Continuum Electrostatics for Ionic Solutions with Nonuniform Ionic Sizes

Transcription

1 Continuum Electrostatics for Ionic Solutions with Nonuniform Ionic Sizes Bo Li Department of Mathematics and Center for Theoretical Biological Physics (CTBP) University of California, San Diego, USA Supported by the US NSF, DOE, and CTBP Discussions with J. Che, V. Chu, J. Dzubiella, and B. Lu ICMSEC, CAS September 18, 2009

2 Outline 1. Introduction 2. The Classical Poisson Boltzmann Theory 3. A Size-Modified Poisson Boltzmann Theory 4. Application Issues 4.1 Generalized Debye Hückel approximations 4.2 Potential monotonicity and charge compensation 4.3 Wall-mediated like-charge attractions 5. Conclusions References [1] B. Li, SIAM J. Math. Anal., 40, , [2] B. Li, Nonlinearity, 22, , 2009 [3] B. Li, unpublished notes, 2009.

3 1. Introduction

4 Electrostatic interactions charges in biomolecules solvent polarization, ions long range main part of solvation free energy G = G 2 G 1 (McCammon s work)

5 Coulomb s law potential U 21 = 1 4πε q 1 q 2 r force F 21 = U 21 (r) = 1 q 1 q 2 4πε r 2 r 21 q 1 r q 2 Poisson s equation ε ψ = 4πρ ψ: electrostatic potential ρ: charge density ε: dielectric coefficient. x Q 2 2 Q 1. x Q. N x N.. Ω m ε.. m Ω εw Implicit-solvent models Γ w Main issue: the electrostatics of induced ions.

6 2. The Classical Poisson Boltzmann Theory

7 Consider an ionic solution occupying a region Ω. ρ f : Ω R: given, fixed charge density c j : Ω R: concentration of jth ionic species c j : bulk constant concentration of jth ionic species q j : charge of jth ionic species β: inverse thermal energy Poisson s equation: Charge density: Boltzmann distributions: ε(x) ψ(x) = 4πρ(x) ρ(x) = ρ f (x) + M q jc j (x) c j (x) = c j e βq jψ(x) The Poisson Boltzmann equation (PBE) M ε ψ + 4π q j cj e βqjψ = 4πρ f

8 M PBE ε ψ + 4π q j cj e βqjψ = 4πρ f Linearized PBE (the Debye Hückel approximation) ε ψ εκ 2 ψ = 4πρ f Here κ > 0 is the ionic strength or the inverse Debye Hückel screening length, defined by κ 2 = 4πβ M i=1 q2 j c j ε Derivation : use the Taylor expansion and Electrostatic neutrality: M q jcj = 0 The sinh PBE for 1:1 salt (q 2 = q 1, c 2 = c 1 ) ε ψ 8πqc 1 sinh(βqψ) = 4πρ f

9 Electrostatic free-energy functional of ionic concentrations 1 M [ G[c] = ρψ + β 1 c j ln(λ 3 c j ) 1 ] M µ j c j 2 dv Ω ρ(x) = ρ f (x) + M q jc j (x) ψ = Lρ : ε ψ = 4πρ and ψ = 0 on Ω Λ : the thermal de Broglie wavelength µ j : chemical potential for the jth ionic species Equilibrium conditions (δg[c]) j = q j ψ+β 1 ln(λ 3 c j ) µ j = 0 Boltzmann distributions Minimum electrostatic free-energy G min = ε M 8π ψ 2 + ρ f ψ β 1 Ω c j ( e βq jψ 1) dv

10 G[c] = 1 M [ ρψ + β 1 c j ln(λ 3 c j ) 1 ] M µ j c j 2 dv Ω Theorem (B.L. 2009). The functional G has a unique minimizer c = (c 1,...,c M ) which is also the unique equilibrium. θ 1 > 0, θ 2 > 0 : θ 1 c j (x) θ 2 a.e. x Ω, j = 1,...,M. The equilibrium concentrations c 1,...,c M and corresponding potential ψ are related by the Boltzmann distributions. The potential ψ is the unique solution to the PBE. Remark. Bounds are not physical! A drawback of the PB theory. Proof. By the direct method in the calculus of variations, using: convexity: G[λu + (1 λ)v] λg[u] + (1 λ)g[v]; lower boundedness of s s(log s α) with α R; superlinearity of s s log s; a lemma (cf. next slide). Q.E.D.

