ANNUAL REVIEWS
Annu.
1990. 19:301-32 1990 by Annual Reviews Inc. All rights reserved
Rev. Biophys. Biophys. Chern.
Copyright ©
Further
Quick links to online content
ELECTROSTATIC Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
INTERACTIONS IN MACROMOLECULES: Theory and Applications Kim A. Sharp and Barry Honig
Department of Biochemistry and Molecular Biophysics, Columbia University, 630 W. 1 68th Street, New York, NY 1 0032 KEY WORDS:
finite
difference
Poisson-Boltzmann,
protein structure, DNA structure.
macromolecular
theory,
CONTENTS
PERSPECTIVE AND OVERVIEW..........................................................................................
301
The Poisson Equation ... . ...... ................... . ...... .. . .. .... . . . . .. ..... . ... . . ...... . ....... . . . ... .... . . . ... . .. Electronic Polarizability Permanent Dipoles .... . .. . . .. . . . ....... . . ..... ....... . . .. . .. . . . . . ............... .... ...... . ... . .. . . . .. .. ............. Mobile Ions ... ...... . . ................. . ....... . . .. ....... . ................... ......... ... ... . . . . . . ..... . ...... . ... .. . .. .
303 303 304 307 309
DIELECTRIC THEORY .... . .... . ...... . .. . .... . ... . .. ............. ....... ................ ... .. . ... ......... ............ . ....
...........................................................................................
METHODS.......................................................................................................................
The Protein Dipole Langevin Dipole Method............................................................ Analytical Solutions t o the Poisson-Boltzmann Equation ...... . ............. ......... .......... ... Numerical Solutions to the Poisson-Boltzmann Equation .......... . ............................... The Finite Difference Poisson-Boltzmann Method ........ ......... .... . ..... . ..... . ... ........ .. . . . ..
APPLICATIONS ............................................ ...................................................................
Electrical Potentials around Macromolecules............................................................ Solvation Energies .. . .......... . . .... . . ......... . . . ...... ...... . . ... .. . . . . .......................... . ................ Catalysis .. ... ..... .. . . . . . ... .. . ... . ... . ..... .. . .. .............. .... . . ............ .. . .. ..... .. ... .... . .................... Acid-Base Equilibria................................................................................................. Redox Potentials ........... . . . . .. .... . .... . . . . .. .......... ........ ........ . ...... .. ........ . . . ...... . .... .... . . .. ... . . Forces in DynamiCS Simulations ........ .. ................... .. .......... .. . .. . . . ....................... . . . .. . . Electrostatic Energies and Stability: Folding, Denaturation, and Binding .. .... . . ..... .... SUMMARY AND PROSPECTS .. .... ................ . . . . . . . . .. . . . . . .. ..... . ........ ..... .... ................. .. . . ...... . .
.
.
. .
310 312 312 3 13 3 14 317 317 319
321
322 324 326 327 329
PERSPECTIVE AND OVERVIEW A recent Annual Review describes a performance at the court of Louis XV
in which an electric shock was administered simultaneously to 700 monks 301 0883-9 1 82/90/061 0-0301$02.00
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
302
SHARP & HONIG
who then jumped in unison to the great amusement of the onlookers (67a) . The point of the story, in the context of the article, was that "electrostatics was not always unfashionable." Given the recent spate of reviews on electrostatics in biological systems (39, 47, 63, 64, 67, 9 1 , 94, 1 30), one might argue instead that the subject has again become fashionable to the point of overexposure. What then justifies another review? Our major motivation has been to describe recent advances in applying the equations and concepts of classical electrostatics to current research areas in molecular biophysics. The fundamentals of classical electrostatics like those of thermodynamics, are well known and can be stated concisely with a few elegant equations. The apparent simplicity of these equations, however, can hide the difficulties of applying them to complex systems. The problem is particularly acute in studies of proteins and nucleic acids owing to the vast amount of structural information about these macro molecules now available. In contrast to traditional models in which pro teins were treated as low dielectric spheres and DNA as a charged cylinder, most current questions of interest are asked at the atomic level. The question of how best to represent atomic and molecular properties within the framework of electrostatic theory poses new conceptual as well as numerical difficulties. It is not uncommon to encounter the opinion that models based on classical electrostatics have been superseded, or even invalidated, by the advent of computer simulations of atomic motions. A criticism sometimes expressed is that classical electrostatics is not valid on a microscopic scale. In fact, excluding quantum mechanical phenomena classical electrostatics is valid on all scales of biological interest. Of course, the theory must be applied in a physically meaningful way to the system being studied. We will demonstrate that this is generally possible and that classical electrostatics provides a rigorous and intuitively satisfying approach to a wide range of microscopic phenomena. The first section gives an overview of the relevant electrostatic theory and discusses the physical origin of each contribution to the dielectric response of solutions of biological macromolecules. We describe the most common representations of the relevant physical processes emphasizing the similarities and differences between them. The second section dis cusses specific theoretical and computational methods that have been de veloped to calculate the electrostatic properties of macromolecular systems. The last section reviews a range of biological problems in which electro statics plays a central role. The framework developed in the earlier sec tions is used to identify the relevant electrostatic quantities for a given bio physical phenomenon and then to outline how these parameters may be calculated. ,
,
303
ELECTROSTATICS IN MACROMOLECULES
DIELECTRIC THEORY The Poisson Equation
The fundamental equation of electrostatics is the Poisson equation:
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
V2¢(r) = -4np(r),
1.
