1. 1 Chemical Bonds are lines Surface is Electrical Potential Red is positive Blue is negative Chemist’s Vie w Ion Channels Proteins with a Hole All Atoms View Figure by Raimund Dutzler ~30 Å

2. 2 Channels form a class of Biological Systems that can be analyzed with Physics as Usual Physics-Mathematics-Engineering are the proper language for Ion Channels in my opinion ION CHANNELS: Proteins with a Hole

3. 3 Physics as Usual along with Biology as Usual Ion Channels can be analyzed with “Why think?. Exhaustively experiment. Then, think” Claude Bernard Appropriate when nothing was known of Inverse Problems! Cited in The Great Influenza, John M. Barry, Viking Penguin Group 2004

4. 4 Channels control flow in and out of cells ION CHANNELS as Biological Objects ~5 µm

5. 5 ~30 Å Ion Channels are the Main Molecular Controllers “Valves” of Biological Function Figure by Raimund Dutzler

6. 6 Channels control flow of Charged Spheres Channels have Simple Invariant Structure on the biological time scale. Why can’t we predict the movement of Charged Spheres through a Hole? ION CHANNELS as Physical Devices

7. 7 Physical Characteristics of Ion Channels Natural Nanodevices Ion channels have Selectivity. K channel selects K + over Na + by ~10 4. Ion channels Gate/Switch in response to pH, voltage, chemical species and mechanical force from conducting to nonconducting state. Ion channels have VERY Large Charge Densities critical to I-V characteristics and selectivity (and gating?) Ion channels allow Mutations that modify conductance, selectivity, and function. Ion channels are Device Elements that self-assemble into perfectly reproducible arrays. Ion channels form Templates for design of bio-devices and biosensors. ~30Å Figure by Raimund Dutzler

8. 8 Channels Control Macroscopic Flow with Atomic Resolution ION CHANNELS as Technological Objects

9. 9 Ion Channels are Important Enough to be Worth the Effort

10. 10 Goal: Predict Function From Structure given Fundamental Physical Laws

11. 11 Current in One Channel Molecule is a Random Telegraph Signal

12. Voltage Step Applied Here (+ 80 mV; 1M KCl) Gating: Opening of Porin Trimer John Tang Rush Medical Center

13. 13 Single Channel Currents have little variance John Tang Rush Medical Center

14. 14 Lipid Bilayer Setup Recordings from a Single Molecule Conflict of Interest

15. 15 Patch clamp and Bilayer apparatus clamp ion concentrations in the baths and the voltage across membranes. Patch Clamp Setup Recordings from One Molecule

16. 16 Voltage in baths Concentration in baths Fixed Charge on channel protein Current depends on John Tang Rush Medical Center OmpF KCl 1M 1M || G119D KCl 1M 1M || ompF KCl 0.05 M 0.05M || G119D KCl 0.05 M 0.05M || Bilayer Setup

17. 17 Current depends on type of ion Selectivity John Tang Rush Medical Center OmpF KCl 1M 1M || OmpF CaCl 2 1M 1M || Bilayer Setup

18. 18 Goal: Predict Function From Structure given Fundamental Physical Laws

19. 19 Structures… Location of charges are known with atomic precision (~0.1 Å) in favorable cases.

20. 20 Charge Mutation in Porin Ompf Structure determined by x-ray crystallography in Tilman Schirmer’s lab Figure by Raimund Dutzler G119D

21. 21 Goal: Predict Function From Structure given Fundamental Physical Laws

22. 22 But … What are the Fundamental Physical ‘Laws’?

23. 23 Verbal Models Are Popular with Biologists but Inadequate

24. 24 James Clerk Maxwell “I carefully abstain from asking molecules where they start… I only count them…., avoiding all personal enquiries which would only get me into trouble.” Royal Society of London, 1879, Archives no. 188 In Maxwell on Heat and Statistical Mechanics, Garber, Brush and Everitt, 1995

25. 25 I fear Biologists indulge themselves with verbal models of molecules where Maxwell abstained

26. 26 Verbal Models are Vague and Difficult to Test

27. 27 Verbal Models lead to Interminable Argument and Interminable Investigation

28. 28 thus, to Interminable Funding

29. 29 and so Verbal Models Are Popular

30. 30 Can Molecular Simulations serve as “Fundamental Physical Laws”? Only if they count correctly !

