STABILITY OF A CYLINDRICAL SOLUTE-SOLVENT INTERFACE: EFFECT OF GEOMETRY, ELECTROSTATICS, AND HYDRODYNAMICS

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1 SIAM J. APPL. MATH. Vol. 75, No. 3, pp c 215 Society for Industrial and Applied Mathematics STABILITY OF A CYLINDRICAL SOLUTE-SOLVENT INTERFACE: EFFECT OF GEOMETRY, ELECTROSTATICS, AND HYDRODYNAMICS BO LI, HUI SUN, AND SHENGGAO ZHOU Abstract. The solute-solvent interface that separates biological molecules from their surrounding aqueous solvent characterizes the conformation and dynamics of such molecules. In this work, we construct a solvent fluid dielectric boundary model for the solvation of charged molecules and apply it to study the stability of a model cylindrical solute-solvent interface. The motion of the solute-solvent interface is defined to be the same as that of solvent fluid at the interface. The solvent fluid is assumed to be incompressible and is described by the Stokes equation. The solute is modeled simply by the ideal-gas law. The viscous force, hydrostatic pressure, solute-solvent van der Waals interaction, surface tension, and electrostatic force are all balanced at the solute-solvent interface. We model the electrostatics by Poisson s equation in which the solute-solvent interface is treated as a dielectric boundary that separates the low-dielectric solute from the high-dielectric solvent. For a cylindrical geometry, we find multiple cylindrically shaped equilibrium interfaces that describe polymodal (e.g., dry and wet) states of hydration of an underlying molecular system. These steady-state solutions exhibit bifurcation behavior with respect to the charge density. For their linearized systems, we use the projection method to solve the fluid equation and find the dispersion relation. Our asymptotic analysis shows that, for large wavenumbers, the decay rate is proportional to the wavenumber whose proportionality is half of the ratio of surface tension to solvent viscosity, indicating that the solvent viscosity does affect the stability of a solute-solvent interface. Consequences of our analysis in the context of biomolecular interactions are discussed. Key words. solute-solvent interfaces, surface energy, electrostatic interactions, dielectric boundary force, solvent hydrodynamics, linear stability, projection method AMS subject classifications. 35B35, 76Z99, 92C5, 92C4 DOI / Introduction. The stability of a solute-solvent interface that separates biomolecules from their surrounding aqueous solvent is crucial to the conformation, dynamics, and function of an underlying biomolecular system. Such a solute-solvent interface is often treated as a dielectric boundary as the dielectric coefficient of a biomolecular region can be as low as 1 4 while that of water can be as high as 8 under normal conditions 25. This boundary is the key quantity in a large class of implicit or continuum solvent models 17, 21, 22, 31, 4 that describe efficiently the effect of aqueous solvent on biomolecular processes, such as protein folding and molecular recognition 21, 27. The geometry of a solute-solvent interface is set largely by the van der Waals (vdw) short-range repulsion between solute particles and solvent molecules, describing the effect of an excluded volume of solute particles. But the interfacial surface energy, which measures the energetic cost of a solute breaking the surrounding network of water molecules, plays an important role in the hydrophobic interaction 5, 9, 3. Received by the editors June 9, 214; accepted for publication (in revised form) February 1, 215; published electronically May 5, 215. This work was supported by the US National Science Foundation (NSF) through grant DMS , the Center for Theoretical Biological Physics through NSF grant PHY , and the National Institutes of Health through grant R1GM Department of Mathematics and NSF Center for Theoretical Biological Physics, University of California, San Diego, 95 Gilman Drive, Mail code: 112, La Jolla, CA (bli@math.ucsd. edu, hus3@math.ucsd.edu, s4zhou@math.ucsd.edu). 97

2 98 BO LI, HUI SUN, AND SHENGGAO ZHOU x Ω w Ω m Γ:r = R(z) n L z y Solute Solvent Fig. 1. A cylindrical solute-solvent interface Γ separates the interior solute region Ω m (m stands for molecules) from the exterior solvent region Ω w (w stands for water). The unit normal n at Γ points from Ω m to Ω w. The system is L-periodic in the z-axis, the axis of the cylinder. Due to the large jump of dielectric coefficient across the dielectric boundary, the electrostatic interaction between biomolecular charges, polarized solvent, and mobile ions generates a strong effective force on the dielectric boundary. It should be noted that electrostatic interactions in biomolecular systems can be efficiently modeled by Poisson s or the Poisson Boltzmann equation 3, 6, 1, 18, 28, 35, 36. Experimental and theoretical studies have also indicated that the solvent shear motion can induce protein conformational changes and the solvent viscosity can affect the kinetics of such changes 1, 24, 26, 32, 33, 37, 38, 39, 41. In this work, we study how these vdw interactions, interfacial surface energy, electrostatics, and solvent fluid motion contribute to the stability of a model cylindrical solute-solvent interface; cf. Figure 1. Such an interface resembles that of a long α-helix, RNA, or DNA. Our model is adapted from that proposed in our previous work 43 (cf. also 44), and consists of the following main elements: (1) The motion of the dielectric boundary Γ(t) (t denotes time) is defined by (1.1) V n = u n, where V n is the normal velocity, u is the velocity field of solvent fluid, and n is the unit normal at the boundary Γ(t) pointing from the solute region Ω m to the solvent region Ω w. (2) The solvent fluid is assumed to be incompressible, and is described by the Stokes equation (1.2) u = and μ w Δu p w =, where μ w and p w are the dynamic viscosity and pressure of the solvent fluid. The pressure p m = p m (t) of solute molecules is assumed to be spatially constant and described simply by the ideal-gas law (1.3) p m (t)vol(ω m (t)) = C m, where Vol (Ω m (t)) is the volume of solute region, and C m is a constant independent of spatial and temporal variables but dependent on the temperature and amount of gas which are assumed to be constant. (3) The electrostatic potential φ solves a boundary-value problem of Poisson s equation (1.4) ε Γ φ = ρ