¢
r
which relates spatial variation of the potential with position to the charge density distribution p, where the permittivity of free space is unity. When the charge distribution can be described in terms of a set of point charges, the solution to the Poisson equation becomes Coulomb's law:
qj ¢(r) = � Ir-r jl' where rj is the position, and q the magnitude of the ith point charge.
2.
j
Essentially all electrostatic models used in studying macromolecules are based on the Poisson equation. Models begin to diverge from one another when choices are made as to the method of applying Equation I to a particular problem. If all charges, whatever their origin, are represented explicitly, all interactions can be viewed as taking place in free space and Coulomb's law can be rigorously applied. If this is not practical, or if the details of specific i nteractions are not of interest, the Poisson equation can be rewritten in more convenient forms. More specifically, if the charge distribution that generates the potential is present in a complex medium, it is possible at times to use spatial averages to account for the response of the medium to the fields generated by the charge distribution. If a region of some material responds in an average, or smeared out, way to the electric field, i .e. it has a uniform susceptibility, X, the polar ization density (induced dipole moment/unit volume) is given by: P XE, where E is the average field in that region. If the entire medium has a uniform susceptibility, thc potentials and fields are screened by a constant factor, the dielectric constant or permittivity: E 4nX + 1 . In this case, the Poisson and Coulomb equations can be written in the forms =
=
V2¢(r)
-4np(r) =
E
3.
and
¢(r)
qj
=
j E r-rj I-I -I'
4.
respectively. If the dielectric varies through space, then Coulomb's law is invalid, while the Poisson equation becomes
304
SHARP & HONIG
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
V' c(r)V¢(r) = -4np(r),
5.
where c is now a function of the position r. Generally, the Poisson equation can be applied in many ways to a particular problem. However, unless a full quantum mechanical treatment is used, there is essentially no way of avoiding averaging an environmental response over some region of space (which might correspond, for example, to a single atom or to an entire protein) . Indeed, Equation 5 says nothing about whether a particular treatment is macroscopic or microscopic. The definition of these terms depends solely on the spatial resolution at which the dielectric response is modelled. The environmental response can broadly be divided into three physical processes that screen the effects of charge: (a) electronic polarization; (b) reorientation of permanent dipoles, i.e. in polar materials; and (c) redistribution of charges, such as the rearrangement of mobile ions in ionic solutions to form electric double layers. Electronic Polarizability
Electronic polarizability describes the reorientation of the electronic cloud around a nucleus in the presence of an electric field. Until recently, elec tronic polarizability has usually been neglected in potential energy force fields used in molecular mechanics simulations because the effects cannot be easily reduced to a set of two-body interactions. For example, if a charge on a particular atom polarizes the electrons on neighboring atoms, those electron clouds will also polarize one another, leading to a complex many-body interaction. There are a number of ways to account for electronic polarizability. The next sections describes three of these. When matter is exposed to radiation of high enough frequency that nuclear reorientation cannot follow the elec tric field, the dielectric response is determined almost entirely by electronic polarization. The high frequency dielectric constant Ceo approximately equals the square of the index of refraction, which is 2 for most polar and nonpolar organic liquids. One way to account for electronic polarizability is to incorporate its effects into a dielectric constant-to assume that all charges and permanent dipoles interact with one another as if they were embedded in a medium that has a dielectric constant of 2. Classical theories of the dielectric constant of polar liquids routinely make this assumption, while treating permanent dipoles explicitly through Equation 3 or 4 (24,
THE UNIFORM DIELECTRIC MODEL
82). A central question concerning the use of a dielectric constant over a
region of space involving many atoms asks if using a single, spatially
ELECTROSTATICS IN MACROMOLECULES
305
invariant parameter that ignores the atomic nature of matter is valid. This problem will be considered after two microscopic models are presented. The most common means of representing electronic polarizability at the molecular level assigns point inducible dipoles (PIDs) to atoms, bonds, or groups (2, 1 4, 1 28) . In the simplest case, the induced dipole moment P is presumed to be linearly related to the field by an isotropic polarizability cc P etE. The fields arising from a charge or a dipole are
INDUCED DIPOLES
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
=
E(q)
=
qr/r3
and
E(p)
=
:!Tp,
6.
respectively, where :!T (3rr r2 J)/r5 is the dipole field tensor. For a collection of charges and PIDs, the field depends on the charges and dipole moments, while each induced dipole moment in turn depends on the field it experiences from the charges and all other dipoles. This leads to a set of simultaneous linear implicit equations for the dipole moments: =
Pi = (XiL [E(q)ij+:!TijPj), Hi
-
7.