31. 31 It is very difficult for Molecular Dynamics to count well enough to reproduce Conservation Laws (e.g., of number, energy) Concentration (i.e., number density) or activity Energy of Electric Field Ohm’s ‘law’ (in simple situations) Fick’s ‘law’ (in simple situations) Fluctuations in number density (e.g., entropy)

32. 32 Can Molecular Simulations serve as “Fundamental Physical Laws”? Only if Calibrated!

33. 33 Calibrated Molecular Dynamics may be possible MD without Periodic Boundary Conditions ─ HNC HyperNetted Chain Pair Correlation Function in Bulk Solution Saraniti Lab, IIT: Aboud, Marreiro, Saraniti & Eisenberg

34. 34 Calibrated Molecular Dynamics may be possible MD without Periodic Boundary Conditions BioMOCA ─ E quilibrium M onte C arlo (ala physical chemistry) Pair Correlation Function in Bulk Solution van der Straaten, Kathawala, Trellakis, Eisenberg & Ravaioli

35. 35 Calibrated MD may be possible, even in a Gramicidin channel Molecular Dynamics without Periodic Boundary Conditions BioMOCA van der Straaten, Kathawala, Trellakis, Eisenberg & Ravaioli Simulations 235ns to 300ns, totaling 4.3 μs. Mean I = 3.85 pA, 24 Na + crossings per 1 μs 16 Na + single channel currents

36. 36 Essence of Engineering is knowing What Variables to Ignore! WC Randels quoted in Warner IEEE Trans CT 48:2457 (2001) Until Mathematics of Simulations is available we take an Engineering Approach

37. 37 What variables should we ignore when we make low resolution models? How can we tell when a model is helpful? Use the scientific method Guess and Check! Intelligent Guesses are MUCH more efficient Sequence of unintelligent guesses may not converge! (e.g., Rate/State theory of channels/proteins)

38. 38 Use the scientific method Guess and Check! Intelligent Guesses are MUCH more efficient When theory works, need few checks ‘guesses’ are almost as good as experiments. Mechanical Engineering, Electricity, Computer Science, Hydrodynamics, in

39. 39 Use Theory of Inverse Problems to replace or optimize “Guess and Check” 1) Measure only what can be measured (e.g., not two resistors in parallel). 2) Measure what determines important parameters. 3) Use efficient estimators. 4) Use estimators with known bias 5) no matter what the theory, Be clever in estimation

40. 40 Channels are only Holes Why can’t we have a fully successful theory? Must know physical basis to make a good theory Physical Basis of Gating is not known Physical Basis of Permeation is known, in my opinion.

41. 41 Proteins Bristle with Charge Cohn (1920’s) & Edsall (1940’s) Ion Channels are no exception We start with Electrostatics because of biology

42. 42 Many atoms in a protein have Permanent Charge ~1e Permanent charge is the (partial) charge on the atom when the local electric field is zero. Active Sites in Proteins have Many Charges in a Small Place

43. 43 −0.55O 0.35HH_22 0.60C 0.50C_z 0.35HH_21 −0.45NH_2 0.35HH_12 −0.45NH_1 0.35HH_11 0.32 Average Magnitude 0.30H_e −0.40N_e 0.10C_d 0.00C_g 0.00C_b 0.10C_a 0.25H −0.40N Charge (units / e) Atom in Arginine Charge according to CHARMM

44. 44 Active Sites of Proteins are Very Charged e.g. 7 charges ~ 20 M net charge Selectivity Filters and Gates of Ion Channels are Active Sites 1 nM = 1.2×10 22 cm -3

