Modeling cells in tissue by connecting. electrodiffusion and Hodgkin Huxley-theory.

Transcription

1 Modeling cells in tissue by connecting electrodiffusion and Hodgkin Huxley-theory. Brigham Young University Department of Mathematics Viktoria R.T. Hsu University of Utah Department of Mathematics

2 Outline Introduction: Motivation and Roadmap. Review: The Classic Hodgkin Huxley Neuron Model. Toward Tissue Modeling: The QSSA of Electrodiffusion. Simpler Models by Further Approximation of the QSSA. Comparing Models of Electrodiffusion to Hodgkin Huxley: approach to equilibrium of a cell with ion channels but no pumps, a measure of performance for comparison of the models, approach to rest after an action potential; channels and pumps. Summary and Conclusions. Future Work.

3 Schematic of a Neuron Cell

4 Different Shapes of Neuron Cells

5 Limbic System of the Human Brain

6 Neuron Cells in the Hippocampus

7 Roadmap HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak)

8 HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak) The Classic Hodgkin Huxley Model

9 The Classic Hodgkin Huxley Neuron Model Assumptions: Infinite buffering: ion concentrations and cell volume remain constant. Passive transport: ion channels pass an Ohmic current with voltage dependent conductance. Active transport: ion pumps maintain homeostasis (living state); represented by reversal potentials. Quantify: the membrane behaves like a leaky capacitor. Parameters are fit to data from giant squid axon.

10 The Classic Hodgkin Huxley Neuron Model The Model Equations: C m dv dt = ḡ K n 4 (V V K ) ḡ Na m 3 h (V V Na )... ḡ L (V V L ) + I app (1) dm dt = α m (1 m) + β m m (2) dn dt = α n (1 n) + β n n (3) dh dt = α h (1 h) + β h h, (4) where α x and β x, for x {m, n, h}, are functions of v = V V, the difference of the cross-membrane potential from the resting potential.

11 The Classic Hodgkin Huxley Neuron Model The Fast-Slow Phase-Plane: m has fastest dynamics, so m m, its resting value. Eliminate h by using FitzHugh s observation that n + h 0.8. V and n then satisfy: C m dv dt = ḡ K n 4 (V V K ) ḡ Na m 3 (0.8 n) (V V Na )... ḡ L (V V L ) + I app (5) dn dt = α n (1 n) + β n n. (6)

12 The Classic Hodgkin Huxley Neuron Model Flow Field in the Fast-Slow Phase-Plane:

13 The Classic Hodgkin Huxley Neuron Model Sub-Threshold Stimulus:

14 The Classic Hodgkin Huxley Neuron Model Signal Generation by Super-Threshold Stimulus:

15 HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak) That s Great! What Improvements Are Needed?

16 A Need for Accurate Modeling of Cells in Tissue Cells in tissue experience variation of their external environment, and volume. Variations can be large under conditions like ischemia (lack of metabolic supply). Recall: present multi-compartment models assume isolated cell in buffer. phenomenological, so electroneutrality is neglected, external concentrations and cell volume are static. Propose: extend multi-compartment models to single-cell micro-environment. physically consistent model for signal generation with varying external ion concentrations and cell volume. Challenge: step away from the model circuit assumption (for now). in finite volume one has to consider mass conservation and electroneutrality. Thus, model charge-carrier transport by electrodiffusion.

17 HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak) From Electrodiffusion To Its QSSA.

18 Charge-Carrier Transport - the Equations Alike Semiconductor-Device Equations: c i t = [D i ( c i + z i ϕc i ) + S i ] (7) (ε ϕ) + i z i c i = N (8) c = species concentration ϕ = electro-static potential D = Diffusion coefficient ε = Dielectric coefficient z = species valency i = species index

19 The Single-Cell Micro-Environment D D D bk m bk bk m bk cell membrane R L internal compartment external compartment

20 Simplifying Assumptions and Reduction to 1D No ionic species are involved in chemical reactions. All medium is homogeneous and neutral. The membrane is of uniform width and is impermeable to some ionic species.