where the subscripts i and j run over all the charges/dipoles. This matrix equation can be solved analytically only for the two-body case because, as pointed out above, electronic polarization involves a many-body inter action that cannot be decomposed into a sum of pairwise interactions. Generally, an iterative procedure is used in which an initial estimate for the dipole moments is substituted into the right side of Equation 7, and the left side yields an improved estimate of the dipole moments. This procedure is repeated until a self-consistent set of fields and dipole moments results (2, 1 28) . The PID model usually assumes that an atom has a uniform polar izability that can be represented by an induced dipole placed at the nucleus. Two difficulties with this model are: (a) atomic polarizabilities taken from experiment or theory on isolated atoms are not necessarily accurate for atoms in molecules (7); and (b) nearby inducible dipoles can mutually increase each other's polarization without limit causing a polarization catastrophe. The ad-hoc exclusion of interactions between neighboring atoms has been used to circumvent this problem (97, 1 2 1 , 1 30) . The relationship between the atomic polarizability and the bulk dielec tric behavior of a material has been the subject of many theoretical studies, which date back over a century to the Clausius-Mossotti relationship for nonpolar materials: x
=
Net (I -4nNct/3),
8.
306
SHARP & HONIG
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
where N is the number density of the polarizable bodies. Although this expression was originally derived for material with low values of N, such as dilute gases, it also holds for denser nonpolar material (2, 14, 1 30) . LOCAL DIELECTRIC CONSTANTS An alternate way of representing the electronic polarizability treats atoms or groups of atoms as polarizable bodies, each with its own local dielectric constant (LDC) (83a, 83b, 1 04, K. Sharp & B. Honig, unpublished work) . Figure I schematically illustrates the relationship between the uniform dielectri c, LDC, and PID rep resentations. The LDC model effectively distributes the dielectric response over the van der Waals volume occupied by the atom's electrons. This model makes fewer approximations than the other two models, since it assumes neither that th e response is uniform throughout space nor that the response arises from infinitesimal dipoles. In the simplest form of the LDC model, each atom is represented as a sphere of constant dielectric, ej. The equivalent point polarizability in the PID model, (Xi, would be ( 1 4) IlC
=
3 V(ej- l ) , 4n(ej+2)
-..,.------c-:
a) Uniform Dielectric
9.
b) Local Dielectric
Indue ible Dipoles
c)
qi-""i
/V
Figure
1
=
(E-1)E/4lt
qje ""j
c=l
E i (1+8mx)V i) (1-4mx iVi) =
Schematic diagram illustrating three different models for the molecular response
to electric fields: (LDC),
(a) Uniform Dielectric Constant (UDC), (b) Local Dielectric Constant (c) Point Inducible Dipoles (PID). For models a and b, the mean induced dipole
density per unit volume at any point
Per) is given by ()l(r) > IV (e(r)-l]E(r)/4n, where B(r) the dielectric constant. In the PID model, the =
E(r) is the Maxwell field at that point and
induced dipole moment at a point i is Pi
=
((iEi, where ((i is the point polarizability. The LDC
model bridges the PID and UDC models through the relation
((i
=
3 Vi(c,-1)/4n(6i+ 2),
relating a point polarizability to the local dielectric constant Gi assigned to a spherical volume Vi around that point.
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
ELECTROSTATICS IN MACROMOLECULES
307
where V is the volume of the sphere. The LDC and PID models are equivalent for two special cases: when the atom is in a homogeneous medium or when it is exposed to a uniform field (K. Sharp and B. Honig, unpublished results) . In general, however, the polarizability response involves higher-order terms than dipoles, and on atomic dimensions the errors in the PID approximation can be quite large ( 1 4, 83a). The LDC model may also be extended to use nonuniform and anisotropic dielectric distributions for each atom (83b) . The LDC and PID models shown in Figure I appear microscopic, while the uniform dielectric model appears macroscopic. As is evident from Figure 1, however, the uniform dielectric and the LDC models differ in the absence of cavities between the atoms in the former and the assumption of the same dielectric constant for each atom. Since atoms in proteins and nucleic acids are closely packed and are neither spherical nor static, it is not unreasonable to consider them as filling space. Moreover, the high frequency dielectric constant of organic liquids depends only weakly on the identity of the solvent molecule. Thus, the use of a single dielectric constant to account for the electronic polarization response of an entire macromolecule appears to be a very reasonable approximation. It should be emphasized that it is not clear which of the three models is actually most appropriate for applications to biological systems. The PID and LDC models are truly microscopic, but they are numerically complex and the PID model in particular entails a number of questionable assump tions. Moreover, both require knowledge of polarizabilities for a large number of atoms in different molecules and thus involve a significant number of parameters. On the other hand, the uniform dielectric model uses a well-known experimental quantity, Boo for organic liquids, to account for electronic polarizability. It should be informative to compare the pre dictions of all three models for different test cases. Permanent Dipoles
The dielectric properties of polar liquids arise primarily from the reori entation of permanent dipoles in an electrical field. For a macromolecule in aqueous solution, it is necessary to account for the permanent dipoles of both the molecule and the solvent. As in electronic polarizability, it is possible in principle to use one or more dielectric constants to describe the response of both the solute and solvent. If a single dielectric constant is used in conjunction with Coulomb's law (Equation 4), one is implicitly assuming that solvent and macromolecule have the same dielectric constant. Water has a dielectric constant of 80 at room temperature, while experimental and theoretical evidence suggest that proteins have an average dielectric response that can be approximated with a dielectric