45. 45 Small Charge in Small Places Large Potentials * Start with electrostatics because * 500 µvolt has significant effect on IV curve in a Sphere of diameter 14 Å 1 charge gives V at psec times; dielectric constant = 2 mV at µsec times; dielectric constant = 80

46. 46 We start without quantum chemistry* (only at first) * i.e., delocalization of orbitals of outer electrons

47. 47 Although we start with electrostatics We will soon add Physical Models of Chemical Effects

48. 48 It is appropriate to be skeptical of analysis in which the only chemistry is physical But give me a chance, ask I will be in Linz for a week! (thanks to Heinz!) Or email beisenbe@rush.edu for the papers

49. 49 Very Different from Traditional Structural Biology which, more or less, ignores Electrostatics

50. 50 Poisson-Nernst-Planck (PNP) Lowest resolution theory that includes Electrostatics and Flux is (probably) PNP, Gouy-Chapman, (nonlinear) Poisson-Boltzmann, Debye-Hückel, are fraternal twins or siblings with similar resolution

51. 51 ION CHANNEL MODELS Poisson’s Equation Continuity Equation & Drift - Diffusion For Derivation …Stay tuned Schuss, Singer, Nadler et al Electrochemical Potential includes Fixed Charge on Protein includes only Mobile Charge

52. 52 One Dimensional PNP Poisson’s Equation Drift-Diffusion & Continuity Equation includes Fixed Charge on Protein includes only Mobile Charge Electrochemical Potential

53. 53 The Problem ionic concentrations, electrostatic potential held constant far from channel membrane with ion channel ionic concentrations, electrostatic potential held constant far from channel

54. 54 Bath Membrane Boundary conditions are Crucial Flared Access to Channel Required in 1D

55. 55 I V INPUT OUTPUT THEORY PNP STRUCTURE Length, diameter, Permanent Charge Dielectric Coefficient EXPERIMENTAL CONDITIONS Bath concentrations Bath Potential Difference PNP Forward Problem

56. 56 Current Voltage Relation Gramicidin 3D PNP Uwe Hollerbach

57. 57 V I PNP -1 Inverse Problem Input Output Theory EXPERIMENTAL CONDITIONS Bath Concentrations Bath Potential Difference EXPERIMENTS ELECTRICAL STRUCTURE Length, diameter, Permanent Charge Dielectric Coefficient PNP -1 Inverse Problem for Mathematicians What measurements best determine electrical structure?

58. 58 Charge Mutation in Porin Ompf Structure determined by x-ray crystallography in Tilman Schirmer’s lab G119D

59. 59 Fit of 1D PNP Current-Voltage 100 mM KCl OmpF G119D Duan Chen John Tang Rush Medical Center

60. 60 Net Charge Difference = 0.13  1.1 =  0.97e Duan Chen John Tang

61. 61 Shielding Dominates Electric Properties of Channels, Proteins, as it does Ionic Solutions Shielding is ignored in traditional treatments of Ion Channels and of Active Sites of proteins Main Qualitative Result

62. 62 Main Qualitative Result Shielding in Gramicidin Uwe Hollerbach Rush Medical Center

63. 63 PNP misfits in some cases even with ‘optimal’ nonuniform D(x) Duan Chen John Tang Rush Medical Center

64. 64 Shielding/PNP is not enough PNP includes Correlations only in the Mean Field PNP ignores ion- ion correlations and discrete particle effects: Single Filing, Crowded Charge Dielectric Boundary Force

65. 65 Neither Field Theory nor Statistical Mechanics easily accommodates Finite Size of Ions and Protein Side-chains How can that be changed? Learn from Mathematicians and/or Physical Chemists Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!