21 Charge-Carrier Transport - 1D Equations Electrodiffusion and Poisson s Equations: c i t = [ ( )] ci D i x x + z ϕ i x c i (9) ( ε ϕ ) + x x i z i c i = 0 (10) c = species concentration ϕ = electro-static potential D = Diffusion coefficient ε = Dielectric coefficient z = species valency i = species index

22 More Assumptions Reduce the System to Its QSSA Diffusion in membrane medium is much slower than diffusion in bulk. The space occupied by membrane medium is small compared to each bulk compartment.

23 The Domain in 1D - End of Membrane Impermeability in C i, =0 mid membrane C i, = internal region external region out + p p p p p (internal bulk) p p (external bulk) p p x L 0 R membrane region

24 The QSSA in Membrane Region PNP System After Assumptions and at Steady-State: ( ) ci J i = D i x + z ϕ i x c i (11) ( ε ϕ ) + x x i z i c i = 0 (12) c = species concentration ϕ = electro-static potential J = species flux density D = Diffusion coefficient ε = Dielectric coefficient z = species valency

25 Dynamic Equations of the QSSA dc R i dt = A c v out J i (13) dc L i dt = A c v in J i (14) From Almost Newton (AN) numerical method: J i = D i c i (R) exp (z i ϕ (R)) c i (L) exp (z i ϕ (L)) R L exp (z iϕ (s)) ds =? (15) ϕ (x) =? (16)

26 HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak) Approximating the QSSA to Obtain Simpler Models of Electrodiffusion.

27 Potential Profile from QSSA

28 Dynamic Equations of the QSSA - with ion channels dc R i dt = A c v out J i, for c i {Na; K; Cl} (17) dx dt dc L i dt = A c v in J i, for c i {Na; K; Cl} (18) = α (V ) (1 X) β (V ) X, for X {m; n; h} (19) J i = D i (gating) c i (R) exp (z i ϕ (R)) c i (L) exp (z i ϕ (L)) R L exp (z iϕ (s)) ds =?, from AN method (20) ϕ = F V R 0 T = ϕ R ϕ L =?, from AN method (21)

29 Dynamic Equations of the CFA dx dt dc out i dt dc in i dt = A c v out J i, for c i {Na; K; Cl} (22) = A c v in J i, for c i {Na; K; Cl} (23) = α (V ) (1 X) β (V ) X, for X {m; n; h} (24) J i = D i (gating) z i ϕ δ ϕ = F V R 0 T = ϕ R ϕ L = cout i e zi ϕ c in i e z i ϕ 1 ( δ v out εa c z i z i c out i ) (25) (26)

30 Relating Parameters of QSSA and CFA to those of HH-type Models With the following assumptions, ln (c i ) x ϕ x (0) 1 δ (ϕ R ϕ L ) = F V δr 0 T (0) 1 δ ( ( ) ( )) ln c out i ln c in 1 i = δ ln ( ) c out i c in i (27) (28) the PNP system yields HH-type currents and voltage dynamics with g i = z2 i D ic i (0) C m ε C m = εa c δ (29) (30)

31 Dynamic Equations of the Hodgkin Huxley-Type Model (HH-plk) HH C m dv dt + i I i = I app dx dt = α (V ) (1 X) β (V ) X, for X {m; n; h} I i = g i (gating) (V V i ), for i {Na; K; Cl} pump leak dc out i dt dc in i dt c in i = A c v out = A c v in I i z i F I i z i F ( ) V i = R 0T z i F ln c out i, for i {Na; K; Cl}

32 HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak) Comparing the Dynamics to Eqlb. of QSSA, CFA, and HH-plk with Ion Channels but No Pumps.