3 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 99 defined in the entire region R 3. Here, ρ is a given, fixed charge density, and ε Γ is the dielectric coefficient defined by { εm = ε mε in Ω m (t), (1.5) ε Γ = ε w = ε w ε in Ω w (t), where ε is the vacuum permittivity, and ε m and ε w are the dielectric coefficients (i.e., relative permittivities) of the solute and solvent, respectively. These are positive constants and satisfy, in general, ε m <ε w. The electrostatic potential determines the normal component of effective dielectric boundary force 7, 8, 14, 29 (1.6) f ele = 1 ( 1 1 ) ε Γ φ n ε w ε m 2 (ε m ε w ) (I n n) φ 2, where I is the 3 3 identity matrix. Note that the electrostatic force always points in the direction from the high-dielectric solvent to low-dielectric solute. (4) At the solute-solvent interface Γ(t), all the viscous force, hydrostatic pressure, surface tension, solute-solvent vdw interaction force, and electrostatic force are balanced: (1.7) 2μ w D(u)n +(p m p w 2γ H + n w U vdw + f ele ) n =, where D(u) =( u +( u) T )/2 is the strain rate tensor, γ is the constant surface tension, H is the mean curvature, n w is the constant number density of solvent molecules, and U vdw is a solute-solvent vdw interaction potential. Typical values of the parameters, such as μ w, n w,andγ, are given in section 4. The magnitude of these forces (per unit area) can be of the order.1 1 k B T/Å with k B the Boltzmann constant and T absolute temperature. We find multiple cylindrically shaped equilibrium interfaces that describe polymodal (e.g., dry and wet) states of hydration of an underlying molecular system 5, 9, 3. For instance, a larger equilibrium cylinder is relatively drier as water molecules are excluded further away from the center line of the cylinder. These steady states exhibit bifurcation behavior with respect to the charge density. We linearize our system around such equilibrium interfaces, and solve the resulting linearized system by a fluid projection method together with special functions for the electrostatic potential. We seek the solutions to the linearized system in the form A k e ωt+ikz, where for any given mode k, A k is a constant, and a negative ω = ω(k) implies the decay of the initial perturbation A k e ikz, indicating the linear stability. Let us denote by R the radius of an underlying cylindrically shaped equilibrium interface. Our calculations lead to the dispersion relation ω(k) = ω air(k)+ω surf (k)+ω vdw (k)+ω ele (k) ω hyd = γ (1.8) k + O(1) as k, 2μ w where ω air (k), ω surf (k), ω vdw (k), ω ele (k), and ω hyd are individual contributions from the solute air pressure (i.e., the term p m ), surface energy, vdw interaction, electrostatics, and hydrodynamics, respectively. They are given by ω air (k) = 2C m πlr 3 χ {k=} (k),

4 91 BO LI, HUI SUN, AND SHENGGAO ZHOU ( ) 1 ω surf (k) =γ R 2 k 2, ω vdw (k) =n w U vdw (R ), ω ele (k) = (ε w ε m ) 2 ε w ε m (ε w + ε m ) ω hyd (k) =2μ w k + O(1), R sρ(s)ds 2 k + O(1), as k,whereχ A denotes the characteristic function of a set A and the symbol denotes an averaged integral (e.g., an integral over a, b divided by b a in one dimension). Except for the viscous force, all the static pressure, surface energy, vdw interaction, and electrostatics that are present in the force balance on the dielectric boundary (cf. (1.7)) are the main components in the recently developed variational implicitsolvent models (VISMs) that have successfully predicted solvation free energies and different conformations of charged molecules. See 13, 15, 16, 19, 2, 23, 34, 42, 47, 48 and 4, 11, 12. VISM centers around a solvation free-energy functional of all possible solute-solvent interfaces or dielectric boundaries Γ that separate the solvent region Ω w from solute region Ω m. A simple form of this functional is given by GΓ = ΔP Vol (Ω m )+γ Area (Γ) + n w Ω w U vdw dv + G ele Γ. Here ΔP is the difference between solvent and solute pressures on the boundary Γ. The term G ele Γ is the electrostatic free energy, defined to be the integral of ρφ/2 over the entire system region with ρ being the charge density and φ the electrostatic potential that solves Poisson s equation (1.4). The negative first variation δ Γ GΓ of the solvation free energy GΓ with respect to the location change of boundary Γ defines the normal component of effective force on Γ: F n = δ Γ GΓ = ΔP 2γ H + n w U vdw + f ele, where f ele = δ Γ G ele Γ is given in (1.6). Therefore, the force balance (1.7) is exactly 2μ w D(u)n + δ Γ GΓn = on the interface with ΔP = p w p m. Note that we have implicitly assumed that the tangential components of the VISM force arising from the free energy GΓ is negligible. In our previous work 14, we studied the linear stability of a cylindrical dielectric boundary in the relaxation of surface energy and electrostatic energy. We found the dispersion relation ω(k) = γ k 2 for k 1. Our current work shows that the viscous effect of solvent fluid changes this dispersion relation to ω(k) = γ 2μ w k for k 1; cf. (1.8). This suggests that viscosity slows down the decay of interface perturbation for large modes k. The rest of the paper is organized as follows: in section 2, we describe in detail the governing equations of our solvent fluid dielectric boundary model. In section 3, we obtain the steady-state solutions and their linearized equations for a cylindrical solute-solvent interface. Then we solve the linearized equations to obtain the dispersion relations. The details of derivation of the linearized system are presented in Appendix A. Some definitions and formulas for the modified Bessel functions are