66. 66 Learning with Mathematicians Zeev Schuss, Boaz Nadler, Amit Singer Dept’s of Mathematics Tel Aviv University, Yale University Molecular Biophysics, Rush Medical College We extend usual chemical treatment to include flux and spatially nonuniform boundary conditions We have concrete results only in the uncorrelated case! We have learned how to derive PNP (by mathematics alone). Count trajectories not states

67. 67 Counting Langevin Trajectories in a Channel (between absorbing boundary conditions ) implies PNP (with some differences) PNP measures the density of trajectories (nearly) Zeev Schuss, Amit Singer: Tel Aviv Univ Boaz Nadler: Yale Univ,

68. 68 Conditional PNP Electric Force depends on Conditional Density of Charge Nernst-Planck gives UNconditional Density of Charge mass friction Schuss, Nadler, Eisenberg

69. 69 Boaz Nadler and Uwe Hollerbach Yale University Dept of Mathematics & Rush Medical Center Dielectric Boundary Force DBF PNP ignores correlations induced by Discovered by Coalson/Kurnikova, Chung/Corry, … not us

70. 70 Counting gives PNP with corrections 1)Correction for Dielectric Boundary Force* 2) NP (diffusion) equation describes probability of location but 3) Poisson equation depends on a conditional probability of location of charge, i.e., probability of location of trajectories, given that a positive ion is in a definite location. 4) Relation of is not known but can be estimated by closure relations, as in equilibrium statistical mechanics. 5) Closure relations involve correlations from single filing, finite volume of ions, boundary conditions, not well understood, … yet. Stay Tuned! *by derivation, not assumption

71. 71 Until mathematics is available, we Follow the Physical Chemists, even if their approximations are ‘irrational’, i.e., do not have error bounds. Bob Eisenberg blames only himself for this approach

72. 72 Physical Chemistry has shown that Chemically Specific Properties of ions come from their Diameter and Charge (much) more than anything else. Physical Models are Enough Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!

73. 73 Physical Theories of Plasma of Ions Determine (±1-2%) Activity* of ionic solutions from Infinite dilution, to Saturated solutions, even in Ionic melts. *Free Energy per Mole Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!

74. 74 } Concentration-independent geometric restrictions solvation (Born) terms Ideal term electrochemical potential of point particles in the electrostatic mean-field includes Poisson equation Excess chemical potential Finite size effects Spatial correlations Chemical potential has three components

75. 75 Simplest Physical Theory MSA Electrostatic Spheres Hard Spheres Ionic radii  I are known values in bulk Learned from Lesser Blum & Doug Henderson … Thanks! We use Simonin/Turq formulation.

76. 76 Properties of Highly Compressible Plasma of Ions Similar Results are computed by many Different theories and Simulations MSA is only simplest. We (and others) have used MSA, SPM, MC, and DFT MSA : M ean S pherical A pproximation SPM: S olvent P rimitive M odel MC: M ont e C arlo Simulation DFT: D ensity F unctiona l T heory of Solutions Most Accurate Atomic Detail Calibrated ! Inhomogeneous Systems

77. 77 back to channels Selectivity in Channels Wolfgang Nonner, Dirk Gillespie University of Miami and Rush Medical Center

78. 78 Binding Curve Wolfgang Nonner

79. 79 O½O½ Selectivity Filter Selectivity Filter Crowded with Charge Wolfgang Nonner

80. 80 Understand Selectivity well enough to Make a Calcium Channel using techniques of molecular genetics, site-directed Mutagenesis Goal: George Robillard, Henk Mediema, Wim Meijberg

81. 81 General Biological Theme (‘adaptation’) Selectivity Arises in a Crowded Space Biological case: 0.1 M NaCl and 1  M CaCl 2 in the baths As the volume is decreased, water is excluded from the filter by crowded charge effects Ca 2+ enters the filter and displaces Na + Biological Case Wolfgang Nonner Dirk Gillespie

82. 82 Sensitivity to Parameters Volume Dielectric Coefficient

83. 83 Trade-offs ‘1. 5’ adjustable parameters Volume Dielectric Coefficient

84. 84 Binding Curves Sensitivity to Parameters 0.75 nm 3 volume 0.20 nm 3 volume

85. 85 At Large Volumes Electrical Potential can Reverse 0.75 nm 3 volume 0.20 nm 3 volume - - 0 Potential- - 0 Potential Positive Negative

86. 86 Competition of Metal Ions vs. Ca ++ in L-type Ca Channel Nonner & Eisenberg

