and (b) the arm of one molecule is aligned with the arm of another, within a certain angle. The hydrogen bond energy depends on the geometric deviations from the ideal bonding distance and alignment (see the caption of Figure 1). Within this model, a nonpolar solute is a circular disk without hydrogen bonding arms. Figure 2 shows the thermodynamic process of interest. We focus on a water molecule that is situated in the first solvation

10.1021/jp0526750 CCC: $30.25 © 2005 American Chemical Society Published on Web 11/17/2005

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Figure 1. Two water molecules in the 2-dimensional Mercedes-Benz model. In the original model, the hydrogen bond energy between the two water molecules is given by uhb(φ1,φ2,d) ) mG(d - lhb)G(cos φ1 - 1)G(cos φ2 - 1), where G(x) ) exp(- x2/2κ2) is a Gaussian function, lhb is the ideal hydrogen bond length, and m < 0 is the maximum hydrogen bond energy. κ ) 0.085 is used in the Monte Carlo simulations and the theoretical calculations presented in this paper.

Figure 2. Transferring a nonpolar solute into water is viewed as a change in the environment of a first-shell (test) water molecule: a neighboring water molecule is replaced by a neighboring nonpolar solute.

shell around the nonpolar solute. Call it the test molecule. We are interested in computing the free energy change that occurs when the test molecule undergoes a change whereby its neighboring water is replaced by a solute molecule. There are n such first-shell water molecules that each undergo this change in environment. The individual-molecule partition function for a water molecule in the bulk is q0, and for a water molecule in the solvation shell is qs. Thus, the free energy of transferring the nonpolar solute into water is the free energy for changing the environment for all n test molecules that comprise the firstsolvation shell,

qs ∆F ) -nkT ln q0

(1)

where k is the Boltzmann constant and T is the temperature. To compute the free energy of solute transfer into water, we need to determine qs and q0. For a solvation-shell water molecule (see A in Figure 3a), the partition function is

qs )

∫02π dθ exp(-uhb(θ)/2kT)

(2)

Figure 3. (a) Defining the critical angle for hydrogen bonding. The lower-right arm of A can form a hydrogen bond with a neighboring water molecule only if φ e φc. (b) Defining tangential and perpendicular orientations of a water next to a solute molecule. (c) Defining tangential and perpendicular orientations of a neighboring water B relative to the test water A.

a hydrogen bond if and only if φ e φc. Second, any arm that satisfies these conditions will form the hydrogen bond with a probability 0 < f(T) < 1. Thus, f(T) is the fraction of hydrogen bonds that are formed. We find the quantity f(T) using previous Monte Carlo simulations of neat MB water (i.e., without the solute). Third, each hydrogen bond has an average energy j(T) < 0. Both f(T) and j(T) depend on temperature. In this model, it is these two temperature dependences that combine to give the temperature dependence of hydrophobic solvation. With these simplifications, the average hydrogen bond energy of A in orientation θ is given by

uhb(θ) ) nhb(θ)f j

(3)

where nhb(θ) ) 0, 1, 2, 3 is the number of arms that can form hydrogen bonds. In the bulk, nhb(θ) ) 3. However, for water molecules adjacent to solute molecules, geometric restrictions may impose a smaller value of this quantity. In this approximation, we have factorized the first-shell water hydrogen bond energy into an energetic component, f(T), that is solute independent, and a steric component, nhb(θ), that is influenced by the solute. If we denote the angle interval in which nhb(θ) ) n as θn, the partition function for water molecule A adjacent to a solute is 3

where the dominant contribution to the water-water interaction is assumed to be the hydrogen bond energy uhb(θ), as a function of the angle of one water molecule relative to its neighbor. The factor of 1/2 corrects for double-counting of hydrogen bonds. To estimate uhb(θ), we make the following approximations. First, an arm of molecule A can hydrogen-bond to its neighboring water only if the neighbor is: (a) aligned with the arm within a threshold angle θc and (b) is located at a distance lhb from A. In Figure 3a, this means that the lower-right arm of A can form

qs )

∑ θn exp(-nf j/2kT)