33 Concentration Dynamics to Equilibrium:

34 Current Density Dynamics to Equilibrium:

35 Electro-Static Potential Dynamics to Equilibrium:

36 HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak) How can one judge which model works best?

37 A Measure of Maintaining Net-Electroneutrality: ( ε ϕ ) x x = i z i c i (31) ( 1 ε ϕ ) 2 2 x (R) 1 ( ε ϕ ) 2 2 x (L) = ε c R i i c L i + 2 ε J i D i (32) i J i D i i c out i c in i 2 ε (33)

38 A Measure of Maintaining Net-Electroneutrality

39 Relative Measure of Maintaining Net-Electroneutrality

40 HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak) Comparing the Dynamics to Rest of CFA, HH-plk, and HH-cls with Ion Channels and Pumps.

41 A Simple Model for Ion Pump Currents Motivation: ( ) ( I i,cls = g i V V NP i + gi V NP i ) Vi,rest NP (34) ( ) I i,plk = g i V V NP i + I pump i (35) Specific concentration gradients are maintained at steady-state by: I pump i = I pump i,rest ( gpump i V NP i V NP i,rest), (36) I pump i,rest, pump current density at rest, equals channel current density at rest. g pump i, conductance of the pump. V NP i, present Nernst potential of species i. Vi,rest NP, Nernst potential of species i at rest, pump is to maintain.

42 Updated Dynamic Equations of CFA and HH-plk dc out i v out dt = A c ( J ch i + J pump ) i, (37) dc in i v in dt = A c ( J ch i + J pump ) i, (38) J pump i = 1 z i F Ipump i J ch i, the previously defined, passive channel flux density of the CFA or HH-plk model.

43 Current Densities by CFA, HH-plk and HH-cls

44 Action Potential by CFA, HH-plk and HH-cls

45 Relative Measure of Maintaining Net-Electroneutrality

46 Summary Electrodiffusion (ED) was approximated to yield several models of ED: QSSA, CFA and HH-plk. The ODE-based models, CFA and HH-plk, were extended to models of electric cell signalling and compared to HH-cls. A measure of the maintenance of net-electroneutrality (net-en) was used to compare the methods performance. No electroneutrality conditions were explicitly enforced. HH cls (classic HH model) HH GHK (GHK currents) QSSA electrodiffusion PNP system model circuit (leaky capacitor) CFA (const. field) HH plk (pump leak)

47 Future Work Incorporate applied currents and volume dynamics (swelling). Perform a full analysis of the CFA model. Study the importance of cell-cell interactions through gap junctions versus other ephaptic means when two cells share a small external environment. Study interactions between neuron and glia cells. Incorporate energetics into the CFA model by, for example, including ATPsensitive, and ATP-consuming, pumps and transporters.

48 Thank you......for Your attention.

49 Questions...???

50 EXTRA SLIDES

51 A Measure of Closeness to the Full PDE: ( ε ϕ ) = z i c i (39) x x i ε ϕ ( ε ϕ ) = ε ϕ z i x x x x c i (40) i ( ( 1 ε ϕ ) ) 2 = ε c i x 2 x x + J i (41) D i i ( 1 ε ϕ ) 2 2 x (out) 1 ( ε ϕ ) 2 2 x (in) = ε c out i c in i + 2 ε J i (42) D i 0 = i i c out i c in i J i D i = i i + 2 ε J i D i (43) c out i c in i 2 ε (44)

52 Hodgkin Huxley Gating Dynamics.

53 Coefficients of the HH-type Gating Variables I Na = g Na m 3 h (V V Na ), I K = g K n 4 (V V K ), I Cl = g L (V V L ) (45) α x and β x, for x {m, n, h}, are the following functions of v = V V, the difference of the cross-membrane potential from the resting potential: α m = v exp( 25 v 10 ) 1 α n = 0.07exp ( v 20 α h = v ) exp( 10 v 10 ) 1 β m = 4exp ( v 18 ) β n = 1 exp( 30 v 10 )+1 β h = 0.125exp ( v 80).