308
SHARP & HONIG
constant of about 4 (30, 39, 74, 1 1 4) . Thus, the system cannot be viewed as uniform and at least two dielectric constants must be used. We now consider how the dielectric response of permanent dipoles can be modelled. The atoms and molecules in a system are standardly represented in detail with potential energy functions in which electrostatic interactions are described by Coulomb's law for charges, or related expressions for higher multipoles. Because all interactions are presumed to be treated explicitly, they can be assumed to occur in free space, so that Equation 2 is valid. If the potential functions are accurate and account for all relevant interactions, when combined with a method to account for the dynamics of the system (i.e. Monte-Carlo or molecular dynamics), they should reliably reproduce electrostatic phenomena. The dielectric properties of macromolecules are difficult to determine experimentally, so the extent to which all-atom models succeed in repro ducing the molecules' behavior is impossible to assess at this time. However, more is known experimentally about the dielectric properties of water. These arise primarily from interactions between permanent dipoles, and include hydrogen-bonding effects. The hydrogen bonds are reasonably well represented with pairwise potential functions such as those of the various water models currently in use [e.g. TTP4P (49a) or SPC (1 1 a) ] . Simulations based on these models produce reasonable bulk dielectric constants for water [50 for TIP4P, about 70 for SPC (5, 7 5)]. However, there remain formidable problems of computational expense, potential function parameterization, convergence behavior, cutoff, and limitations of the current pairwise point potential formalisms (39) . The long-range nature of electrostatic interactions compounds the difficulty. Moreover, water models that account for electronic polarization are in their earliest stages of development and introduce significant additional computational requiremen ts. The problem is exacerbated when simulating waters around macro molecules. In this case, the size of the macromolecule limits the number of water layers that can be included in a simulation. This limitation and the use of boundary conditions whose effects are unclear introduce con siderable uncertainty about the dielectric response of waters that appear in the simulations, even if the potential functions produce reasonable dielectric constants for bulk water.
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
EXPLICIT ATOM REPRESENTATIONS
STATISTICAL MECHANICS REPRESENTATIONS The Langevin model The Langevin equation relates the mean orientation e of a freely rotating dipole to the magnitude of the directing field Ed:
1 0.
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
ELECTROSTATICS IN MACROMOLECULES
309
This equation represents the balance between the polarizing effect of the field and the randomizing effect of thermal fluctuations, where T is the absolute temperature and k is Boltzmann's constant. Onsager (82) used this model to calculate bulk dielectric behavior. In principle, the Langevin model is applicable only to freely rotating polar molecules, i .e. those whose only interactions are electrostatic. Moreover, because it ignores correlated motion between solvent molecules, it cannot account for the dielectric constant of hydrogen bonding liquids without some type of correction ( 1 4), nor does it properly represent the effect of dielectric saturation or electrostriction around ions (48 ) . However, by suitable parameterization it has been used to represent even a highly associated liquid like water ( 1 30) . Kirkwood-Frohlich theory Many detailed statistical-mechanical treat ments of orientational polarization of polar dielectrics are based on the Kirkwood-Frohlich theory (28, 52). Here, the material is considered as a collection of permanent dipoles embedded in a continuum with a high frequency dielectric constant of eoo. The effective dipole moment of each dipole thus increases from the gas phase value due to polarization by its reaction field. Thc thcory treats a spherical region of material in molecular detail by statistical mechanics, representing the rest of the material as a continuum: 9kT(e-eoo)(2e+eoo) 4nNs(soo +2)2
=
2
gf-l ,
11.
where c is the bulk dielectric constant, N is the dipole number density, f-l is the gas phase dipole moment, and 9 is the Kirkwood correlation factor that accounts for cooperative effects orienting neighboring dipoles. Kirk wood-Frohlich analysis has been used to theoretically estimate the dielec tric constant of a protein. Assuming a hypothetical oc-helical protein of infinite extent (to avoid boundary problems) , Gilson & Honig (30) esti mated a value between 3 and 4. Nakamura & Nishida (73) extended this analysis by calculating a dielectric constant as a function of position in the protein Pancreatic Trypsin Inhibitor (PTI) . Numbers ranging between 3 and 4 were also found for the protein itself. Although Nakamura & Nishida (73) obtained much higher effective dielectric constants when they averaged the dielectric constant of water into the values assigned to atoms on the protein surface, the essential conclusion in both studies is that proteins, when viewed as pure materials, have a low dielectric constant. Mobile Ions THE POISSON-BOLTZMANN EQUATION In this mean-field approach, mobile ions in ionic solvents are not represented explicitly. Instead the chemical
3lO
SHARP & HONIG
potential of each ion is assumed to be uniform throughout solution. The entropic and electrostatic contributions to the chemical potential of an ion at any point rare kTln C(r) and q
Annu.