(4)

n)0

Let us first consider a solute molecule having the same diameter, lhb, as a water molecule. The water molecule can either straddle the solute, making all 3 of its possible hydrogen bonds with nearby waters, or it can point one of its arms toward the solute, leaving only 2 arms for hydrogen bonding with other waters. We refer to the former as the tangential orientation, Ot, and the

Water’s Hydrogen Bonds

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latter as the perpendicular orientation, Op (see Figure 3b). Geometry considerations show that the angle interval for the tangential orientation is θ3 ) 6φc, and for perpendicular orientation is θ2 ) 2π - 6φc (see the section on solute size effect below). Thus, the partition function for a first-shell water molecule is

qs ) (2π - 6φc) exp(-f j/kT) + 6φc exp(-3f j/2kT)

(5)

Now consider a water molecule, A, in the bulk. We consider two situations, depending on the orientation of a neighboring water molecule B (see Figure 3c): (1) B cannot form a hydrogen bond with A (the “tangential state” Bt), or (2) B can form a hydrogen bond with A (the “perpendicular state” Bp). When molecule B is in the tangential configuration, it cannot form a hydrogen bond with A. In that situation, B is no different than a nonpolar solute, from A’s perspective. Therefore, the partition function for an A molecule in tangential bulk configurations is

q0,t ) (2π - 6φc) exp(-f j/kT) + 6φc exp(-3f j/2kT) (6) However, when molecule B is in the perpendicular state, A and B can form a hydrogen bond. This situation is different than if molecule B were a nonpolar solute. In this case, the average hydrogen bond energy of A in orientation θ is

u′hb(θ) ) n′hb(θ)f j + (θ)

(7)

where n′hb(θ) is the number of hydrogen bonds that A can form with other water molecules except for B, and (θ) is the hydrogen bond energy between A and B. When B is in this perpendicular state, A can either form 3 hydrogen bonds with water molecules other than B, or form 2 hydrogen bonds with other water molecules and 1 hydrogen bond with B. This yields the partition function bulk water A when B is in the perpendicular state Bp

q0,p ) 3

π/3-φ dθ exp(- (2fj + (θ))/2kT) + ∫-π/3+φ c

c

6φc exp(-3f j/2kT) (8) The total partition function for a bulk water molecule A is the weighted sum of the partition functions over the tangential and perpendicular states:

q0 ) G(gq0,t + q0,p)

(9)

where g is the relative density of states of tangential vs perpendicular states of water B,

g)

(2π - 6φc) 6φc

∆F ) -nkT ln[((θ2 exp(-f j/kT) + θ3 exp(-3f j/2kT))/G)/ (gθ2 exp(-f j/kT) + (1 + g)θ3 exp(-3f j/2kT)] + 3

π/3-φ dθ exp(-(2f j + (θ))/2kT))] ∫-π/3+φ c

(11)

c

Because the first shell water molecules form a hexagonal cage around a water-sized solute molecule having diameter lhb, we use n ) 6. To evaluate this very simple expression, we need values for the threshold angle φc, the hydrogen bond energy (θ), the bulk hydrogen-bonding fraction, f(T), and the average energy of a hydrogen bond j(T). To estimate the temperature-dependent hydrogen bonding probability, f(T), we use Muller’s simple 2-state model.27 We divide the distribution of hydrogen bonding into two states, formed or broken:

H bond (broken) S H bond (formed)

(12)

The equilibrium constant is given by Khb ) f/(1 - f). Its temperature dependence is found from -kT ln Khb ) ∆h T∆s, where ∆h is the enthalpy of hydrogen bond formation and ∆s is the corresponding entropy. So, we have

f(T) )

1 1 + exp(∆h/kT) exp(-∆s/k)

(13)

We obtain ∆h and ∆s from a previous Monte Carlo study of the fraction of broken hydrogen bonds in neat liquid of MB water;34 see Table 1. It was previously found that this 2-state approximation gives an accurate fit of Monte Carlo results, even though the latter involve a continuum of hydrogen bonding states, not just two. For the hydrogen bond energy, (θ), we assume a squarelaw potential,

(θ) ) min (m + ksθ2,0)