5 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 911 collected in Appendix B. In section 4, we present numerical examples to show the contribution of each component to the stability. Finally, in section 5, we draw our conclusions and discuss several possible directions of future work. 2. Governing equations. We describe the cylindrical solute-solvent interface at time t by a smooth function r = R(z,t) and assume it is L-periodic in z for some constant L>, where, as usual, r = x 2 + y 2. With the cylindrical coordinates (r, θ, z), we define the solute-solvent interface Γ(t), the solute region Ω m (t), and the solvent region Ω w (t) by Γ(t) ={(r, θ, z) R 3 : r = R(z,t)}, Ω m (t) ={(r, θ, z) R 3 : r<r(z,t)}, Ω w (t) ={(r, θ, z) R 3 : r>r(z,t)}, respectively; cf. Figure 1. Our governing equations are given by (1.1) (1.4) and (1.7). We use the following boundary conditions (x denotes (x, y, z)): all R(z, t), u(x, t), p w (x,t), and φ(x,t)arel-periodic in z; andu =, p w = p, and φ =atr = and for all z R. The viscosity μ w of solvent fluid, the number density of solvent molecules n w, the pressure of solvent fluid p > at infinity, and the surface tension γ > of the solute-solvent interface are all known constants. In the ideal-gas law, C m is a positive constant independent of time t and spatial points. The dielectric coefficient ε Γ is defined in (1.5), where the relative permittivities ε m for the solute and ε w for the solvent are known constants, satisfying <ε m <ε w. We assume that the solutesolvent vdw interaction potential U vdw = U vdw (r) is a given smooth function that depends only on r and that satisfies (2.1) U vdw () = and U vdw ( ) =. We also assume that the charge density ρ = ρ(r) :Ω R in Poisson s equation is a known, continuous function that depends only on r. It is compactly supported: there exists R c > such that ρ(r) =ifr>r c. Moreover, it satisfies the condition of charge neutrality: (2.2) Rc rρ(r) dr =. We now rewrite all the governing equations using cylindrical coordinates. Let us first denote as usual by i, j, andk the Cartesian coordinate vectors. For any point x = x i + y j + z k with cylindrical coordinates (r, θ, z), where r = x 2 + y 2 and θ = arctan y/x, the local cylindrical basis (or coordinate) vectors are e r =cosθ i +sinθ j, e θ = sin θ i +cosθ j, ande z = k. Note that x = re r + ze z. We assume that the velocity u, pressure p w, and electrostatic potential φ at x are given by (2.3) u(x,t)=u(r, z, t) e r + w(r, z, t) e z, p w = p w (r, z, t), and φ = φ(r, z, t), respectively. These, together with r = R(z,t), are all the unknown functions in our equations. Note that the pressure p m (t) is eliminated as it can be readily expressed in terms of R(z,t). Our governing equations and boundary conditions for these unknown functions in the cylindrical coordinates are the following:

6 912 BO LI, HUI SUN, AND SHENGGAO ZHOU Equations for interface motion: { t R(z,t) =u(r(z,t),z,t) w(r(z,t),z,t) z R(z,t) z R, (2.4) R(z,t) isl-periodic in z. Equations for solvent fluid: μ w ( rru r ru 1 ) r 2 u + 2 zzu r p w = ifr>r(z,t), μ w ( rrw ) r rw + zzw 2 z p w = ifr>r(z,t), (2.5) r u + 1 r u + zw = ifr>r(z,t), u(r, z, t), w(r, z, t), and p w (r, z, t) arel-periodic in z, u(,z,t)=w(,z,t)= and p w (,z,t)=p z R. Poisson s equation for electrostatics: (2.6) ε m ( 2 rr φ + 1 r rφ + 2 zz φ ) = ρ in Ω m (t), ε w ( 2 rr φ + 1 r rφ + 2 zz φ ) = ρ in Ω w (t), φ(r(z,t),z,t)=φ(r(z,t) +,z,t) z R, ε m r φ(r(z,t),z,t) z R(z,t) z φ(r(z,t),z,t) = ε w r φ(r(z,t) +,z,t) z R(z,t) z φ(r(z,t) +,z,t) z R, φ = φ(r, z, t) isl-periodic in z, φ(,z,t)= z R, where for a function h = h(x), h(x ) = lim y x h(y)andh(x + ) = lim y x+ h(y) are the left and right limits, respectively. Equations of force balance: 2μ w r 1+( z R) 2 u z R( z u + r w)+( z R) 2 z w 1 + C L m R(s, t) ds 2 p w π { } 1 γ R 1+( z R) 2 zzr 2 + n 1/2 1+( z R) 2 3/2 w U vdw (R) (2.7) + 1 ( 1 1 ) εγ(t) ( r φ z R z φ) 2 2 ε w ε m 1+( z R) (ε m ε w ) ( zr r φ + z φ) 2 1+( r R) 2 = z R, z R ( r u z w)+ 1 1 ( z R) 2 ( z u + r w)= z R. 2

7 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 913 We now derive (2.4) (2.7) from(1.1) (1.7). If x = x(t) Γ(t) has the cylindrical coordinates (R(z,t),θ,z), then (2.8) n(x) = e r z R e z 1+( z R) 2, n = zr e r + e z 1+( z R), and e 2 θ are orthonormal with n(x) the unit normal to Γ(t). Since x(t) =R(z,t) e r + z e z,the normal velocity at x is V n (x) =ẋ n(x) = t R/ 1+( z R) 2, where a dot denotes the time derivative. This, together with (1.1) and (2.3), leads to the first equation in (2.4). By (2.3) and a series of calculations, we obtain for x Ω w (t) the expression of Δu(x,t)and p w (x,t) as linear combinations of e r and e z. They lead to the Stokes equation, the first two equations, in (2.5). Similar calculations lead to the third equation in (2.5) for the incompressibility equation. The gradient of φ = φ(r, z, t) atx = re r + ze z is φ(x,t)= r φ e r + z φ e z. If x = x(t) Γ(t) has the cylindrical coordinates (R(z, t), θ, z), then by (2.8) the normal derivatives of φ are given by (2.9) n φ(x,t)= φ(x,t) n(x) = rφ(r(z,t) s,z,t) z R(z,t) z φ(r(z,t) s,z,t), 1+ z R(z,t) 2 where s = or +. Consequently, by these and the expression of the Laplacian in cylindrical coordinates, we can rewrite the boundary-value problem of Poisson s equation (1.4) for the electrostatic potential φ = φ(r, z, t) as the elliptic interface problem (2.6) 14, 28. Note that the third and fourth equations in (2.6) are the continuities of the electrostatic potential φ and the normal component of electric displacement ε Γ(t) n φ, respectively, across the dielectric boundary r = R(z,t). To finally derive (2.7), we note that the volume Vol (Ω m (t)) of the cylindrical region Ω m (t) means that for one period <z<l.since the cylindrical surface is represented by r = R(z,t), the equation (1.3) for the ideal-gas law for the solute region is L (2.1) p m (t) R(z,t) 2 dz = C m π. If x = x(t) Γ(t) has the cylindrical coordinates (R(z,t),θ,z), then direct calculations lead to (2.11) D(u(x))n(x) = 1 ( r u 1 ) 1+( z R) 2 2 zr( z u + r w) e r ( ) ( zu + r w) z R z w e z, 2 (2.12) zz 2H(x) = R 1+( z R) /2 R 1+( z R) 2. 1/2 By (2.8), (2.9), and (1.6), we have (2.13) f ele (x) = 1 ( 1 1 ) εγ(t) ( r φ z R z φ) 2 2 ε w ε m 1+( z R) (ε m ε w ) ( zr r φ + z φ) 2 1+( r R) 2.