87. 87 Similar Results have been found by Henderson, Boda, et al. Hansen, Melchiona, Allen, et al., Nonner, Gillespie, Eisenberg, et al., Using MSA, SPM, MC and DFT for the L-type Ca Channel MSA: Mean Spherical Approximation SPM: Solvent Primitive Model MC: Monte Carlo Simulation DFT: Density Functional Theory of Solutions Atomic Detail Calibrated! Most Accurate Inhomogeneous Systems

88. 88 Best Result to Date with Atomic Detail Monte Carlo, including Dielectric Boundary Force Dezso Boda, Dirk Gillespie, Doug Henderson, Wolfgang Nonner Na + Ca ++

89. 89 Other Properties of Ion Channels are likely to involve more subtle physics including orbital delocalization and chemical binding Selectivity apparently does not! Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!

90. 90 Ionic Selectivity in Protein Channels Crowded Charge Mechanism Simplest Version: MSA How does Crowded Charge give Selectivity?

91. 91 General Biological Theme (‘adaptation’) Selectivity Arises in a Crowded Space Biological case: 0.1 M NaCl and 1  M CaCl 2 in the baths As the volume is decreased, water is excluded from the filter by crowded charge effects Ca 2+ enters the filter and displaces Na + Biological Case Wolfgang Nonner Dirk Gillespie

92. 92 Ionic Selectivity in Protein Channels Crowded Charge Mechanism 4 Negative Charges of glutamates of protein DEMAND 4 Positive Charges nearby either 4 Na + or 2 Ca ++

93. 93 Ionic Selectivity in Protein Channels Crowded Charge Mechanism Simplest Version: MSA 2 Ca ++ are LESS CROWDED than 4 Na +, Ca ++ SHIELDS BETTER than Na +, so Protein Prefers Calcium

94. 94 2 Ca ++ are LESS CROWDED than 4 Na +

95. 95 What does the protein do? Selectivity arises from Electrostatics and Crowding of Charge Certain MEASURES of structure are Powerful DETERMINANTS of Function e.g., Volume, Dielectric Coefficient, etc. Precise Arrangement of Atoms is not involved in the model, to first order.

96. 96 Protein provides Mechanical Strength Volume of Pore Dielectric Coefficient/Boundary Permanent Charge Precise Arrangement of Atoms is not involved in the model, to first order. but Particular properties (‘measures’) of the protein are crucial! What does the protein do?

97. 97 Mechanical Strength Volume of Pore Dielectric Coefficient/Boundary Permanent Charge But not the precise arrangement of atoms Implications for Artificial Channels Design Goals are

98. 98 Implications for Traditional Biochemistry Traditional Biochemistry focuses on Particular locations of atoms

99. 99 Traditional Biochemistry assumes Rate Constants Independent of Concentration & Conditions

100. 100 Implications for Traditional Biochemistry Traditional Biochemistry (more or less) Ignores the Electric Field

101. 101 But Rate Constants depend steeply on Concentration and Electrical Properties* because of shielding, a fundamental property of matter, independent of model, in my opinion. *nearly always

102. 102 Electrostatic Contribution to ‘Dissociation Constant’ is large and is an Important Determinant of Biological Properties Change of Dissociation ‘Constant’ with concentration is large and is an Important Determinant of Biological Properties

103. 103 Traditional Biochemistry ignores Shielding and Crowded Charge although Shielding Dominates Properties of Ionic Solutions and cannot be ignored in Channels and Proteins in my opinion

104. 104 Make a Calcium Channel using techniques of molecular genetics, site-directed Mutagenesis How can we use these ideas? George Robillard, Henk Mediema, Wim Meijberg BioMaDe Corporation, Groningen, Netherlands

105. 105 More?

106. 106 Function can be predicted From Structure given Fundamental Physical Laws (sometimes, in some cases).