(14)

where m < 0 is the hydrogen bond energy for an optimal hydrogen bond and ks is the spring constant. As a matter of finding the best way to compare our model results to previous Monte Carlo simulations of the MB model, we choose a value of ks that minimizes, over the angles for which m + ksθ2 e0, the mean standard deviation between our quadratic potential and the full Gaussian potential used in the Monte Carlo simulations. The threshold angle φc is chosen so that the cutoff hydrogen bond energy is 1/3 of the maximum value, i.e., (φc) ) m/3. For our quadratic potential, that means

φc ) (-2m/3ks)1/2

(15)

Since the hydrogen bonding energy in our model is quadratic, equipartition applies, so

(10)

and G is the relative density of states between the water in bulk and the water in the solvation shell. G will change with different solutes, but we assume that it is independent of temperature. We choose G ) 0.363 based on best fits to the Monte Carlo results of T∆S in nonpolar solvation. The heat capacity ∆C does not depend on this parameter. Combining the individual-molecule partition functions for the solvation shell (eq 5) and the bulk (eq 9) gives the free energy of transferring a solute into water

j(T) ) m + kT/2

(16)

Table 1 gives the numerical values of our parameters. The only free fitting parameter is G; other quantities are obtained from the Monte Carlo simulation of the MB model of neat water, as indicated above. Since G does not depend on temperature, our predictions for the heat capacity, ∆C ) T∂∆S/∂T, are independent of this parameter. Figure 4 compares the predictions of our model with experiments and with previous Monte Carlo simulations of the MB model. Figure 4 shows that the model captures well the

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TABLE 1: Model Parametersa ks/|m|

φc

g

G

∆h/|m|

∆s/k

n

2.39

0.528

0.983

0.363

-0.710

-3.001

6

a

ks: the force constant for the quadratic angle-dependent potential of a hydrogen bond. φc: the cutoff angle of a hydrogen bond. g: the relative density of states between the tangential state and the perpendicular state of a neighboring water molecule. G: the relative density of states between a water molecule in bulk and a water molecule in the solvation shell. ∆h and ∆s: the enthalpy and the entropy of the hydrogen bond reaction. n: the number of water molecules in the first solvation shell.

Figure 5. (a) Increasing temperature increases the population of perpendicular waters around the solute, and decreases the population of tangential waters. p(θi) is the probability of being in orientation θi; Ωt ) 2π-6φc and Ωp ) 6φc are the angle intervals of the tangential state and the perpendicular state. (b) Loss of orientational entropy ∆So ) - k[(2π - 6φc)p(θt) ln p(θt) + 6φcp(θp) ln p(θp)] as a function of the temperature.

Figure 4. Free energy ∆F (0), the enthalpy ∆H (4), and the entropy T ∆S (]) of nonpolar solvation. (a) Experiments on the transfer of argon from the gas phase into liquid water at different temperatures T. (b) Transfer of a circular nonpolar disk of diameter lhb into MercedesBenz water. The results of Monte Carlo simulations on the MB model are shown by the symbols, and the solid lines are calculated using the present theoretical model. (c) Heat capacity ∆C ) T∂∆S/∂Tfor the same process. The points are the results of Monte Carlo simulations, and the solid line is the result of the theoretical model. The anomalous peak in heat capacity at kT/|m| ≈ 0.2 is not observed experimentally. We conjecture that it is an artifact of the two-dimensional model, as it is present in both the Monte Carlo simulations and our analytical model.

temperature dependence of the free energy, entropy, energy, and heat capacity of nonpolar solvation. The Temperature Dependence In our model, the temperature dependence of the hydrophobic effect is a consequence of two properties of hydrogen bonds. First, increasing the temperature breaks hydrogen bonds, both in the bulk and the first shell; the fraction (1 - f) increases. Second, hydrogen bonds weaken with increasing temperature (i.e., the average energy, j(T), of a formed hydrogen bond increases; eq 16). As the temperature increases, both factors lead to freer rotations of first-shell waters around nonpolar solutes, and thus to increased orientational entropy. The large