8 914 BO LI, HUI SUN, AND SHENGGAO ZHOU These, together with (2.11) and (2.1), allow us to rewrite the equation of force balance (1.7) on the boundary Γ(t) into those in (2.7) by dotting (1.7) with n and n, respectively. Note that each term in (2.7) is orthogonal to e θ. The correspondence between terms in (1.7) and(2.7) is through the parameters μ w, γ, n w,etc. 3. Steady-state solutions and their linear stabilities. Steady-state solutions with cylindrical solute-solvent interfaces r = R with constant radii R >, together with their corresponding fluid velocity fields u,w, and pressures p w,and the electrostatic potentials φ are given by r = R z R, u = w = forr>r, p w = p for r>r, 1 r 1 s σρ(σ)dσds + C 2 if <r<r, ε m s φ (r) = 1 r 1 s σρ(σ)dσds + C 3 log r + C 4 if r>r, ε w R s R f(r ):= C m πlr 2 p γ + n w U vdw (R ) R + 1 ( 1 1 ) 2 R (3.1) sρ(s) ds =, 2 ε w ε m where C 2 C 3 log R C 4 = 1 R 1 s σρ(σ)dσ ds, ε m s C 3 = 1 R sρ(s)ds, ε w 1 r 1 s C 4 = lim σρ(σ)dσ ds C 3 log r. r ε w R s R All R, u, w, p w, and φ satisfy the governing equations for the interface motion, fluid velocity and pressure, and electrostatics, and the corresponding boundary conditions. Equation (3.1) is the force balance equation. The existence of R > such that f(r ) = follows from (2.1) together with the fact that p > and <ε m <ε w. Note that if we consider the ordinary differential equation Ṙ = f(r) for R = R(t), then the sign of f (R ) determines the stability of R. The constants C 2,C 3,andC 4 are determined by the interface jump conditions and the boundary condition for the electrostatic potential φ (cf. the third, fourth, and last equations in (2.6)). Note that the limit C 4 exists by the fact that the charge density ρ = ρ(r) is compactly supported and by the charge neutrality (2.2). We now linearize our system around a steady-state solution r = R, u = w =, p w = p,andφ = φ (r) defined above. We assume R(z,t) =R + δr 1 (z,t) for z R, u(r, z, t) =δu 1 (r, z, t)+o(δ 2 ) for r>r(z,t), (3.2) w(r, z, t) =δw 1 (r, z, t)+o(δ 2 ) for r>r(z,t), p w (r, z, t) =p + δp w1 (r, z, t)+o(δ 2 ) for r>r(z,t), φ(r, z, t) =φ (r)+δφ 1 (r, z, t)+o(δ 2 ) for r, z R,

9 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 915 where δ R is such that δ 1. All functions are L-periodic in z. Note that the expansion of the velocity field is u(x,t) = δu 1 (x,t) + O(δ 2 ) = δ u 1 (r, z, t) e r + w 1 (r, z, t) e z +O(δ 2 ), where e r, e θ,ande z are the local basis vectors at x that has cylindrical coordinates (r, θ, z). The linearized equations and boundary conditions for R 1, u 1, w 1, p w1,andφ 1 are summarized below, and their derivations are detailed in Appendix A: (3.3) t R 1 (z,t) =u 1 (R,z,t) z R, μ w ( rr 2 u r ru 1 1 ) r 2 u 1 + zz 2 u 1 r p w1 = forr>r,z R, μ w ( rr 2 w ) r rw 1 + zz 2 w 1 z p w1 = forr>r,z R, (3.4) r u r u 1 + z w 1 = forr>r,z R, u 1 (,z,t)=w 1 (,z,t)=p w1 (,z,t)= z R, 2 rrφ r rφ 1 + zz φ 1 = ifr<r, (3.5) (3.6) 2 rrφ r rφ 1 + zz φ 1 = ifr>r, φ 1 (R +,z,t) φ 1(R,z,t)= R 1(z,t)φ (R+ ) φ (R ) z R, ε m r φ 1 (R,z,t)=ε w r φ 1 (R +,z,t) z R, φ 1 (,z,t)= z R, L 2μ w r u 1 (R,z,t) 2C m πlr 3 R 1 (s, t) ds p w1 (R,z,t) ( ) ε 2 ε w ε wφ (R + ) R 1 (z,t)φ (R + )+ rφ 1 (R +,z,t) m ( + γ zzr R ) 1 + n w U vdw(r )R 1 = z R, R 2 r w 1 (R,z,t)+ z u 1 (R,z,t)= z R. We now seek solutions to the linearized system (3.3) (3.6) of the form R 1 (z,t) =e ωt e ikz, (u 1 (r, z, t),w 1 (r, z, t),p w1 (r, z, t),φ 1 (r, z, t)) = e ωt e ikz (û 1 (r), ŵ 1 (r), ˆp w1 (r), ˆφ 1 (r)), where k = 2πk /L and k is a nonnegative integer and all the functions û 1 = û 1 (r), ŵ 1 = ŵ 1 (r), ˆp w1 = ˆp w1 (r), and ˆφ 1 = ˆφ 1 (r) are to be determined. The sign of the growth rate ω = ω(k) determines the linear stability of an underlying steady state. The system (3.3) (3.6) reduces to (3.7) ω =û 1 (R ), μ w û ( ) 1 r û 1 r 2 + k2 û 1 ˆp w1 = forr>r, ( μ w ŵ ) r ŵ 1 k 2 ŵ 1 ikˆp w1 = forr>r, (3.8) û r û1 + ikŵ 1 = forr>r, û 1 ( ) =ŵ 1 ( ) =ˆp w1 ( ) =,