1990. 19:301-32 1990 by Annual Reviews Inc. All rights reserved
Rev. Biophys. Biophys. Chern.
Copyright ©
Further
Quick links to online content
ELECTROSTATIC Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
INTERACTIONS IN MACROMOLECULES: Theory and Applications Kim A. Sharp and Barry Honig
Department of Biochemistry and Molecular Biophysics, Columbia University, 630 W. 1 68th Street, New York, NY 1 0032 KEY WORDS:
finite
difference
Poisson-Boltzmann,
protein structure, DNA structure.
macromolecular
theory,
CONTENTS
PERSPECTIVE AND OVERVIEW..........................................................................................
301
The Poisson Equation ... . ...... ................... . ...... .. . .. .... . . . . .. ..... . ... . . ...... . ....... . . . ... .... . . . ... . .. Electronic Polarizability Permanent Dipoles .... . .. . . .. . . . ....... . . ..... ....... . . .. . .. . . . . . ............... .... ...... . ... . .. . . . .. .. ............. Mobile Ions ... ...... . . ................. . ....... . . .. ....... . ................... ......... ... ... . . . . . . ..... . ...... . ... .. . .. .
303 303 304 307 309
DIELECTRIC THEORY .... . .... . ...... . .. . .... . ... . .. ............. ....... ................ ... .. . ... ......... ............ . ....
...........................................................................................
METHODS.......................................................................................................................
The Protein Dipole Langevin Dipole Method............................................................ Analytical Solutions t o the Poisson-Boltzmann Equation ...... . ............. ......... .......... ... Numerical Solutions to the Poisson-Boltzmann Equation .......... . ............................... The Finite Difference Poisson-Boltzmann Method ........ ......... .... . ..... . ..... . ... ........ .. . . . ..
APPLICATIONS ............................................ ...................................................................
Electrical Potentials around Macromolecules............................................................ Solvation Energies .. . .......... . . .... . . ......... . . . ...... ...... . . ... .. . . . . .......................... . ................ Catalysis .. ... ..... .. . . . . . ... .. . ... . ... . ..... .. . .. .............. .... . . ............ .. . .. ..... .. ... .... . .................... Acid-Base Equilibria................................................................................................. Redox Potentials ........... . . . . .. .... . .... . . . . .. .......... ........ ........ . ...... .. ........ . . . ...... . .... .... . . .. ... . . Forces in DynamiCS Simulations ........ .. ................... .. .......... .. . .. . . . ....................... . . . .. . . Electrostatic Energies and Stability: Folding, Denaturation, and Binding .. .... . . ..... .... SUMMARY AND PROSPECTS .. .... ................ . . . . . . . . .. . . . . . .. ..... . ........ ..... .... ................. .. . . ...... . .
.
.
. .
310 312 312 3 13 3 14 317 317 319
321
322 324 326 327 329
PERSPECTIVE AND OVERVIEW A recent Annual Review describes a performance at the court of Louis XV
in which an electric shock was administered simultaneously to 700 monks 301 0883-9 1 82/90/061 0-0301$02.00
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
302
SHARP & HONIG
who then jumped in unison to the great amusement of the onlookers (67a) . The point of the story, in the context of the article, was that "electrostatics was not always unfashionable." Given the recent spate of reviews on electrostatics in biological systems (39, 47, 63, 64, 67, 9 1 , 94, 1 30), one might argue instead that the subject has again become fashionable to the point of overexposure. What then justifies another review? Our major motivation has been to describe recent advances in applying the equations and concepts of classical electrostatics to current research areas in molecular biophysics. The fundamentals of classical electrostatics like those of thermodynamics, are well known and can be stated concisely with a few elegant equations. The apparent simplicity of these equations, however, can hide the difficulties of applying them to complex systems. The problem is particularly acute in studies of proteins and nucleic acids owing to the vast amount of structural information about these macro molecules now available. In contrast to traditional models in which pro teins were treated as low dielectric spheres and DNA as a charged cylinder, most current questions of interest are asked at the atomic level. The question of how best to represent atomic and molecular properties within the framework of electrostatic theory poses new conceptual as well as numerical difficulties. It is not uncommon to encounter the opinion that models based on classical electrostatics have been superseded, or even invalidated, by the advent of computer simulations of atomic motions. A criticism sometimes expressed is that classical electrostatics is not valid on a microscopic scale. In fact, excluding quantum mechanical phenomena classical electrostatics is valid on all scales of biological interest. Of course, the theory must be applied in a physically meaningful way to the system being studied. We will demonstrate that this is generally possible and that classical electrostatics provides a rigorous and intuitively satisfying approach to a wide range of microscopic phenomena. The first section gives an overview of the relevant electrostatic theory and discusses the physical origin of each contribution to the dielectric response of solutions of biological macromolecules. We describe the most common representations of the relevant physical processes emphasizing the similarities and differences between them. The second section dis cusses specific theoretical and computational methods that have been de veloped to calculate the electrostatic properties of macromolecular systems. The last section reviews a range of biological problems in which electro statics plays a central role. The framework developed in the earlier sec tions is used to identify the relevant electrostatic quantities for a given bio physical phenomenon and then to outline how these parameters may be calculated. ,
,
303
ELECTROSTATICS IN MACROMOLECULES
DIELECTRIC THEORY The Poisson Equation
The fundamental equation of electrostatics is the Poisson equation:
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
V2¢(r) = -4np(r),
1.
