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2008, 112, 10791–10794 Published on Web 08/12/2008
Reformulation of Maxwell’s Equations to Incorporate Near-Solute Solvent Structure Pei-Kun Yang† and Carmay Lim*,†,‡ Institute of Biomedical Sciences, Academia Sinica, Taipei 115, Taiwan R.O.C., and National Tsing Hua UniVersity, Hsinchu 300, Taiwan R.O.C ReceiVed: March 27, 2008
Maxwell’s equations, which treat electromagnetic interactions between macroscopic charged objects in materials, have explained many phenomena and contributed to many applications in our lives. Derived in 1861 when no methods were available to determine the atomic structure of macromolecules, Maxwell’s equations assume the solvent to be a structureless continuum. However, near-solute solvent molecules are highly structured, unlike far-solute bulk solvent molecules. Current methods cannot treat both the near-solute solvent structure and time-dependent electromagnetic interactions in a macroscopic system. Here, we derive “microscopic” electrodynamics equations that can treat macroscopic time-dependent electromagnetic field problems like Maxwell’s equations and reproduce the solvent molecular and dipole density distributions observed in molecular dynamics simulations. These equations greatly reduce computational expense by not having to include explicit solvent molecules, yet they treat the solvent electrostatic and van der Waals effects more accurately than continuum models. They provide a foundation to study electromagnetic interactions between molecules in a macroscopic system that are ubiquitous in biology, bioelectromagnetism, and nanotechnology. The general strategy presented herein to incorporate the near-solute solvent structure would enable studies on how complex cellular protein-ligand interactions are affected by electromagnetic radiation, which could help to prevent harmful electromagnetic spectra or find potential therapeutic applications. Introduction Electromagnetic (EM) interactions comprise one of the four fundamental interactions in nature.1,2 In the gas phase where polarized/magnetized molecules are few, Gauss’s law, Ampe`re’s law with Maxwell’s correction, and Faraday’s law of induction (Table 1) can be used to compute the EM interactions between charged molecules directly. In materials, treating the numerous polarized/magnetized molecules as sources to evaluate the EM field is currently impossible. Consequently, Maxwell treated the electromagnetic effect of discrete polarized/magnetized molecules as an uniform distribution of electric/magnetic dipole moments, p/m, and assumed the induced electric/magnetic dipole moment per unit volume, P/M, to be proportional to the electric/ magnetic field, E/B. By space-averaging E and B over a region containing many thousands of atoms,1 he derived a set of equations (Table 1) that describe the EM interactions between two widely separated molecules in materials.1,2 These equations govern classical macroscopic electrodynamics. Maxwell’s equations, however, cannot treat EM interactions between two closely interacting molecules, for example, a solute and a nearby solvent molecule; such interactions are ubiquitous in nanotechnology,3 bioelectromagnetism,4,5 and biological systems.6,7 This is mainly because in Maxwell’s equations, the solvent density F is assumed to be constant, equal to the bulk solvent density Fbulk, an assumption that is valid only in the far-solute region.1,2 However, F near a solute differs dramatically * To whom correspondence should be addressed. E-mail: carmay@ gate.sinica.edu.tw. † Academia Sinica. ‡ National Tsing Hua University.
10.1021/jp805705m CCC: $40.75
from Fbulk (Figure 1a, compare dashed and gray curves); thus, the macroscopic solvent electric dipole density P averaged over a continuum solvent cannot replace the EM effects stemming from structured solvent molecules near a solute. Yet, continuum models are widely used to compute hydration free energies of molecules.8,9 With advances in observational tools, a major concern is how to incorporate the EM effects of the near-solute solvent molecules and thus accurately treat the EM interactions between molecules in a macroscopic system. Although several methods such as quantum mechanics, molecular dynamics (MD), and Monte Carlo simulation can capture the microscopic features of the near-solute solvent molecules,10,11 they cannot treat time-dependent EM fields and all of the atoms in a macroscopic system with Avogadro’s number of solvent molecules. Thus, most simulations employ a continuum model to treat solvent molecules outside of the explicit solvent region.12-14 However, such simulations cannot accurately treat the explicit-implicit solvent interface and are inefficient at studying problems involving large conformational changes owing to the numerous explicit water molecules needed to hydrate the flexible protein. Notably, they cannot treat timedependent EM fields, which are not considered in the conventional explicit-implicit solvent boundary12 and periodic boundaries;15 moreover, simulation times exceeding the EM wave period are needed. Hence, current methods cannot accurately/ efficiently treat time-dependent EM interactions between molecules in a macroscopic system or relate the time-dependent electric field to the magnetic field in the near-solute region. Consequently, they cannot address intriguing questions such as whether EM radiation from mobile phones, wireless networks, 2008 American Chemical Society
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TABLE 1: Macroscopic versus “Microscopic” Electrodynamics Equationsa Maxwell’s equations Gauss’s law Faraday’s law of induction Ampe`re-Maxwell’s law electric displacement field, D magnetic field H electric dipole density, P magnetic dipole density, M solvent molecular density distribution, g Lorentz force, F
“microscopic” electrodynamics equations
∇•D ) Ffree ∇•B ) 0 ∇ × E ) -∂B/∂t ∇ × H ) J + ∂D/∂t D ) ε0E + P H ) B/µ0 - M P ) ε0χeE M ) χmB/µ0 g)1 F ) q(E + W × B)
D ) ε0E - ∫(∇′•P′)∇(1/4πR)d3r′ P ) ε0χegE 3g/g ) F/kBT F ) q(E + W × B) + FvdW
a P′: perturbed electric dipole density; q: charge of the particle; W: velocity of the particle; χe: electric susceptibility; χm: magnetic susceptibility; ε0: vacuum permittivity; µ0: vacuum magnetic permeability; Ffree: free electric charge density.