10 916 BO LI, HUI SUN, AND SHENGGAO ZHOU ˆφ r ˆφ 1 k 2 ˆφ1 = forr<r, (3.9) (3.1) ˆφ r ˆφ 1 k 2 ˆφ1 = forr>r, ˆφ 1 (R + ) ˆφ 1 (R )= φ (R+ ) φ (R ), ε m ˆφ 1 (R )=ε ˆφ w 1(R + ), ˆφ 1 ( ) =, 2μ w û 1(R ) 2C ( ) m 1 πlr 3 χ {k=} ˆp w1 (R )+γ R 2 k 2 + n w U vdw(r ) ( ) ε 2 ε w ε wφ (R) s φ (R)+ s ˆφ 1(R) s =, s = or +, m ŵ 1 (R )+ikû 1 (R )=. We shall use the modified Bessel functions of order n of the first and second kind I n (x) andk n (x). The definition and basic properties of these functions are recalled in Appendix B. Lemma 3.1. The first three equations in (3.8) together with the boundary conditions (3.7), the fourth equation in (3.8), and the second equation in (3.1) have a unique solution û 1 =û 1 (r), ŵ 1 =ŵ 1 (r), ˆp w1 =ˆp w1 (r), given by, for k =: for k>: û 1 (r) = R ω r, ŵ 1(r) =, and ˆp w1 (r) =; û 1 (r) = ωkrk (kr) + 1 kr K (kr ) ωk1 (kr) K 1 (kr ) K 1 (kr ) K 1 (kr ), ŵ 1 (r) = iωk (kr) 1+ kr K (kr ) iωkrk 1(kr) K 1 (kr ) K 1 (kr ) K 1 (kr ), ˆp w1 (r) = 2ωμ wkk (kr). K 1 (kr ) Proof. The case k = can be proved by direct verifications. For the case k >, we use a projection method to find the solution in three steps. Step 1. Let a, b R. Consider the equations for the velocity components ũ 1 = ũ 1 (r) and w 1 = w 1 (r): ũ r ũ 1 ( 1 r 2 + k2 ) ũ 1 = forr>r, By (B.5), the solution is given by w r w 1 k 2 w 1 = forr>r, ũ 1 (R )=a, w 1 (R )=b, ũ 1 ( ) = w 1 ( ) =. (3.11) ũ 1 (r) =a K 1(kr) K 1 (kr ) and w 1 (r) =b K (kr) K (kr ). Step 2. Find the general solution p 1 = p 1 (r) to the equation (3.12) p w1 + 1 r p w1 k 2 p w1 =ũ r ũ1 + ik w 1

11 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 917 and p w1 ( ) =. By (3.11) and (B.2), we can simplify the right-hand side of (3.12) as ak K 1 (kr ) + ikb K (kr). K (kr ) Since K (kr) andi (kr) are two linearly independent solutions to the homogeneous equation associated with (3.12) and their Wronskian is 1/r by (B.3) (the factor k does not appear), the general solution to (3.12) is then given by K (kr) p w1 (r) =Ĉ1 K (kr ) + I (kr) Ĉ2 I (kr ) + ak K 1 (kr ) + ikb K (kr ) r r K (kr) si (ks)k (ks)ds + I (kr) s(k (ks)) 2 ds R R K (kr) = Ĉ1 K (kr ) + I (kr) Ĉ2 I (kr ) + ak K 1 (kr ) + ikb K (kr ) { rk 1(kr) + R2 2k 2 I (kr )K (kr )+I 1 (kr )K 1 (kr ) K (kr) R2 (K (kr )) 2 (K 1 (kr )) 2 } I (kr), 2 where Ĉ1 and Ĉ2 are constants and where in the last step we used the formulas si (s)k (s)ds = s2 2 I (s)k (s)+i 1 (s)k 1 (s), sk 2 s2 (s)ds = K 2 2 (s) K1 2 (s), together with (B.2) and (B.3). These formulas can be directly verified by taking the derivatives of the right-hand sides and using (B.2) and (B.3). It now follows from (B.5) and p w1 ( ) =that Ĉ 2 = R2 I (kr ) 2 Hence, ak K 1 (kr ) + (3.13) p w1 (r) = C K (kr) 1 K (kr ) where C 1 = Ĉ1 + R2 K (kr ) 2 ikb K (kr ) a 2K 1 (kr ) + (K (kr )) 2 (K 1 (kr )) 2. ib rk 1 (kr), 2K (kr ) ak K 1 (kr ) + ikb K (kr ) I (kr )K (kr )+I 1 (kr )K 1 (kr ) is a constant. Step 3. Define û 1 (r) =ũ 1 (r) p w1 (r), ŵ 1 (r) = w 1 (r) ik p w1 (r), ˆp w1 (r) = μ w p w1 (r)+1 r p w1 (r) k2 p w1 (r).

12 918 BO LI, HUI SUN, AND SHENGGAO ZHOU It is easy to verify that û 1, ŵ 1,andˆp w1 satisfy the first three equations in (3.8). By (B.2) and (B.5), we can also verify that û 1 ( ), ŵ 1 ( ), and ˆp w1 ( ) allequal. If we set τ 1 = akk (kr )/K 1 (kr )+ikb and τ 2 = b ik C 1, then we can verify by (3.11) (3.13), together with (B.2), that 2K 1 (kr)+rkk (kr) K 1 (kr) û 1 (r) = τ 1 + iτ 2 2kK (kr ) K (kr ), irk 1 (kr) ŵ 1 (r) =τ 1 2K (kr ) + τ K (kr) 2 K (kr ), K (kr) ˆp w1 (r) = μ w τ 1 K (kr ). By (3.7) and the second equation in (3.1), we have K1 (kr ) kk (kr ) + R ik 1 (kr ) τ 1 τ 2 2 K (kr ) = ω, ir 2 τ 1 + K 1(kR ) K (kr ) τ 2 = iω. Solving these two equations, we get τ 1 = 2kω K (kr ) K 1 (kr ) and K (kr ) τ 2 = ωi K 1 (kr ) + kr (K (kr )) 2 (K 1 (kr )) 2 Plugging τ 1 and τ 2 into û 1, ŵ 1,andˆp w1 above, we obtain the desired solution expressions for the case k>. Lemma 3.2. The solution ˆφ 1 = ˆφ 1 (r) to the system of equations and boundary conditions (3.9) for the electrostatic potential is given by, { φ (R + for k =: ˆφ1 (r) = ) φ (R ) if <r<r, if R <r< ; for k>: ˆφ1 (r) =. ε w φ (R + ) φ (R ) K 1 (kr ) ε m K (kr )I 1 (kr )+ε w I (kr )K 1 (kr ) I (kr) if <r<r, ε m φ (R + ) φ (R ) I 1 (kr ) ε m K (kr )I 1 (kr )+ε w I (kr )K 1 (kr ) K (kr) if R <r<. Proof. Direct verifications prove the result for the case k =. Fix k>. The solution ˆφ 1 (r) in each of the regions r<r and r>r is a linear combination of I (kr) andk (kr). By the last equation in (3.9) and by the asymptotic properties (B.4) and (B.5), we then have { C1 I (kr) if r<r, ˆφ 1 (r) = C 2 K (kr) if R <r<, where C 1 and C 2 are two constants. They are determined by the third and fourth equations in (3.9): C 2 K (kr ) C 1 I (kr )= φ (R+ ) φ (R ), C 1 ε m I 1 (kr )+ C 2 ε w K 1 (kr )=,