¢
r
which relates spatial variation of the potential with position to the charge density distribution p, where the permittivity of free space is unity. When the charge distribution can be described in terms of a set of point charges, the solution to the Poisson equation becomes Coulomb's law:
qj ¢(r) = � Ir-r jl' where rj is the position, and q the magnitude of the ith point charge.
2.
j
Essentially all electrostatic models used in studying macromolecules are based on the Poisson equation. Models begin to diverge from one another when choices are made as to the method of applying Equation I to a particular problem. If all charges, whatever their origin, are represented explicitly, all interactions can be viewed as taking place in free space and Coulomb's law can be rigorously applied. If this is not practical, or if the details of specific i nteractions are not of interest, the Poisson equation can be rewritten in more convenient forms. More specifically, if the charge distribution that generates the potential is present in a complex medium, it is possible at times to use spatial averages to account for the response of the medium to the fields generated by the charge distribution. If a region of some material responds in an average, or smeared out, way to the electric field, i .e. it has a uniform susceptibility, X, the polar ization density (induced dipole moment/unit volume) is given by: P XE, where E is the average field in that region. If the entire medium has a uniform susceptibility, thc potentials and fields are screened by a constant factor, the dielectric constant or permittivity: E 4nX + 1 . In this case, the Poisson and Coulomb equations can be written in the forms =
=
V2¢(r)
-4np(r) =
E
3.
and
¢(r)
qj
=
j E r-rj I-I -I'
4.
respectively. If the dielectric varies through space, then Coulomb's law is invalid, while the Poisson equation becomes
304
SHARP & HONIG
Annu. Rev. Biophys. Biophys. Chem. 1990.19:301-332. Downloaded from www.annualreviews.org by University of Sussex on 03/12/13. For personal use only.
V' c(r)V¢(r) = -4np(r),
5.
where c is now a function of the position r. Generally, the Poisson equation can be applied in many ways to a particular problem. However, unless a full quantum mechanical treatment is used, there is essentially no way of avoiding averaging an environmental response over some region of space (which might correspond, for example, to a single atom or to an entire protein) . Indeed, Equation 5 says nothing about whether a particular treatment is macroscopic or microscopic. The definition of these terms depends solely on the spatial resolution at which the dielectric response is modelled. The environmental response can broadly be divided into three physical processes that screen the effects of charge: (a) electronic polarization; (b) reorientation of permanent dipoles, i.e. in polar materials; and (c) redistribution of charges, such as the rearrangement of mobile ions in ionic solutions to form electric double layers. Electronic Polarizability
Electronic polarizability describes the reorientation of the electronic cloud around a nucleus in the presence of an electric field. Until recently, elec tronic polarizability has usually been neglected in potential energy force fields used in molecular mechanics simulations because the effects cannot be easily reduced to a set of two-body interactions. For example, if a charge on a particular atom polarizes the electrons on neighboring atoms, those electron clouds will also polarize one another, leading to a complex many-body interaction. There are a number of ways to account for electronic polarizability. The next sections describes three of these. When matter is exposed to radiation of high enough frequency that nuclear reorientation cannot follow the elec tric field, the dielectric response is determined almost entirely by electronic polarization. The high frequency dielectric constant Ceo approximately equals the square of the index of refraction, which is 2 for most polar and nonpolar organic liquids. One way to account for electronic polarizability is to incorporate its effects into a dielectric constant-to assume that all charges and permanent dipoles interact with one another as if they were embedded in a medium that has a dielectric constant of 2. Classical theories of the dielectric constant of polar liquids routinely make this assumption, while treating permanent dipoles explicitly through Equation 3 or 4 (24,
THE UNIFORM DIELECTRIC MODEL
82). A central question concerning the use of a dielectric constant over a
region of space involving many atoms asks if using a single, spatially
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305
invariant parameter that ignores the atomic nature of matter is valid. This problem will be considered after two microscopic models are presented. The most common means of representing electronic polarizability at the molecular level assigns point inducible dipoles (PIDs) to atoms, bonds, or groups (2, 1 4, 1 28) . In the simplest case, the induced dipole moment P is presumed to be linearly related to the field by an isotropic polarizability cc P etE. The fields arising from a charge or a dipole are
INDUCED DIPOLES
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=
E(q)
=
qr/r3
and
E(p)
=
:!Tp,
6.
respectively, where :!T (3rr r2 J)/r5 is the dipole field tensor. For a collection of charges and PIDs, the field depends on the charges and dipole moments, while each induced dipole moment in turn depends on the field it experiences from the charges and all other dipoles. This leads to a set of simultaneous linear implicit equations for the dipole moments: =
Pi = (XiL [E(q)ij+:!TijPj), Hi
-
7.