positive heat capacity of nonpolar solvation is a consequence of this increase in energy and entropy with temperature. Figure 5a shows the probability distributions of the orientations of a water molecule in the solvation shell at three different temperatures. As the temperature increases, solvation-shell waters prefer tangential orientations less; they increasingly point their hydrogen bonding arms toward the solute. This is accompanied by increased orientational entropy, as shown in Figure 5b. Our model resolves an interesting puzzle. The experimental data in Figure 4a show that, in hot water, the entropy of nonpolar solvation becomes positiVe. Guillot et al. have investigated this phenomenon using molecular dynamics simulations with SPC/E water model.36 Monte Carlo simulations of the MB model give the same result. That is, inserting a solute into hot water increases the entropy, in contrast to the well-known result in cold water that solvation is opposed by entropy. How can we understand the solvent disordering that the solute induces in hot water? It means that solvation-shell waters must be more disordered than bulk water molecules. The entropy of solvation, ∆S, arises from the difference in freedom of the test water molecule adjacent to a solute vs adjacent to another water. In cold water, the test water molecule is more constrained by a neighboring solute than by a neighboring water molecule. In hot water, a test water molecule must be more constrained by a neighboring water molecule than by a neighboring solute. Our model readily explains this result. Cold water is dominated by the energy-lowering drive to make hydrogen bonds; hot water is dominated by the entropy-increasing drive to break them. In cold water, an adjacent solute molecule restricts the freedom of the test water in order to make hydrogen bonds to other waters. In hot water, an adjacent water molecule restricts the freedom of the test water to break its hydrogen bonds. Figure 6 shows the temperature dependence of the orientational entropy of a water molecule A when a neighboring water molecule B is fixed and pointing one arm toward A. This shows that B restricts the orientations of A with increasing temperature.

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J. Phys. Chem. B, Vol. 109, No. 49, 2005 23615

Figure 6. Orientational entropy of a water molecule A, when B is fixed and pointing a hydrogen bonding arm toward A (see Figure 3c). This conditional entropy decreases with temperature, indicating that neighboring waters are restrictive in hot water.

Figure 8. Dependence of solvation thermodynamic quantities on the size of the solute. All thermodynamic quantities, ∆F, ∆H (∆U), and T∆S, are normalized by the surface area (perimeter) of the cavity, A ) π(lhb + 2R). (a) Results from Monte Carlo simulations using particle insertion methods. (b) Predictions from the present analytical model. The parameters in the model are the same as listed in Table 1, except for n, which is now a function of the solute size as given in eq 19.

collision with the solute, so that the arm points at it within angle φc. θ1, θ2, and θ3 are simple functions of γ and therefore of solute radius R. They are given by the following: (1) If γ e π/3 Figure 7. Definition of the angle γ, which is used in defining θ1, θ2, and θ3 for treating the size dependence (see eqs 17 and 18).

Our model may be extended to understand the sign of the entropy in a common class of solvation processes, where the solute weakly interacts with the solvent, but the solvent strongly interacts with each other. The strong interactions between the solvent molecules restrict the freedom of each other and lower the entropy. The introduction of the solute usually disrupts this interaction, and consequently liberates the solvent molecules from each other’s restriction. Therefore, the entropy of such solvation process is usually positive. However, when the strong interaction between the solvents is directional, the solvent can accommodate the solute without breaking their interactions by adopting preferential orientations, which leads to negative entropy, as observed in hydrophobic solvation. At high temperatures, the directional interactions are diminished, and solvation incurs positive entropy as in the usual case. Solute Size Effects So far, we have treated only a small solute, of the size of a water molecule. MB model simulations have shown that increasing the radius of a spherical solute changes the thermodynamics of nonpolar solvation.35 The present model can account for these effects. In our model, this size dependence comes from the angles θ1, θ2, and θ3 (eq 4), and from the number of water molecules in the first solvation shell (eq 1). Consider the angle γ ) arccos(lhb/(lhb + 2R)), where R is the radius of the spherical nonpolar solute. This is the angle at which a hydrogen bonding arm will point along a tangent to the solute (Figure 7). In our model, a hydrogen bonding arm is exposed to the solvent, thus capable of forming hydrogen bonds, if its orientation falls outside the interval [-(γ - φc),γ - φc]. Outside this interval, another water molecule can be placed free of steric