13 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 919 where we used (B.2) in deriving the second equation. Solving for C 1 and C 2,wehave the desired solution. Proposition 3.3. The growth rate ω = ω(k) is given by ω(k) = ω air(k)+ω surf (k)+ω vdw (k)+ω ele (k) (3.14) ω hyd (k) where the asymptotics is with respect to k and ω air (k) = 2C m πlr 3 χ {k=}, ( ) 1 ω surf (k) =γ R 2 k 2, ω vdw (k) =n w U vdw (R ), ω ele (k) = ( 1 ε w 1 ε m + (ε w ε m ) 2 ε w ε m ) R ρ(r ) = (ε w ε m ) 2 ε w ε m (ε w + ε m ) R R sρ(s)ds = γ 2μ w k + O(1), sρ(s)ds 1 ) 2 R ( sρ(s)ds R 2 ki 1 (kr )K 1 (kr ) ε m K (kr )I 1 (kr )+ε w I (kr )K 1 (kr ) sρ(s)ds 2 k + O(1), ω hyd (k) =2μ w 1 R + k 2 R k2 R (K (kr )) 2 (K 1 (kr )) 2 =2μ w k + O(1). We remark that the zeroth-mode perturbation (k = ) is only in the r- direction. In this case, we have by taking the limit as k and using (B.4) that ω ele () = ( 1 1 ) R ρ(r ) ε w ε m ω hyd () = 2μ w. R The growth rate is then exactly ω() = R f (R ) 2μ w = R 2μ w 2C m πlr 3 1 {k=} + γ R 2 + n w U vdw (R ) ( ) R ρ(r ) ε w ε m where f is defined in (3.1). Proof of Proposition 3.3. By Lemma 3.1, 2μ w û 1(R ) ˆp w1 (R )= 2μ w ω sρ(s)ds 1 ) 2 R ( sρ(s)ds, R sρ(s)ds 1 R ( sρ(s)ds R 1 + k 2 R k2 R (K (kr )) 2 R (K 1 (kr )) 2 ) 2,.

14 92 BO LI, HUI SUN, AND SHENGGAO ZHOU For the steady-state electrostatic potential φ = φ (r), we have By Lemma 3.2, R ε w φ (R+ )=ε mφ (R )= rρ(r) dr. ˆφ 1(R + )= ε mk φ (R+ ) φ (R ) I 1 (kr )K 1 (kr ) ε m K (kr )I 1 (kr )+ε w I (kr )K 1 (kr ). Plugging these and (A.2) into the first equation of (3.1) with s = +, we then obtain 1 2μ w ω + k 2 R k2 R (K (kr )) 2 R (K 1 (kr )) 2 2C m πlr 3 1 {k=} ( ) 1 + γ R 2 k 2 + n w U vdw(r ) ( ) R ρ(r ) sρ(s)ds 1 ) 2 R ( sρ(s)ds ε w ε m R + (ε w ε m ) 2 2 R ki 1 (kr )K 1 (kr ) sρ(s)ds ε w ε m ε m K (kr )I 1 (kr )+ε w I (kr )K 1 (kr ) =. This leads to the expression of growth rate ω = ω(k) as desired. Notice that ω hyd (k) > for all k>, since <K (x) <K 1 (x) for any x>2. By (B.5), K (kr ) K 1 (kr ) =1 1 + O(k 2 ) 2kR and I (kr ) I 1 (kr ) = O(k 2 ) for k 1. 2kR Thus, 1 ω hyd (k) =2μ w + k 2 R k2 R (K (kr )) 2 R (K 1 (kr )) 2 =2μ w k + O(1). Moreover, Therefore, ki 1 (kr )K 1 (kr ) ε m K (kr )I 1 (kr )+ε w I (kr )K 1 (kr ) k ε w ε m = 1 + O(k 1 ). ε m + ε w 2kR (ε w + ε m ) ω ele (k) = (ε w ε m ) 2 ε w ε m (ε w + ε m ) R sρ(s)ds 2 k + O(1) as k. Combining all these, we obtain the desired asymptotics of ω(k).

15 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE R (Å) 9 7 R (Å) (a) 5e+4 6e+4 7e+4 8e+4 ρ (e Å 1 ) (b) 5 1e+4 3e+4 5e+4 6e+4 ρ (e Å 1 ) Fig. 2. Equilibrium radii R versus the parameter ρ in the charge density ρ and their stabilities for the mode k =with (a) R c =1and (b) R c =11.5. Multiple values of R are found for ρ in certain ranges. Solid and dotted lines indicate the stable and unstable branches of R, respectively. The small magenta solid circles correspond to the saddle-node bifurcation points. In (b), the marker points in red, blue, and black from different branches correspond to the plots of dispersion relations in Figures 3 and Numerical results. We use the following parameters that are typical values in molecular dynamics simulations: T = 298 K, μ w =.156 k B T ps/å 3,n w =.331 Å 3, p = k B T/Å 3, γ =.175 k B T/Å 2, ε = e 2 /(k B T Å), ε m = ε, ε w =8ε, where e is the elementary charge. We set the period L = 1 Å and the constant C m =.65 k B T. We define (σ ) 11 ( σ ) 5 U vdw (r) =κ r r with κ =5. k B T/Å 3 and σ =1. Å. This potential is obtained by integrating a Lennard Jones (LJ) 12-6 potential along the z-axis. We use an artificial charge density of the form (in the units e/å 3 ) R 4 ( ρ(r) =ρ c(2r R c ) 8r 4 (r R c ) 3 exp R 4 ) c 16r 2 (r R c ) 2 χ { r Rc}(r), where ρ (in the units e/å) and R c (in the units Å) are two positive numbers to be chosen later. One verifies that this charge density satisfies the charge neutrality condition (2.2), since r ( Rc 4 ρ(s)s ds = ρ exp 16r 2 (r R c ) 2 ) χ {r Rc}(r). Note that the parameter ρ sets the magnitude of the charge distribution and hence the electrostatic force. The parameter R c determines the spatial range of electrostatic force. If we set R c to be close to the LJ parameter σ, then a small variation of the value of R c can result in a dramatical change of equilibrium radii. We use Newton s method to solve (3.1) for R (in the units Å), the radius of a cylindrically shaped equilibrium interface. For certain values of the parameter ρ, we find multiple values R with different initial guesses in Newton s iteration. In Figure 2, we plot multiple branches of solutions R versus ρ for (a) R c =1Åand (b) R c =11.5 Å, respectively. Solid and dotted lines indicate stable and unstable branches of R, respectively, with respect to the zeroth-mode (k =)perturbation. For R c = 1, there is a bifurcation at ρ = For R c =11.5, there are two