where the subscripts i and j run over all the charges/dipoles. This matrix equation can be solved analytically only for the two-body case because, as pointed out above, electronic polarization involves a many-body inter action that cannot be decomposed into a sum of pairwise interactions. Generally, an iterative procedure is used in which an initial estimate for the dipole moments is substituted into the right side of Equation 7, and the left side yields an improved estimate of the dipole moments. This procedure is repeated until a self-consistent set of fields and dipole moments results (2, 1 28) . The PID model usually assumes that an atom has a uniform polar izability that can be represented by an induced dipole placed at the nucleus. Two difficulties with this model are: (a) atomic polarizabilities taken from experiment or theory on isolated atoms are not necessarily accurate for atoms in molecules (7); and (b) nearby inducible dipoles can mutually increase each other's polarization without limit causing a polarization catastrophe. The ad-hoc exclusion of interactions between neighboring atoms has been used to circumvent this problem (97, 1 2 1 , 1 30) . The relationship between the atomic polarizability and the bulk dielec tric behavior of a material has been the subject of many theoretical studies, which date back over a century to the Clausius-Mossotti relationship for nonpolar materials: x
=
Net (I -4nNct/3),
8.
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SHARP & HONIG
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where N is the number density of the polarizable bodies. Although this expression was originally derived for material with low values of N, such as dilute gases, it also holds for denser nonpolar material (2, 14, 1 30) . LOCAL DIELECTRIC CONSTANTS An alternate way of representing the electronic polarizability treats atoms or groups of atoms as polarizable bodies, each with its own local dielectric constant (LDC) (83a, 83b, 1 04, K. Sharp & B. Honig, unpublished work) . Figure I schematically illustrates the relationship between the uniform dielectri c, LDC, and PID rep resentations. The LDC model effectively distributes the dielectric response over the van der Waals volume occupied by the atom's electrons. This model makes fewer approximations than the other two models, since it assumes neither that th e response is uniform throughout space nor that the response arises from infinitesimal dipoles. In the simplest form of the LDC model, each atom is represented as a sphere of constant dielectric, ej. The equivalent point polarizability in the PID model, (Xi, would be ( 1 4) IlC
=
3 V(ej- l ) , 4n(ej+2)
-..,.------c-:
a) Uniform Dielectric
9.
b) Local Dielectric
Indue ible Dipoles
c)
qi-""i
/V
Figure
1
=
(E-1)E/4lt
qje ""j
c=l
E i (1+8mx)V i) (1-4mx iVi) =
Schematic diagram illustrating three different models for the molecular response
to electric fields: (LDC),
(a) Uniform Dielectric Constant (UDC), (b) Local Dielectric Constant (c) Point Inducible Dipoles (PID). For models a and b, the mean induced dipole
density per unit volume at any point
Per) is given by ()l(r) > IV (e(r)-l]E(r)/4n, where B(r) the dielectric constant. In the PID model, the =
E(r) is the Maxwell field at that point and
induced dipole moment at a point i is Pi
=
((iEi, where ((i is the point polarizability. The LDC
model bridges the PID and UDC models through the relation
((i
=
3 Vi(c,-1)/4n(6i+ 2),
relating a point polarizability to the local dielectric constant Gi assigned to a spherical volume Vi around that point.
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ELECTROSTATICS IN MACROMOLECULES
307
where V is the volume of the sphere. The LDC and PID models are equivalent for two special cases: when the atom is in a homogeneous medium or when it is exposed to a uniform field (K. Sharp and B. Honig, unpublished results) . In general, however, the polarizability response involves higher-order terms than dipoles, and on atomic dimensions the errors in the PID approximation can be quite large ( 1 4, 83a). The LDC model may also be extended to use nonuniform and anisotropic dielectric distributions for each atom (83b) . The LDC and PID models shown in Figure I appear microscopic, while the uniform dielectric model appears macroscopic. As is evident from Figure 1, however, the uniform dielectric and the LDC models differ in the absence of cavities between the atoms in the former and the assumption of the same dielectric constant for each atom. Since atoms in proteins and nucleic acids are closely packed and are neither spherical nor static, it is not unreasonable to consider them as filling space. Moreover, the high frequency dielectric constant of organic liquids depends only weakly on the identity of the solvent molecule. Thus, the use of a single dielectric constant to account for the electronic polarization response of an entire macromolecule appears to be a very reasonable approximation. It should be emphasized that it is not clear which of the three models is actually most appropriate for applications to biological systems. The PID and LDC models are truly microscopic, but they are numerically complex and the PID model in particular entails a number of questionable assump tions. Moreover, both require knowledge of polarizabilities for a large number of atoms in different molecules and thus involve a significant number of parameters. On the other hand, the uniform dielectric model uses a well-known experimental quantity, Boo for organic liquids, to account for electronic polarizability. It should be informative to compare the pre dictions of all three models for different test cases. Permanent Dipoles
The dielectric properties of polar liquids arise primarily from the reori entation of permanent dipoles in an electrical field. For a macromolecule in aqueous solution, it is necessary to account for the permanent dipoles of both the molecule and the solvent. As in electronic polarizability, it is possible in principle to use one or more dielectric constants to describe the response of both the solute and solvent. If a single dielectric constant is used in conjunction with Coulomb's law (Equation 4), one is implicitly assuming that solvent and macromolecule have the same dielectric constant. Water has a dielectric constant of 80 at room temperature, while experimental and theoretical evidence suggest that proteins have an average dielectric response that can be approximated with a dielectric