()(

θ1 0 θ2 ) 6(γ - φc) θ3 2π - 6(γ - φc)

(2) If π/3 < γ < π/2

()(

θ1 6(γ - φc) - 2π θ2 ) 4π - 6(γ - φc) θ3 0

)

)

(17)

(18)

The number of water molecules in the first solvation shell is proportional to the surface area of the spherical cavity

n(R) ) π/arcsin(lhb/(lhb + 2R))

(19)

Using these simple geometric relationships, we calculated the size dependence of the hydrophobic thermodynamic quantities and compared them to the Monte Carlo simulation results for MB water35 (Figure 8). This model captures the trend of the size dependence and predicts the sign reversal in energy ∆U and entropy ∆S as the solute radius increases. Hydrogen Bonding vs Small Solvent Size in Hydrophobicity Water’s ability to form hydrogen bonds and its small size have both been indicated to be the primary cause of hydrophobicity,1,37 and much debate has ensued to decide which is the more important. We suggest that depending on the concerned thermodynamic quantity of hydrophobic solvation, the two properties of water contribute differently. In most cases, it is even impossible to separate the two. For instance, in many

23616 J. Phys. Chem. B, Vol. 109, No. 49, 2005 successful phenomenological theories of hydrophobicity,10,12,13 the radial distribution function of liquid water is used as the input of the model, which then implicitly contains contributions from both hydrogen bonds and small water size. To the lowest order, the entropy of solvation can be decomposed into two parts: the translational entropy from the density fluctuations around the solute, and the orientational entropy from the orientational ordering near the solute. Despite its omission of the translational part, our model, based on hydrogen bonding, well captures the temperature dependence of the entropy. As a result, we propose that although both water’s hydrogen bonding ability and its small size contribute to the entropy of hydrophobic solvation, its hydrogen bonding ability accounts primarily for the entropy’s temperature dependence. Because of enthalpy-entropy compensation,1,38 the free energy ∆G is insensitive to the temperature. This suggests that a model devoid of water’s hydrogen bonds may be capable of reproducing the solvation free energies at various temperatures. Similarly, the entropy loss due to orientational restrictions may be compensated by the enthalpy gain due to stronger hydrogen bonds for water in the solvation shell. When this cancellation holds, the excluded volume effect will successfully account for the solvation free energies of various nonpolar solutes. Our model, however, suggests that, for solutes of different sizes, there are still sizable contributions from hydrogen bonds to the total free energies (see Figure 8). This contribution may be particularly significant for solute with complicated surfaces, such as biomolecules, where perturbations to water’s hydrogen bonds will strongly depend on the solute’s surface topography. We recognize that the omission of the translational entropy in our model may yield quantitatively inaccurate results, as reflected in Figure 4c and more so in Figure 8. In a separate study, however, we find that the translational part is strongly correlated with the orientational part, so that the trends in the solvation thermodynamics can be captured by our model without additional parameters. Inclusion of the translational entropy in our model will imply a readjustment of the only fitting parameter, G, to produce good quantitative agreement. Nevertheless, the most significant conclusion from our models that water’s hydrogen bonding propensity gives rise to the temperature dependence of the hydrophobic effectsremains unchanged. Summary We have developed a simple analytical model of the hydrophobic effect. Our model explicitly treats water hydrogen bonds and orientations. The free energy of solvation is found by focusing on a test water molecule in the first solvation shell; it “sees” a solute molecule that has replaced a water molecule. The heat capacity of hydrophobic solvation is found to be large in this model because increasing temperature breaks hydrogen bonds and increases the orientational entropy of the first-shell waters, consistent with the conventional “iceberg” picture,29 but not predicting the high degrees of ordering in clathrate structures. The present model also gives an explanation for the positive entropy of solvation at high temperatures in experiments. Compared to a neighboring nonpolar solute, a neighboring water molecule provides extra hydrogen bonding opportunities for a test water molecule. In cold water, it means extra orientational freedom for the test molecule. In hot water, it reduces the chance to break hydrogen bonds to gain orientational entropy.

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