107. 107 Strategy Use site-directed mutagenesis to put in extra glutamates and create an EEEE locus in the selectivity filter of OmpF Site-directed mutagenesis R132 R82 E42 E132 R42 A82 Wild typeEAE mutant E117 D113 George Robillard, Henk Mediema, Wim Meijberg BioMaDe Corporation, Groningen, Netherlands

108. 108 Zero-current potential or reversal potential = measure of ion selectivity Henk Mediema Wim Meijberg

109. 109 Ca 2+ over Cl - selectivity (P Ca /P Cl ) recorded in 1 : 0.1 M CaCl 2 SUMMARY OF RESULTS (1) Conclusions: - Taking positive charge out of the constriction zone (  = -3, see control mutant AAA) enhances the cation over anion permeability. - Putting in extra negative charge (  = -5, see EAE mutant) further increases the cation selectivity. Henk Mediema Wim Meijberg

110. 110 Ca 2+ over Na + selectivity (P Ca /P Na ) recorded in 0.1 M NaCl : 0.1 M CaCl 2 SUMMARY OF RESULTS (2) Conclusion: - Compared to WT, EAE shows just a moderate increase of the Ca 2+ over Na + selectivity. - To further enhance P Ca /P Na may require additional negative charge and/or a change of the ‘dielectric volume’. Work in Progress! Henk Mediema Wim Meijberg

111. 111 Selectivity Differs in Different Types of Channels Wolfgang Nonner Dirk Gillespie Other Types of Channels

112. 112 Ca channel Na channel Cl channel K channel prefers Small ions Ca 2+ > Na + prefers Small ions Na + > Ca 2+ Na + over K + prefers Large ions prefers K + > Na + Selectivity filter EEEE 4 − charges Selectivity filter DEKA 2 −, 1+ charge Selectivity filter hydrophobic partial charges Selectivity filter single filing partial charges PNP/DFTMonte CarloBulk ApproxNot modeled yet Selectivity of Different Channel Types The same crowded charge mechanism can explain all these different channel properties with surprisingly little extra physics.

113. 113 Sodium Channel (with D. Boda, D. Busath, and D. Henderson) Related to Ca ++ channel removing the positive lysine (K) from the DEKA locus makes calcium-selective channel High Na + selectivity 1 mM CaCl 2 in 0.1 M NaCl gives all Na + current (compare to calcium channel) only >10 mM CaCl 2 gives substantial Ca ++ current Monte Carlo method is limited (so far) to a uniform dielectric Stay tuned…. Wolfgang Nonner Dirk Gillespie

114. 114 Na + Ca ++ Na + /Ca 2+ Competition in the Sodium Channel Biological Region Ca ++ in bath (M) Wolfgang Nonner Dirk Gillespie

115. 115 Model gives small-ion selectivity. Result also applies to the calcium channel. Na + /Alkali Metal Competition in Na + Channel Biological Region Wolfgang Nonner Dirk Gillespie

116. 116 ‘New’ result from PNP/SPM combined analysis Spatial Nonuniformity in Na + Channel Wolfgang Nonner Dirk Gillespie

117. 117 Na + vs K + Selectivity Na + K + Channel Protein Na + Channel Wolfgang Nonner Dirk Gillespie

118. 118 Na + Channels Select Small Na + over Big K + because (we predict) Protein side chains are small allowing Small Na + to Pack into Niches K + is too big for the niches! Wolfgang Nonner Dirk Gillespie Summary of Na + Channel

119. 119 Sodium Channel Summary Na + channel is a Poorly Selective Highly Conducting Calcium channel, which is Roughened so it prefers Small Na + over big K + Wolfgang Nonner Dirk Gillespie

120. 120 Cl − Selective Channel Selective for Larger Anions The Dilute Channel Wolfgang Nonner Dirk Gillespie

121. 121 Chloride Channel Channel prefers large anions in experiments, Low Density of Charge (several partial charges in 0.75 nm 3 ) Selectivity Filter contains hydrophobic groups these are modeled to (slightly) repel water this results in large-ion selectivity Conducts only anions at low concentrations Conducts both anions and cations at high concentration Current depends on anion type and concentration Wolfgang Nonner Dirk Gillespie Doug Henderson Dezso Boda