54 More Intuitive Gating Equations Defining the new functions x and τ x for x {m, n, h} according to x = α x α x + β x and τ x = 1 α x + β x (46) allows us to write the original gating equations in a more intuitive form, namely τ m (v) dm dt = m (v) m (47) τ n (v) dn dt = n (v) n (48) τ h (v) dh dt = h (v) h. (49)

55 What is a Quasi Steady-State Approximation?

56 Quasi Steady-State Approximation, in general εx t = F (x, y) y t = G(x, y) (50) 0 = F (x, y) y t = G(x, y) x = f(y) y t = G(f(y), y) (51) (52)

57 Detailed Derivation of the QSSA.

58 The fully transient PDE model in 1D Electro-Diffusion and Poisson s Equations: ( c i t = DB i c i ( x + z i ϕ x c i) for L x < m x 2 c i t = DM i c i x + z i ϕ x c i) for m x 2 x m 2 c i t = DB i ( c i x + z i ϕ x c i) x for m 2 < x R (53) (ε B ϕ x ) x = i zi c i for L x < m 2 (ε M ϕ x ) x = i zi c i for m 2 x m 2 (ε B ϕ x ) x = i zi c i for m 2 < x R, (54) It is understood that the natural length scales of each compartment are R m 2 = v out A and L m 2 = v in A.

59 Rescale time and space for different time scales ( ) ( ) Assume that max i D i M mini D i B. Rescale space by x = 2x m and time by t = ( ) 2 2 m D min M t, where Dmin M = min ( ) i D i M, then σb i cī t = ( c ī x + z i ϕ x c i) x for 2L m x < 1 σm i cī t = ( c ī x + z i ϕ x c i) x for 1 x 1 σb i cī t = ( (55) c ī x + z i ϕ x c i) x for 1 < x 2R m, in which σm i = Dmin M DM i = O (1) and σb i = Dmin M DB i 1.

60 Relaxation of membrane region to steady-state Neglect small terms σb i cī t and approximate the dynamics by 0 = ( c ī x + z i ϕ x c i) x for 2L m x < 1 σm i cī t = ( c ī x + z i ϕ x c i) x for 1 x 1 0 = ( (56) c ī x + z i ϕ x c i) x for 1 < x 2R m, which implies in turn that c i ( t, x) = c i in for 2L m x < 1 σm i cī t = ( c ī x + z i ϕ x c i) x for 1 x 1 (57) c i ( t, x) = c i out for 1 < x 2R m.

61 Membrane at steady-state and bulk changes slowly Assume that 1 2R m 2L m bulk compartment, d d t 1 2L m c i ( t, x) dx = Rescaling time once more by τ = where γ i out = σi M σ max M γin i dc i in dτ and track the total mass by integrating over each ( ) 2L dc i m 1 in d t σ max M t ( 2R m 1 ) = 1 σ i M ( c ī x + z i ϕ x c i) x= 1. (58), where σmax M = max i ( σ i M ) = 1, then = ( c ī x + z i ϕ x c i) x= 1 γm i ci τ = ( c ī x + z i ϕ x c i) x for 1 x 1 γout i dc i out dτ = ( c ī x + z i ϕ x c i), x=1 (59) = O (1), γin i = σi M( 2L m 1 ) σm max ( 2R m 1 ) = O (1), and γi M = σi M σm max ( 2R m 1 ) 1.

62 Membrane at steady-state and bulk obeys ODEs Neglect the small terms γm i ci τ and approximate the dynamics by γin i dc i in dτ = ( c ī x + z i ϕ x c i) x= 1 0 = ( c ī x + z i ϕ x c i) x for 1 x 1 γout i dc i out dτ = ( c ī x + z i ϕ x c i). x=1 (60)

63 QSSA for Relaxation to Donnan Equilibrium, I Equations (62) and (64) are to be satisfied together with Poisson s equation. Concentration profiles of permeant species in the membrane region are c ī x + z i ϕ x c i = ci oute zi ϕ(1) c i in ezi ϕ( 1) 1 1 ezi ϕ( x) d x = const. for 1 x 1 or (61) c i ( x) = e zi ϕ( x) ci in ezi ϕ( 1) 1 x ezi ϕ( x) d x + c i oute zi ϕ(1) x 1 1 ezi ϕ( x) d x 1 ezi ϕ( x) d x, (62) while species impermeant to the membrane have Boltzmann densities, c ī x + z i ϕ x c i = 0 for 1 x 1 or (63) { c i c ( x) = i in ezi (ϕ( 1) ϕ( x)) for 1 x < 0 c i oute zi (ϕ(1) ϕ( x)) (64) for 0 < x 1.