11 G[c] = 1 M [ ρlρ + β 1 c j ln(λ 3 c j ) 1 ] M µ j c j 2 dv Ω ρ(x) = ρ f (x) + M q jc j (x) ψ = Lρ : ε ψ = 4πρ and ψ = 0 on Ω Lemma (B.L. 2009). Given c. There exists ĉ satisfying: ĉ is close to c; G[ĉ] G[c]; θ 1 > 0, θ 2 > 0 : θ 1 ĉ j (x) θ 2 a.e. x Ω, j = 1,...,M. Proof. By construction using the fact that the entropic change is very large for c j 0 and c j 1. Q.E.D. slns O s

12 3. A Size-Modified Poisson Boltzmann Theory

13 Electrostatic free-energy functional of ionic concentrations G[c] = 1 M [ ρψ + β 1 c j ln(a 3 2 j c j ) 1 ] M µ j c j dv Ω j=0 ρ(x) = ρ f (x) + M q jc j (x) Lρ = ψ : ε ψ = 4πρ and ψ = 0 on Ω. [ c 0 (x) = a0 3 1 ] M i=1 a3 i c i(x) a j (1 j M): linear size of ions of jth species a 0 : linear size of a solvent molecule c 0 : local concentration of solvent

14 G[c] = 1 M [ ρlρ + β 1 c j ln(a 3 2 j c j ) 1 ] M µ j c j dv Ω ρ = ρ f + M i=1 q ic i j=0 Theorem (B.L. 2009). The functional G has a unique minimizer c = (c 1,...,c M ) which is also the unique local minimizer. It is characterized by the following two conditions: Bounds. There exist θ 1, θ 2 (0, 1) such that θ 1 a 3 j c j (x) θ 2 a.e.x Ω, j = 0, 1,...,M; Equilibrium conditions (i.e.,(δg[c]) j = 0 for j = 1,...,M) a 3 j a 3 0 log ( a 3 0c 0 ) log ( a 3 j c j ) = β (qj ψ µ j ) a.e. Ω, j = 1,...,M. Remark. The bounds are non-physical microscopically! Proof. Similar. Use the convexity and a lemma. Q.E.D.

15 The convexity. ( )[ ( ) ] M M h(u 1,...,u M ) = 1 ak 3 u k log 1 ak 3 u k 1 M ui uj h(u)v i v j = i, k=1 ( M i=1 a3 i v i) 2 1 M k=1 a3 k u k 0 k=1 v 1,...,v M R Lemma (B.L. 2009). Given c = (c 1,...,c M ). There exists ĉ = (ĉ 1,...,ĉ M ) that satisfies the following: ĉ is close to c; G[ĉ] G[c]; there exist θ 1 and θ 2 with 0 < θ 1 < θ 2 < 1 such that θ 1 a 3 j ĉ j (x) θ 2 a.e.x Ω, j = 0, 1,...,M. Proof. By construction. First, treat c 0 = a0 3 (1 M i=1 a3 i c i). Then, treat c j (j = 1,...,M). Q.E.D.

16 Uniform size: Generalized Boltzmann distributions Assume: a 0 = a 1 = = a M =: a. Equilibrium conditions: c j = c 0 a 3 c j e βq jψ (j = 1,...,M) Definition of c 0 : a 3 M j=0 c j = 1 The generalized Boltzmann distributions c j = The generalized PBE cj e βq jψ 1 + a, j = 1,...,M 3 M i=1 c i e βq iψ ε ψ + 4π M q jc j e βq jψ 1 + a 3 M i=1 c i e βq iψ = 4πρ f A variational principle: ψ minimizes the convex functional [ ( )] ε M I[φ] = 8π φ 2 ρ f φ + β 1 a 3 log 1 + a 3 ci e βq iφ dx Ω i=1

17 Nonuniform size: Implicit Boltzmann distributions Equilibrium conditions a 3 j a 3 0 log ( a 3 0c 0 ) log ( a 3 j c j ) = β (qj ψ µ j ), a.e. Ω, j = 1,...,M. An implicit Boltzmann distributions: c j (x) = B j (ψ(x)), Electrostatic neutrality: M q jb j (0) = 0. x Ω, j = 1,...,M. Define V : R R so that V (ψ) = M q jc j = M q jb j (ψ). Implicit PBE and the related variational principle ε ψ 4πV (ψ) = 4πρ f, J[φ] = [ ε Ω 8π φ 2 ρ f φ + V (φ) ] dx.