16 922 BO LI, HUI SUN, AND SHENGGAO ZHOU 5e+3 (a) (b) ρ =2.5e+4, R = 8.96 ρ =4e+4, R = 9.61 (c) ρ =5e+4, R = 9.8 ω (ps 1 ) 1e+4 2e+4 2e+4 4e+4 ρ =2.5e+4, R = 8.3 ρ =4e+4, R = 6.88 ρ =5e+4, R = 6.5 ρ =5.5e+4, R = e e+4 2e+4 4e k (Å 1 ) 5 2e+4 2e+4 4e+4 ρ =2e+4, R = 9.99 ρ =2.5e+4, R = 9.98 ρ =4e+4, R = 9.96 ρ =5e+4, R = 9.91 Fig. 3. Plots of the dispersion relations for multiple solutions R with R c =11.5. The four curves in (a) correspond to the four data points on the bottom curve in Figure 2(b). The three curves in (b) correspond to the three data points on the middle curve in Figure 2(b). The four curves in (c) correspond to the four data points on the top curve in Figure 2(b). Plots in each inset are for a large range of k. bifurcation points at ρ = and ρ = , respectively. All these are saddle-node bifurcations. We plot in Figure 3 the dispersion relation ω = ω(k) given by (3.14)forR c =11.5. Figure 3(a), (b), and (c) correspond to the data points on the bottom, middle, and top curves in Figure 2(b), respectively. Let us first examine the stability and instability for the zeroth mode k =. From Figure 3(a) and (c), we observe the zeroth-mode linear stability for all the parameter choices along the bottom or top curve in Figure 2(b), and the instability for those along the middle curve in Figure 2(b). Consider the increase of ρ from the left bifurcation point from which the bottom and middle curves branch out in Figure 2(b). We can see from Figure 3(a) and (b) that the value of ω() along the bottom branch becomes more negative, indicating that the corresponding equilibrium shape is more stable. Along the middle branch, the value of ω() begins with at the bifurcation point but rapidly increases to a large positive number and then gradually decreases to a small positive number near the right bifurcation point in Figure 2(b). Consider now the decrease of ρ from the right bifurcation point in Figure 2(b) from which the top curve and middle curve branch out. We observe from Figure 3(c) and (b) that the value of ω() becomes more negative along the top curve and more positive along the middle curve. Let us now examine the general, nonzero mode k. Consider first the data points on the bottom curve in Figure 2(b)fromlefttorightasρ increases. Figure 3(a) indicates that both the largest value of ω(k) and the range of the small-wavenumber instability increase. Consider now the data points on the middle branch in Figure 2(b) from left to right as ρ increases. We find from Figure 3(b) the small-wavenumber instability and the large-wavenumber stability. As ρ increases along the middle branch further to reach the right bifurcation point meeting with the top branch, the small-wavenumber instability diminishes and the interface switches its stability at the bifurcation. Finally, consider the top curve in Figure 2. Weobservefrom Figure 2(c) the linear stability for all wavenumbers k and for each data point on this curve. Moreover, for each k, ω(k) is more negative further away from the right bifurcation point. Note that the top branch terminates at the right bifurcation point

17 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 923 ω (ps 1 ) R = 8.3 R = 8.96 R = ωhyd(k ) ω vdw(k ) k (Å 1 ) 4 2 ω ele(k ) (a) ω surf (k ) ω (ps 1 ) ω hyd(k ) 2e+5 1e ω ele(k ) ω vdw(k ) e k (Å 1 ) R = 8.3 R = 8.96 R = (b) 2e+5 ω surf (k ) 2e+5 Fig. 4. The dispersion relations for multiple solutions of R at ρ = and R c =11.5 that correspond to the same marker points in Figure 2(b). The insets indicate different contributions to the growth rate ω(k) :(a)small range of k; (b) large range of k. with ρ = in Figure 2(b). This indicates that for ρ larger than this value, and R c =11.5, there is no equilibrium radius R that is linearly stable for all wavenumbers. Figure 4 displays the dispersion relations and contributions from different components of force corresponding to the three data points at ρ = on the three branches, respectively, near but on the right of the left bifurcation point, in Figure 2(b). Figure 4(a) and (b) are for k 3and k 15, respectively. We observe for large k the linear decay of ω(k), the quadratic decay of ω surf (k), and the linear growth of ω hyd (k) andω ele (k), respectively, consistent with our asymptotic analysis summarized in Proposition 3.3. In the range of small wavenumbers k, the growth rate ω(k) increaseswithk for the top and bottom branches, but decreases with k for the middle branch. It is readily seen from Figure 4(a) that, in the small k regime, the dispersion relation is dominated by ω vdw (k) andω ele (k). The part ω surf (k) is slightly positive for small k due to the cylindrical geometry but then drops quadratically. The part ω ele (k) grows almost linearly with k except for k where there is some subtle nonlinearity. Hence ω vdw (k)+ω ele (k) can be approximated as a linear function with a positive slope due to ω ele (k). By diving ω vdw (k)+ω ele (k) by ω hyd (k), we find that the dispersion relation ω(k) can be approximately described by an inverse power law ω(k) (c 1 + c 2 /k)/μ w, where c 1 and c 2 are constants depending on ρ and R c. This approximation is valid for k away from zero. For k close to zero, ω hyd 2μ w /R. The effect of air pressure ω air (k) is only reflected at k =,andit causes a negligibly small discontinuity at k =. 5. Conclusions. We have constructed a solvent fluid dielectric boundary model to describe the effect of solvent fluid on conformations of charged molecules in aqueous solution. In addition to the Stokes equation for the incompressible solvent fluid, we use the ideal-gas law to model solutes. Moreover, on the moving dielectric boundary, the viscous force balances the VISM forces that include hydrostatic pressures, surface tension, solute-solvent vdw interactions, and electrostatic force. We have analyzed the linear stability of a steady-state cylindrical dielectric boundary. We find multiple such steady states that correspond to polymodal states (e.g., dry and wet) of hydration of charged molecules, as captured by our previous level-set variational implicit-solvent modeling 15, 16, 34, 42, 47, 48. Such nonuniqueness results from the competition between the surface tension, electrostatics, and solute-solvent