64 QSSA for Relaxation to Donnan Equilibrium, II A set of ODEs in time governs the dynamics of the bulk concentrations, c i ( x) = c i in for 2L m x < 1 and ci ( x) = c i out for 1 < x 2R m, namely γin i dc i in dτ = ci oute ziϕ(1) c i in eziϕ( 1) 1 1 ezi ϕ( x) d x (65) γout i dc i out dτ = ci oute zi ϕ(1) c i in ezi ϕ( 1) 1 1 ezi ϕ( x) d x, (66) where γ i out = σi M σ max M and γin i = σi M( 2L m 1 ) σm max ( 2R m 1 ).

65 Relaxation Time to Donnan Equilibrium Reconnecting the time τ with the original time t delivers an estimate for the relaxation time to Donnan equilibrium. In particular, τ = αt with α = m 2 DM min ( ) R m (67) and we approximate the dynamic approach to Donnan equilibrium of the bulk concentrations by c i in (t) = c i in ( ) ( c i in ( ) c i in (0) ) e αt (68) c i out (t) = c i out ( ) ( c i out ( ) c i out (0) ) e αt, (69) where c i in,out (0) are the initial bulk concentrations, and ci in,out ( ) are the final bulk concentrations at Donnan equilibrium. 2

66 QSSA: PNP Equation and Boundary Conditions.

67 Setup for Mid-Membrane Impermeability in C i, =0 mid membrane C i, = internal region external region out + p p (internal bulk) p p (external bulk) p p x L 0 R boundary layer and membrane

68 The Electro-Diffusion and Poisson System in 1D Recall that: c i t = [ ( )] ci D i x x + z ϕ i x c i (70) ( ε ϕ ) + x x i z i c i = 0 (71) c = species concentration ϕ = electro-static potential D = Diffusion coefficient ε = Dielectric coefficient z = species valency i = species index

69 Poisson-Nernst-Planck System in Membrane Region After Assumptions and at Steady-State: ( ) ci J i = D i x + z ϕ i x c i (72) ( ε ϕ ) + x x i z i c i = 0 (73) c = species concentration ϕ = electro-static potential J = species flux density D = Diffusion coefficient ε = Dielectric coefficient z = species valency

70 Solving Nernst-Planck s Equation Smoothness and continuity of ϕ at mid-membrane: c i (R) e ziϕ(r) c i (L) e z iϕ(l) J i = D i R L exp (z, (74) iϕ (s)) ds Species permeant to the membrane obey: c i (x) = e z iϕ(x) c i (L) e z iϕ(l) R x ez iϕ(s) ds + c i (R) e z iϕ(r) x L ez iϕ(s) ds R L ez iϕ(s) ds (75) Species impermeant to the membrane have Boltzmann densities: c i (x) = c i (L) e z i(ϕ(x) ϕ(l)) for x < 0 c i (R) e z i(ϕ(x) ϕ(r)) for 0 < x. (76)

71 The Poisson-Nernst-Planck (PNP) Equation Notation: α x j = c i (x) τ x j = c i (x) (77) permeant i z i = j trapped i z i = j Poisson-Nernst-Planck (PNP) Equation, to be Solved with an Almost-Newton (AN) Method: ( ε ϕ ) = je [τ jϕ(x) j L e jϕ(l) H ( x) + τj R e jϕ(r) H (x) +... x x all j... + αl j ejϕ(l) R x ejϕ(s) ds + α ] j Rejϕ(R) x L ejϕ(s) ds R, (78) L ejϕ(s) ds