18 Equilibrium conditions a 3 j a 3 0 log ( a 3 0c 0 ) log ( a 3 j c j ) = β (qj ψ µ j ), a.e. Ω, j = 1,...,M. Set D M = {u ( = (u 1,...,u M ) R M : u j > 0, j = 0, 1,...,M}, u 0 = a0 3 1 ) M a3 j u j, f j (u) = aj 3a 3 0 log ( a0 3u ) ( ) 0 log aj 3u j, j = 1,...,M. Lemma (B.L. 2009). The mapping f : D M R M is C and bijective. Proof. It is clear that f is C. f is injective: det( f ) 0 by det(i + v w) = 1 + v w. f is surjective: f j (u) = r j all z/ u j = f j (u) r j = 0, where z(u) = M [ ( ) ] M u j log a 3 j u j 1 + r j u j. j=0 By constructions, min DM z < min DM z. Hence, z/ u j = 0 for all j = 1,...,M. Q.E.D.

19 Set g = (g 1,...,g M ) = f 1 : R M D M, B j (φ) = g j (β(q 1 φ µ 1 ),...,β(q M φ µ M )), [ B 0 (φ) = a0 3 1 ] M a3 j B j(φ). Electrostatic neutrality: M q jb j (0) = 0. Define M φ V(φ) = q j B j (ξ)dξ 0 φ R Lemma (B.L. 2009). The function V is strictly convex. Moreover, M > 0 if φ > 0, V (φ) = q j B j (φ) = 0 if φ = 0, < 0 if φ < 0, and V(φ) > V(0) = 0 for all φ R with φ 0. Proof. Direct calculations using the Cauchy Schwarz inequality to show that V > 0. Also, use the neutrality. Q.E.D.

20 G[c] = 1 M [ ρlρ + β 1 c j ln(a 3 2 j c j ) 1 ] M µ j c j dv Ω ρ = ρ f + M i=1 q ic i Theorem (B.L. 2009). j=0 The equilibrium concentrations (c 1,...,c M ) and corresponding potential ψ are related by the implicit Boltzmann distributions c j (x) = B j (ψ(x)) x Ω, j = 1,...,M. The potential ψ is the unique solution of the boundary-value problem of the implicit PBE ε ψ 4πV (ψ) = 4πρ f. This is the Euler Lagrange equation of the convex functional [ ε ] J[φ] = 8π φ 2 ρ f φ + V (φ) dx φ H0(Ω). 1 Ω

21 4. Application Issues

22 4.1 Generalized Debye Hückel approximations The implicit PBE: ε ψ 4πV (ψ) = 4πρ f V (φ) V (0) + V (0)φ = V (0)φ for φ 1. Electrostatic neutrality = V (0) = M q jb j (0) = 0. It is shown (B.L. 2009) M V (0) = β qi 2 B i (0) Proposal i=1 [ M ] 2 β i=1 a3 i q ib i (0) M i=0 a6 i B i(0) Solve nonlinear algebraic equations to get B j (0) (j = 0, 1,...,M) and hence V (0). Solve the generalized Debye Hückel approximation ε ψ 4πV (0)ψ = 4πρ f

23 4.2 Potential monotonicity and charge compensation Q O R εm ε w ε w ψ = V (ψ) (r > R) ψ(r) : given ψ( ) = 0 Assume that V : R R is strictly convex and V (0) = 0. Strict monotonicity of the potential: ψ (r) > 0 in (R, ) or ψ (r) < 0 in (R, ). Charge compensation: The induced charge g( R) = R< x < R M q j c j (x)dv decreases (if Q > 0) or increases (if Q < 0) from g(r) = 0 to g( ) = Q Vol(B(0, R)).

24 4.3. Wall-mediated like-charge attractions ψ = V (ψ) ψ is a constant on the walls inside walls and outside balls ψ is a constant on the boundary of balls Assume that V : R R is strictly convex and V (0) = 0. The electrostatic surface force is given by F = 1 2 balls ( nψ) 2 n ds Force repulsive: F {unit horizontal vector to the center} > 0.

25 5. Conclusions

26 Summary Electrostatic free-energy minimization provides bonds on concentrations, a limitation of the PB theory. A size-modified PB theory is developed. Generalized explicit and implicit Boltzmann distributions are obtained for the case of uniform and nonuniform ionic and molecular sizes, respectively. A generalized Debye Hückel approximation is obtained that can be used for numerical calculations. The new theory has the essential features same as the classical one. Thus it is not able to explain the geometry-mediated like-charge attraction.

27 Outlook Bridge the continuum and atomistic models of electrostatics. Statistical mechanics basis of the continuum theory with nonuniform ionic sizes. Potential monotonicity and charge compensation for general domains. The relation between the PB theory and the generalized Born model.

28 Thank you!