Unformatted text preview:

Key Concepts for this sectionElectroneutrality Gel Electrophoresis1: Lorentz force law, Field, Maxwell’s equation2: Ion Transport, Nernst-Planck equation3: (Quasi)electrostatics, potential function,4: Laplace’s equation, Uniqueness5: Debye layer, electroneutralityGoals of Part II: (1) Understand when and why electromagnetic (E and B) interaction is relevant (or not relevant) in biological systems.(2) Be able to analyze quasistatic electric fields in 2D and 3D. Key Concepts for this sectionElectroneutralityProtein A++-+++Protein B+-----~ Debye length (κ-1)Buffer counterions0()xxeκ−Φ=Φ01/ 222001e tanh224() ln ,1e tanh4xxzFzFcRTRTxzFzF RTRTκκκε−−⎡⎤Φ⎛⎞+⎜⎟⎢⎥⎛⎞⎝⎠⎢⎥Φ= =⎜⎟Φ⎛⎞⎢⎥⎝⎠−⎜⎟⎢⎥⎝⎠⎣⎦When 0zF RTΦ<<0 0.5 1 1.5 2 2.5 30246810120 0.5 1 1.5 2 2.5 300.511.52()zF xRTΦ()zF xRTΦxκxκ02zFRTΦ=010zFRTΦ=D-HexactexactD-HExact solutionDebye-Huckel approximation0 0.5 1 1.5 2 2.5 3024680 0.5 1 1.5 2 2.5 30.90.9511.051.11.150()cxcxκxκ02zFRTΦ=00.1zFRTΦ=0()cxcc-(counterion)c+ (co-ion)c+ (co-ion)c-(counterion)When 00()zF RT ze kTΦ<< Φ<<thermal energy >> electrical potential energy(diffusion dominates.)When 00()zF RT ze kTΦ>> Φ>>thermal energy << electrical potential energy(drift dominates. significant charge accumulation)0.1Msucrose0.01MsucroseMembrane permeable to sucrose0.1MKCl0.01MKClMembrane permeable only to K+??c1=0.1MKClc2= 0.01MKCl+++++-----δNernst Equilibrium Potentialc: K+ concentrationVm+-[]00212212 1 2110ln ( ) ( 0)ln lnxxxxdc dDEuc Edx dxdc dDudxcdxdc dDudxcdxcDuxxcccDRTuc zFcδδδ====Φ−+⋅⋅= =−Φ−=⋅Φ−=⋅⎛⎞−=Φ=−Φ=⎜⎟⎝⎠⎛⎞ ⎛⎞ΔΦ = Φ − Φ = =⎜⎟ ⎜⎟⎝⎠ ⎝⎠∫∫Nernst Equilibrium potentialDiffusion of charged particles -> generate electric field-> stops diffusion of ionsMembrane permeable only to K+Quasi-Electrostatics()eEερ∇⋅ =JG0B∇⋅=JG1eEBJtεμ∂∇× = +∂GGGBEt∂∇× =−∂GG000E∇× =G()0SCEds Edl∇×⋅= ⋅=∫∫GGGGvv1C2ab221( ) 1( )0abEdl Edl⋅−⋅=∫∫GGGGElectrostatic force : conservativePotential function Φ can be defined.21(2) (1)EdlΦ−Φ =− ⋅∫GG2() ( )(' )eeEEPoisson s Equationεερρε=−∇Φ ∇⋅ =∇⋅ − ∇Φ =∇Φ=−GG2cct∂∇=∂0q∇⋅ =GqkT=− ∇G20c∇=(Fick’s second law)(steady-state diffusion)(Fourier’s law for heat conduction)(conservation law for heat)20T∇=(steady heat flow)However, biomolecules in the system do not generate E-field, since they are shielded by counterions (electroneutrality)…….It all comes down to solving…..20( ' )Laplace s Equation∇Φ=Φ1=0ElectrostaticsΦ2=0Φ3=0Φ4=0Φ5=0Φ=?c4=0Steady state diffusionc3=0c5=0c1=0c2=0c=0 20c∇=T4=0Thermal conductionT3=0T5=0T1=0T2=0T=0 20T∇=Uniqueness of Solution22;;eaaiiebbiion Son Sρερε∇⋅Φ=− Φ=Φ∇⋅Φ=− Φ=Φ20; 0dabddon SΦ=Φ−Φ∇⋅Φ = Φ =i(satisfy Laplace Eq.)Let’s assume two different solutions, Φaand ΦbThen defineS1S2S3S4S5200ddion all S∇⋅Φ =Φ=0dΦ=Answer:for everywhere 0ab∴Φ−Φ=Gel ElectrophoresisΦ=V0Pt electrodeGel (ε, σ)Plastic (σ =0)biomolecules()0JEσ∇⋅ =∇⋅ =JGJG0( )BE electrostaticst∂∇× =− =∂GG0eJtρ∂∇⋅ =− =∂JG(steady state, no charge accumulation)0E∇⋅=GE=−∇ΦG20∇Φ=0J∇⋅ =JG00yyyorWJEyσ=∂Φ===∂xyLWΦ=V0 when x=LΦ=0 when x=0(no charge accumulation)00yy=∂Φ=∂0yWy=∂Φ=∂J=0 (insulator)ˆxJJx=JGBoundary Conditions (For EQS approximation)()eEερ∇⋅ =JG0E∇× =GJtρ∂∇⋅ =−∂JG11 2 2ˆ()snE Eεεσ⋅−=JJGJJG121 2tangential tangentialˆˆ()nE nE E E×=× =JJGJJGGG11 2 2ˆ()snE Etσσσ∂⋅− =−∂JJGJJGFrom H&MFigure 5.3.1 (a) Differential contour intersecting surface supporting surface charge density. (b) Differential volume enclosing surface charge on surface having normal n.Courtesy of Herman Haus and James Melcher. Used with permission.Source: http://web.mit.edu/6.013_book/www/220()dxaxbdxΦ=→Φ=+22220xy∂Φ ∂Φ+=∂∂1D case: 2D case: (1,)nmΦ−(, )nmΦ(, 1)nmΦ+(, 1)nmΦ−(1,)nmΦ+x (n)y (m)1(,)(1,)(,)2nm nm nmx∂Φ+=Φ+−Φ∂1(,)(,)(1,)2nm nm nmx∂Φ−=Φ−Φ−∂2211(, ) ( , ) ( , )22( 1,) ( 1,)2(,)nm n m n mxx xnm nm nm∂Φ ∂Φ ∂Φ=+−−∂∂ ∂=Φ + +Φ − − Φ2222(, ) (, )( 1,) ( 1,) (, 1) (, 1)4(,) 0nm nmxyn m n m nm nm nm∂Φ ∂Φ+=∂∂Φ + +Φ − +Φ + +Φ − − Φ =( 1, ) ( 1, ) (, 1) (, 1)(, )4nm nm nm nmnmΦ + +Φ − +Φ + +Φ


View Full Document

MIT 20 330J - Laplace’s equation

Download Laplace’s equation
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Laplace’s equation and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Laplace’s equation 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?