72 Boundary Conditions?

73 Charge-Carrier Transport in Various Disciplines Neumann and Dirichlet BCs on el. potential Mathematical Device: PNP Equations Dirichlet BCs on el. potential Charge Carrier Transport Natural Device: ionic species cell membranes current and el. potential caused by carrier concentration gradient Physical Device: holes and electrons semiconductors current caused by applied el. potential

74 Natural Boundary Conditions by Gauss Law Integrating Poisson s equation over the entire domain and using that the system is net-electroneutral, ε R L 2 ϕ x 2 dx = i ε ϕ x (R) ε ϕ x Since L represents the interior of the cell, ϕ x (L) = 0 = ϕ x R z i c i dx (79) L (L) = 0. (80) (R). (81) Two Neumann boundary conditions do not define a well-posed problem!

75 Almost-Newton Method for Solving the QSSA.

76 The PNP Equation - Linearize for a Newton-Type Iteration Scheme given ϕ, solve for δ: ε 2 ( ϕ + δ) = x2 all j je j ϕ(x) [A j (x) (1 jδ (x)) + jb j (x) δ (L) + jc j (x) δ (R) jd j (x) x L δ (s) e j ϕ(s) ds + je j (x) R x ] δ (s) e j ϕ(s) ds, (82) where A j through E j are highly nonlinear.

77 Coefficients of the Linearized PNP Equation A j (x) = B j (x) + C j (x) (83) B j (x) = e j ϕ(l) (τ L j H ( x) + α L j R x ej ϕ(s) ds R L ej ϕ(s) ds ) (84) C j (x) = e j ϕ(r) (τ R j H (x) + α R j x ) L ej ϕ(s) ds R L ej ϕ(s) ds (85) D j (x) = αr j ej ϕ(r) α j L R L ej ϕ(s) ds ej ϕ(l) R x ej ϕ(s) ds R L ej ϕ(s) ds (86) E j (x) = αr j ej ϕ(r) α j L R L ej ϕ(s) ds ej ϕ(l) x L ej ϕ(s) ds R L ej ϕ(s) ds (87)

78 Coefficients of the Almost-Newton (AN) Iteration Scheme D j (x) replaced by D j = αr j R L ej ϕ(s) ds ej ϕ(r) (88) E j (x) replaced by E j = αl j R L ej ϕ(s) ds. (89) ej ϕ(l) Comparing the equations defining AN and FN implies: R L δ (s) e jϕ(s) ds = 0 for all j. (90)

79 Comparison of the FN, MG and AN Methods FN faces problems with catastrophic cancellation, especially as flux densities become large. (D j and E j are similar terms with opposite signs, and proportional to flux densities.) MG uses A j through C j only, D j and E j are neglected. Thus, the system matrix is sparse but terms proportional to flux densities are neglected. MG as well faces problems of convergence when flux densities become large. AN uses modified, simpler D j and E j, which arise when the PNP equation is linearized under the assumption that R L ejϕ(s) = const.. AN requires two Neumann BCs and performs well even for large flux densities.

80 Results: Convergence of PNP-solvers.

81 Results: Steady-State Study of PNP-solvers.

82 Summary for Almost-Newton Method (AN) The solution by AN is at least as accurate as solutions by MG or FN. Physiological bulk concentrations result in large negative flux densities, at which AN converges more efficiently than MG or FN. For any flux density, AN converges with roughly the same, low number of iterations.