solution from “microscopic” Gauss’s law reflects the near-solute solvent structure and the oscillatory behavior of F and solvent P observed in MD simulations (Figure 1). Theory The near-solute structured solvent molecules can be distinguished from the far-solute bulk solvent molecules by the solvent molecular and orientational distribution. The solvent molecular distribution g, defined as F/Fbulk, can be computed from the mean force 〈F〉 acting on a solvent molecule according to Kirkwood’s equation16
∇g/g ) 〈F 〉 /kBT ) (〈FvdW 〉 + 〈Fele 〉 )/kBT
Figure 1. Comparison of the solvent molecular density F/Fbulk and the solvent dipole density P from “microscopic” and macroscopic Gauss’s law and MD simulations. The F/Fbulk (a) and 4πR2P (b) as a function of distance R from a charged solute atom were derived from “microscopic” Gauss’s law without explicit solvent using the p-vdWsphere solvent model (black solid curve), macroscopic Gauss’s law (gray line), and MD simulations (black dashed curve).
power transmission lines, or microwave ovens is harmful to humans, whether such EM radiation alters protein-ligand cellular networks, and if so, whether EM radiation of a specific frequency and amplitude affects binding between a protein and its ligand. Because Maxwell’s equations using a continuum model cannot account for the electrostatic effects of near-solute solvent molecules while quantum mechanics and molecular simulations cannot handle macroscopic systems and time-dependent EM fields, our aim is to derive a set of “microscopic” electrodynamics equations that can capture the near-solute F and solvent P as in molecular dynamic simulations but can treat timedependent EM interactions and macroscopic systems just like Maxwell’s equations. We first present a general strategy to incorporate the near-solute solvent structure. We hypothesize that the near-solute F is not equal to Fbulk, as assumed in Maxwell’s equations, largely because of excluded solvent volume effects, that is, each solvent molecule occupies a volume that excludes other solvent molecules. The latter is incorporated by treating each solvent molecule as a perfect dipole moment p at the center of a vdW sphere instead of a point dipole as in Maxwell’s equations. On the basis of this p-vdW-sphere solvent model, we show that the F and solvent P can be computed from the mean force 〈F〉 and electric field E acting on a solvent molecule. We then derive the microscopic equivalent of Maxwell’s equations in solution (Table 1) and show that the
(1)
where kB is Boltzmann’s constant, T is the absolute temperature, while 〈Fvdw〉 and 〈Fele〉 are the mean vdW and electric force components, respectively (see Supporting Information). The solvent orientational distribution can be described by timeaveraging the solvent P over a microscopic rather than a macroscopic region so that P ) Fp ) Fbulk(F/Fbulk)p ) Fbulkgp. Assuming that p is proportional to the E acting on the solvent molecule2 such that p ) ε0γeE, where ε0 is the vacuum permittivity and γe is the solvent molecular polarizability, the solvent P can be computed from
P ) Fbulkgp ) Fbulkg(ε0γeE) ) ε0χegE
(2)
where the “microscopic” electric susceptibility is ) Fbulkγe. To evaluate the solvent P, we need to compute g according to eq 1 and the E acting on the solvent molecule.17 Since a solvent molecule at r will exclude other solvent molecules from being located at r and perturb the distribution of the surrounding solvent molecules, E can be decomposed into contributions from the solute charge density Ffree(r′) and the perturbed solvent dipole density P′(r′;r) due to the solvent dipole at r, that is χe
E(r) ) -
∫ [Ffree(r′) - ∇ ′•P′(r′;r)] ∇ (1/4πε0R)d3r′ (3a)