308
SHARP & HONIG
constant of about 4 (30, 39, 74, 1 1 4) . Thus, the system cannot be viewed as uniform and at least two dielectric constants must be used. We now consider how the dielectric response of permanent dipoles can be modelled. The atoms and molecules in a system are standardly represented in detail with potential energy functions in which electrostatic interactions are described by Coulomb's law for charges, or related expressions for higher multipoles. Because all interactions are presumed to be treated explicitly, they can be assumed to occur in free space, so that Equation 2 is valid. If the potential functions are accurate and account for all relevant interactions, when combined with a method to account for the dynamics of the system (i.e. Monte-Carlo or molecular dynamics), they should reliably reproduce electrostatic phenomena. The dielectric properties of macromolecules are difficult to determine experimentally, so the extent to which all-atom models succeed in repro ducing the molecules' behavior is impossible to assess at this time. However, more is known experimentally about the dielectric properties of water. These arise primarily from interactions between permanent dipoles, and include hydrogen-bonding effects. The hydrogen bonds are reasonably well represented with pairwise potential functions such as those of the various water models currently in use [e.g. TTP4P (49a) or SPC (1 1 a) ] . Simulations based on these models produce reasonable bulk dielectric constants for water [50 for TIP4P, about 70 for SPC (5, 7 5)]. However, there remain formidable problems of computational expense, potential function parameterization, convergence behavior, cutoff, and limitations of the current pairwise point potential formalisms (39) . The long-range nature of electrostatic interactions compounds the difficulty. Moreover, water models that account for electronic polarization are in their earliest stages of development and introduce significant additional computational requiremen ts. The problem is exacerbated when simulating waters around macro molecules. In this case, the size of the macromolecule limits the number of water layers that can be included in a simulation. This limitation and the use of boundary conditions whose effects are unclear introduce con siderable uncertainty about the dielectric response of waters that appear in the simulations, even if the potential functions produce reasonable dielectric constants for bulk water.
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EXPLICIT ATOM REPRESENTATIONS
STATISTICAL MECHANICS REPRESENTATIONS The Langevin model The Langevin equation relates the mean orientation e of a freely rotating dipole to the magnitude of the directing field Ed:
1 0.
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ELECTROSTATICS IN MACROMOLECULES
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This equation represents the balance between the polarizing effect of the field and the randomizing effect of thermal fluctuations, where T is the absolute temperature and k is Boltzmann's constant. Onsager (82) used this model to calculate bulk dielectric behavior. In principle, the Langevin model is applicable only to freely rotating polar molecules, i .e. those whose only interactions are electrostatic. Moreover, because it ignores correlated motion between solvent molecules, it cannot account for the dielectric constant of hydrogen bonding liquids without some type of correction ( 1 4), nor does it properly represent the effect of dielectric saturation or electrostriction around ions (48 ) . However, by suitable parameterization it has been used to represent even a highly associated liquid like water ( 1 30) . Kirkwood-Frohlich theory Many detailed statistical-mechanical treat ments of orientational polarization of polar dielectrics are based on the Kirkwood-Frohlich theory (28, 52). Here, the material is considered as a collection of permanent dipoles embedded in a continuum with a high frequency dielectric constant of eoo. The effective dipole moment of each dipole thus increases from the gas phase value due to polarization by its reaction field. Thc thcory treats a spherical region of material in molecular detail by statistical mechanics, representing the rest of the material as a continuum: 9kT(e-eoo)(2e+eoo) 4nNs(soo +2)2
=
2
gf-l ,
11.
where c is the bulk dielectric constant, N is the dipole number density, f-l is the gas phase dipole moment, and 9 is the Kirkwood correlation factor that accounts for cooperative effects orienting neighboring dipoles. Kirk wood-Frohlich analysis has been used to theoretically estimate the dielec tric constant of a protein. Assuming a hypothetical oc-helical protein of infinite extent (to avoid boundary problems) , Gilson & Honig (30) esti mated a value between 3 and 4. Nakamura & Nishida (73) extended this analysis by calculating a dielectric constant as a function of position in the protein Pancreatic Trypsin Inhibitor (PTI) . Numbers ranging between 3 and 4 were also found for the protein itself. Although Nakamura & Nishida (73) obtained much higher effective dielectric constants when they averaged the dielectric constant of water into the values assigned to atoms on the protein surface, the essential conclusion in both studies is that proteins, when viewed as pure materials, have a low dielectric constant. Mobile Ions THE POISSON-BOLTZMANN EQUATION In this mean-field approach, mobile ions in ionic solvents are not represented explicitly. Instead the chemical
3lO
SHARP & HONIG
potential of each ion is assumed to be uniform throughout solution. The entropic and electrostatic contributions to the chemical potential of an ion at any point rare kTln C(r) and q