83 Including Sources in the QSSA

84 The Generalized PNP Equation, Including Sources Notation: σ j = permeant i z i = j S i D i σ j = trapped i z i = j S i D i (91) Generalized Poisson-Nernst-Planck (PNP) Equation: ( ε ϕ ) = x ) je [(τ jϕ(x) j L e jϕ(l) σ j (s) e jϕ(s) ds H ( x) +... x x L all j ( R ) τj R e jϕ(r) + σ j (s) e jϕ(s) ds H (x) +... x ( x α j L e jϕ(l) L ( R α j R e jϕ(r) + x ) R σ j (s) e jϕ(s) x ds ejϕ(s) ds R L ejϕ(s) ds +... ) x ] σ j (s) e jϕ(s) L ds ejϕ(s) ds R L ejϕ(s) ds

85 The generalized PNP Equation - Linearized, Including Sources given ϕ, σ, and σ, solve for δ: ε 2 ( ϕ + δ) = x2 all j je j ϕ(x) [A j (x) (1 jδ (x)) + jb j (x) δ (L) + jc j (x) δ (R) jd j (x)... +jf j (x)... +jk j (x) x L x L x L δ (s) e j ϕ(s) ds + je j (x) δ (s) σ (s) e j ϕ(s) ds + jg j (x) δ (s) σ (s) e j ϕ(s) ds + jm j (x) where A j through M j are highly nonlinear. R x R x R x δ (s) e j ϕ(s) ds δ (s) σ (s) e j ϕ(s) ds ] δ (s) σ (s) e j ϕ(s) ds

86 Validity of the QSSA.

87 1st Set: Sharp Initial Conditions

88 Flux Density Dynamics to Donnan Eq. - sharp ICs:

89 Electro-Static Potential Dynamics to Donnan Eq. - sharp ICs:

90 2nd Set: Steady-State Initial Conditions

91 Flux Density Dynamics to Donnan Eq. - St.State ICs:

92 Electro-Static Potential Dynamics to Donnan Eq. - St.State ICs:

93 3rd Set: Far From Eqlb. Initial Conditions

94 Flux Density Dynamics to Donnan Eq. - Far From Eq. ICs:

95 Electro-Static Potential Dyn. to Donnan Eq. - Far From Eq. ICs:

96 Summary for Approach to Donnan Equilibrium Boundary layer is established within a few ms. Quasi steady-state assumption holds well for the kinetic approach to Donnan equilibrium. (accuracy!) An ODE system based on AN takes a few seconds to solve, whereas the full PDE takes 32 hours to solve. (efficiency!) Thus, the implementation of the QSSA using AN yields an accurate and efficient means of modeling electrodiffusion and, in particular, the kinetic approach to Donnan equilibrium.

97 The Constant Field Approximation, CFA.

98 Deriving the Constant Field Approximation (CFA) The corresponding CFA uses and thus dϕ dx (x) = ϕ (x) ϕ (L) = dϕ εa c dx (0) = v in z i c in i, (92) i 0, for x < m 2 v in εa c i z ic in i, for m 2 x m 2 (93) 0, for m 2 < x 0, for x < m 2 ( ) x + m vin 2 εa c i z ic in i, for m 2 x m 2 (94) m v in i z ic in i, for m 2 < x. εa c

99 Deriving the Constant Field Approximation (CFA) With ϕ (x) piecewise linear, the steady-state flux-density, J i can be simplified: J i = D i c i (R) e z iϕ(r) c i (L) e z iϕ(l) R L exp (z iϕ (s)) ds (95) J i = D i z i ϕ x (0) cout i e zi ϕ c in i e z i ϕ 1 (96) J i = D i z i ϕ δ cout i e zi ϕ c in i e z i ϕ 1 (97)

100 Verify that the CFA of the QSSA Holds at Far-From-Eqlb. Steady-States.

101 Potential Profiles by AN and CFA, No Species Trapped

102 Error in CFA Potential Profile, No Species Trapped

103 Closeness of Bulk Profile to Eqlb. Profile, No Species Trapped

104 Difference of Membrane Profile to Eqlb. Profile, No Species Trapped

105 Summary: CFA for End-of-Membrane Impermeability Bulk regions are almost equilibrated, even at far-from-equilibrium steady-states (relative error of order ). The membrane region is far from equilibrated at far-from-equilibrium steadystates (relative error of order 10 1 ). The steady-state potential profile and cross-membrane potential difference of the QSSA are approximated reasonably well by the CFA (relative error within 5%).

106 THE END