18 924 BO LI, HUI SUN, AND SHENGGAO ZHOU vdw interactions. The multiple steady states bifurcate with respect to the charge distributions. Our stability analysis shows that the surface tension is the dominant effect in high-wavenumber stability. But the solvent viscosity weakens such stabilization as it changes the quadratic tail for large wavenumbers to a linear one. The electrostatics always destablizes the system. The most important stability parameter we find is γ /(2μ w ), half of the ratio of the surface tension γ to the solvent fluid viscosity μ w. The instability of a solute-solvent interface is in fact a generic property of a real biomolecular system. For instance, the interface separating a protein from water molecules that are trapped inside the protein is metastable, as such water molecules lose their hydrogen bonding to the network of bulk water molecules 45, 46. Our preliminary analysis only indicates qualitatively the stabilizing factors, but does not characterize various scales at which the stability or instability occurs. It will be interesting to connect the small-wavenumber instability analyzed here with the experiment and computer simulations on the metastable water molecules inside a protein. The inclusion of solvent fluid flow is one step further to make an implicit-solvent approach more accurate in describing the important solvent effect. The development of numerical methods to solve our modeling system of equations is, however, nontrivial. One difficulty is to impose proper boundary conditions to allow solutes to change their volumes as occurs during the biomolecular conformational change. Usual boundary conditions, such as the periodic boundary condition imposed on the boundary of a computational region, will always keep the volume of solute region a constant because of the incompressibility of solvent fluid. A different issue is that the description of solutes by the ideal-gas law is over simplified. An improved approach is to include the solute molecular mechanical motion 16 and the coupling of such motion with the solvent fluid motion. Finally, our proposed solvent fluid model allows us to describe the solvent fluctuations through the Stokes equation with a stochastic stress tensor 43. Such fluctuations are important in conformational changes of biomolecules. The mathematical description and numerical computation of such fluctuations will be challenging tasks. Appendix A. In this appendix, we derive the linearized system (3.3) (3.6). Equations (3.3) and (3.4) can be obtained simply by plugging the expansions (3.2) into (2.4) and (2.5), and matching the O(δ) terms. The first, second, and fifth equations in (3.5) can be obtained similarly using the corresponding equations in (2.6). At the interface Γ(t) defined by r = R(r, t) with R(z,t) given in (3.2), we have φ(r, z, t) =φ (R + δr 1 (z,t),z,t,) = φ (R + δr 1 (z,t)) + δφ 1 (R + δr 1 (z,t),z,t)+o(δ 2 ) = φ (R)+δ s R 1 (z,t)φ (R)+φ s 1 (R,z,t) s + O(δ 2 ) for s = or +. Comparing O(1)-terms, we obtain the already known continuity φ (R )=φ (R + ); cf. the third equation in (2.6) with φ and R replacing φ and R(z,t), respectively. Comparing O(δ)-terms, we obtain the third equation in (3.5), which is the first interface condition for φ 1 = φ 1 (r, z, t). By (2.9), we have at either side s = or + on the interface r = R(z,t) defined in (3.2) that n φ(r(z,t) s,z,t)= rφ(r(z,t) s,z,t) z R(z,t) z φ(r(z,t) s,z,t) 1+ z R(z,t) 2 (A.1) = φ (Rs + δr 1(z,t)) + δ r φ 1 (R s,z,t)+o(δ2 ) = φ (Rs )+δr 1(z,t)φ (Rs )+ rφ 1 (R s,z,t) + O(δ2 ).

19 STABILITY OF CYLINDRICAL SOLUTE-SOLVENT INTERFACE 925 Therefore, the O(1)-terms in the continuity of ε Γ(t) n φ at r = R(z,t) (cf. the fourth equation in (2.6)) leads to the known continuity ε m φ (R )=ε wφ (R+ ). The O(δ)- terms lead to ε m R1 (z,t)φ (R )+ rφ 1 (R,z,t) = ε w R1 (z,t)φ (R + )+ rφ 1 (R +,z,t). From the expression of the steady-state electrostatic potential φ,wehave (A.2) ε m φ (R )=ε wφ (R + )= 1 R rρ(r) dr ρ(r ). R These imply the fourth equation in (3.5) which is the second interface condition for φ 1. To linearize the force balance equations (2.7), we first have by (3.2) and (2.1) that 1 p m (t) = C L m (R + δr 1 (s, t)) 2 ds = C m π πr 2L δ 2C L m πlr 3 R 1 (s, t) ds + O(δ 2 ). Now, at the interface r = R(z,t), where a point has the cylindrical coordinates (R(z,t),θ,z), we have by (3.2) and (2.12), and routine calculations, that α(r(z,t),z,t)=δ α 1 (R,z,t)+O(δ 2 ), = r or z and α = u or w, p w (R(z,t),z,t)=p + δp w1 (R )+O(δ 2 ), 2H(R(z,t),θ,z)= 1 ( δ zz 2 R R 1 + R ) 1 R 2 + O(δ 2 ), n w U vdw (R(z,t)) = n w U vdw (R )+δn w U vdw (R )R 1 (z,t)+o(δ 2 ). Similarly, we have by (3.2) and (2.13) that f ele (R(z,t),z) + 1 ( 1 1 ) εγ(t) ( r φ z R z φ) 2 2 ε w ε m 1+( z R) (ε m ε w ) ( zr r φ + z φ) 2 1+( r R) 2 = 1 ( 1 1 ) εγ(t) r φ(r(z,t),z,t) 2 + O(δ 2 ) 2 ε w ε m = 1 ( 1 1 ) ε 2 w φ 2 ε w ε (R(z,t) + )+δ r φ 1 (R(z,t) +,z,t) 2 + O(δ 2 ) m = 1 ( 1 1 ) ε 2 w φ 2 ε w ε (R + )+δr 1(z,t)φ (R+ )+δ rφ 1 (R +,z,t) 2 + O(δ 2 ) m = 1 ( 1 1 ) εw φ 2 ε w ε (R + ) 2 m ( 1 + δ 1 ) ε 2 w ε w ε φ (R+ ) R 1 (z,t)φ (R+ )+ rφ 1 (R +,z,t) + O(δ 2 ). m By the continuity of φ at r = R, we can change ε w and the sign + to ε m and the sign, respectively. All these, together with (2.7), imply the first equation in (3.6). The second equation in (3.6) can be obtained by straightforward calculations using (3.2).

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