R is the distance between the solvent cavity center at r and the solute charge or solvent dipole at r′. The perturbed solvent P′(r′;r) can be written as Fbulkg′(r′;r)p(r′;r), where g′(r′;r) and p(r′;r) are, respectively, the perturbed solvent molecular distribution and perturbed solvent dipole at r′ due to the solvent dipole at r. The perturbed g′(r′;r) can be approximated by the unperturbed g(r′) scaled by a cavity function C(R): g′(r′;r) ≈ C(R)g(r′).10 For the p-vdWsphere solvent model, the cavity function is estimated by a Heaviside/unit step function, C(R) ) u(R - Rc), where Rc is the solvent cavity radius defined by the solvent-accessible surface (as opposed to the cavity defined by the solvent-excluded or molecular surface). Assuming that the solvent dipole at r
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negligibly polarizes the surrounding solvent molecules so that p(r′;r) ∼ p(r′), the perturbed solvent P′(r′;r) can be computed from the cavity function C(R) and the corresponding unperturbed P(r′) since P′(r′;r) ) FbulkC(R)g(r′)p(r′) ) u(R - Rc)P(r′). Hence, eq 3a can rewritten as
E(r) ) -
∫ [Ffree(r′) - ∇ ′•(u(R - Rc)P(r′))] ∇ (1/4πε0R)d3r′ (3b)
“Microscopic” Gauss’s Law is obtained by taking the divergence of E (eq 3a) and defining the “microscopic” electric displacement D such that its divergence is the solute charge density, 3•D ) Ffree. Whereas D ) ε0E + P in “macroscopic” Gauss’s Law, it is related to E in “microscopic” Gauss’s Law by
D(r) ) ε0E(r) -
∫ [∇′•P′(r, r′)] ∇ (1/4πR)d3r′
(4)
“Microscopic” Ampe`re-Maxwell’s equation is obtained by replacing the macroscopic D in Ampe`re-Maxwell’s equation (∇ × H ) J + ∂D/∂t) with the “microscopic” D in eq 4. It relates the “microscopic” magnetic field H to the total electric current J and the “microscopic” D, which, in turn, is related to the solvent molecular distribution g from eqs 2-4. The differences between the macroscopic and “microscopic” electrodynamics equations are summarized in Table 1. The former equation neglects the solvent structure (g ) 1) and yields a macroscopic E averaged over many thousands of solvent molecules, while the latter equation takes into account the nearsolute solvent structure (variable g) and yields a “microscopic” E acting on a single solvent molecule. Results and Discussion To verify that the solution from “microscopic” Gauss’s law can capture the near-solute solvent structure, we compared the g and solvent P from a MD simulation of a monocation in explicit water with those obtained from “microscopic” Gauss’s law using an iterative procedure, as described in the Appendix. Figure 1 shows that the solution from “microscopic” Gauss’s law can indeed yield the near-solute water structure. Both g and 4πR2P as a function of distance from the solute exhibit an oscillatory behavior near the solute, similar to those derived from the MD simulation employing explicit water. To verify that this qualitative agreement is not fortuitous, we have derived approximate analytical solutions of g and P for a spherical solute (Yang, P.-K.; Lim, C. manuscript in preparation). They explain why g and 4πR2P derived from the microscopic electrodynamics equations exhibit an oscillatory behavior near the solute and show how the amplitude and wavelength of the microscopic solvent P depends on the solvent cavity radius Rc. In sharp contrast, the solution from “macroscopic” Gauss’s law yields a constant g(R) and 4πR2Pmacro(R) (Figure 1, gray curve). Differences between the amplitude and wavelength of the g and 4πR2P curves computed from “microscopic” Gauss’s law and those derived from the MD simulation are expected. They are due to (a) the one-site p-vdW-sphere solvent model used in “microscopic” Gauss’s law versus the three-site TIP3P solvent model used in the MD simulation and (b) the approximations made in computing the perturbed solvent P′(r′;r). Because the microscopic and macroscopic solvent P distributions differ in the near-solute region (Figure 1b, compare solid black and gray curves), the microscopic D and H would also likely differ from their macroscopic counterparts.
Conclusion Whereas Gauss’s law, Ampe`re-Maxwell’s law, and Faraday’s law govern the EM interactions between charged molecules in vacuum, while Maxwell’s equations treat the EM interactions between macroscopic objects in materials, the equations derived herein can deal with EM and vdW interactions between molecular solutes in a macroscopic system containing Avogadro’s number of solvent molecules. Notably, the equations derived are independent of the nature of the solute, that is, whether it is a single spherical atom or a multiatom nonspherical molecule. Our strategy to incorporate the near-solute solvent structure into the “microscopic” electrodynamics equations is general. Rather than treating each atom of a solvent molecule as in MD simulations, the entire solvent molecule is treated as a unit, and the distribution of solvent molecules (g) and their orientations at given positions (P) are used to describe the distribution of the solvent atoms. Instead of averaging all of the solvent molecules over a macroscopic region, as in Maxwell’s equations, solvent molecules whose centers remain in the same position for a period of time are averaged over a microscopic region. Consequently, the near-solute solvent structure is taken into account via g and P, which are evaluated, respectively, by the mean force and electric field acting on the solvent molecule. Although any solvent model can be used to compute g and P, the p-vdW-sphere solvent model was used to illustrate the importance of excluded solvent volume effects. One of the applications of “microscopic” Gauss’s law is to provide an analytical solution for the hydration free energy of a spherical solute (Yang, P.-K.; Lim, C. J. Phys. Chem. B., submitted) or a numerical solution for the solvation free energy of a nonspherical solute. This is analogous to using “macroscopic” Gauss’s law to derive the Born free energy for a spherical ion or converting it to the Poisson equation, which can then be solved by finite difference methods or surface boundary element techniques.8,9 Future studies will be focused on developing methods to obtain a numerical solution to “microscopic” Gauss’s law for a multiatom solute and evaluating the accuracy and speed of this method as compared to MD simulations in the presence of explicit water and current continuum dielectric methods. With the rapid development of nanotechnology, concern over EM radiation effects to humans, and the inefficiency/inadequacy of current methods to study time-dependent EM interaction between molecular solutes in a macroscopic system, the strategy presented herein to derive “microscopic” electrodynamics equations appears timely and fills an important gap. MD Simulations. The g and P were computed as the timeaveraged number of water oxygen atoms and the water dipole moment, respectively, in the annular region from r to r + dr divided by the respective volume from an 8 ns MD simulation of an atom of charge q ) +1e solvated by 2185 TIP3P water molecules18 in a 25 Å sphere using the CHARMM program.19 The TIP3P ε (0.1521 kcal/mol) and Rmin (1.7682 Å) values for the water oxygen were also used for the solute atom. Details of the simulation are given in Supporting Information. Macroscopic Gauss’s Law. For a solute atom with radius RBorn immersed in water characterized by a dielectric constant εr ) 80, g ) 1, and 4πR2P ) q(1 - 1/εr) u(R - RBorn).1 “Microscopic” Gauss’s Law. An iterative strategy exploiting the radial symmetry of the spherical solute was used to obtain the g and P (Figure 2). Using an initial guess of g(0) ) 1 and P(0) ) q(1 - 1/εr)/(4πR2) from macroscopic Gauss’s law, the E acting on the solvent molecule at r was evaluated using eq 3b with Rc ) 2.8 Å. Knowing E(1) and g(0), the mean electric
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Letters Supporting Information Available: Details of (1) the simulation protocol, (2) how the solvent molecular distribution is computed from the mean vdW and electrostatic forces acting on the solvent molecule, and (3) how the “microscopic” electric susceptibility of water is estimated. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes
Figure 2. Flowchart showing how g(r) and P(r) can be evaluated, respectively, from the mean force and the E acting on the solvent molecule at r.
F(1)ele and vdW F(1)vdW force components acting on the solvent molecule at r can be evaluated using eqs S2 and S3, respectively, as described in Supporting Information. The sum of F(1)vdW and F(1)ele was used in eq 1 to yield a new estimate of g(1), which was used together with E(1) to compute a new estimate of P(1) from eq 2. With the new g(1) and P(1) values, the above procedure was repeated until the g and P values converged. This took on the order of 103 iterations. For a nonspherical solute, such an iterative strategy would be inefficient, and other methods are needed to obtain converged g and P values, which is not the focus herein. Hence, to verify if the solution from “microscopic” Gauss’s law can capture the near-solute solvent structure, we employed a spherical solute and a simple iterative strategy. Acknowledgment. We thank M. Karplus for the CHARMM program. This work was supported by Academia Sinica and the National Science Council, Taiwan.
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