122. 122 Biological Case Chloride Channel Selectivity depends Qualitatively on concentration Wolfgang Nonner Dirk Gillespie

123. 123 The Dilute Channel: Anion Selective Channel protein creates a Pressure difference between bath and channel Large ions like Cl – are Pushed into the channel more than smaller ions like F – Wolfgang Nonner Dirk Gillespie

124. 124 Key: Hydrophobic Residues Repel Water giving Large-ion selectivity (in both anion and cation channels). Peculiar non-monotonic conductance properties and IV curves observed in experiments Hydrophobic repulsion can give gating. ‘Vacuum lock’ model of gating (M. Green, D. Henderson; J.-P. Hansen; Mark Sansom; Sergei Sukarev) Chloride Channel Wolfgang Nonner Dirk Gillespie

125. 125 Conclusion Each channel type is a variation on a theme of Crowded Charge and Electrostatics, Each channel types uses particular physics as a variation. Wolfgang Nonner Dirk Gillespie

126. 126 Function can be predicted From Structure given Fundamental Physical Laws (sometimes, in some cases).

127. 127 More? DFT

128. 128 Density Functional and Poisson Nernst Planck model of Ion Selectivity in Biological Ion Channels Dirk Gillespie Wolfgang Nonner Department of Physiology and Biophysics University of Miami School of Medicine Bob Eisenberg Department of Molecular Biophysics and Physiology Rush Medical College, Chicago

129. 129 We (following many others) have used Many Theories of Ionic Solutions as Highly Compressible Plasma of Ions with similar results MSA, SPM, MC and DFT MSA : M ean S pherical A pproximation SPM: S olvent P rimitive M odel MC: M ont e C arlo Simulation DFT: D ensity F unctiona l T heory of Solutions Most Accurate Atomic Detail Inhomogeneous Systems

130. 130 Density Functional Theory HS excess chemical potential is from free energy functional Energy density depends on “non-local densities” Nonner, Gillespie, Eisenberg

131. 131 HS excess chemical potential is Free energy functional is due to Yasha Rosenfeld and is considered more than adequate by most physical chemists. The double convolution is hard to compute efficiently. Nonner, Gillespie, Eisenberg We have extended the functional to Charged Inhomogeneous Systems with a bootstrap perturbation method that fits MC simulations nearly perfectly.

132. 132 Example of an Inhomogeneous Liquid A two-component hard- sphere fluid near a wall in equilibrium (a small and a large species). Near the wall there are excluded-volume effects that cause the particles to pack in layers. These effects are very nonlinear and are amplified in channels because of the high densities. small species large species

133. 133 The Problem We are interested in computing the flux of ions between two baths of fixed ionic concentrations. Across the system an electrostatic potential is applied. Separating the two baths is a lipid membrane containing an ion channel. ionic concentrations and electrostatic potential held constant far from channel membrane with ion channel ionic concentrations and electrostatic potential held constant far from channel

134. 134 Modeling Ion Flux The flux of ion species i is given by the constitutive relationship The flux follows the gradient of the total chemical potential. where D i is the diffusion coefficient  i is the number density  i is the total chemical potential of species i

135. 135 } concentration-independent geometric restrictions solvation (Born) terms ideal term electrochemical potential of point particles in the electrostatic mean-field includes Poisson equation excess chemical potential the “rest”: the difference between the “real” solution and the ideal solution The chemical potential has three components

136. 136 When ions are charged, hard spheres the excess chemical potential is split into two parts } } Electrostatic Component describes the electrostatic effects of charging up the ions Hard-Sphere Component describes the effects of excluded volume the centers of two hard spheres of radius R cannot come closer than 2R

137. 137 Density Functional Theory HS excess chemical potential is derived from free energy functional Energy density depends on “non-local densities” Nonner, Gillespie, Eisenberg

138. 138 HS excess chemical potential is Free energy functional is due to Yasha Rosenfeld and is considered more than adequate by most physical chemists. The double convolution is hard to compute efficiently. Nonner, Gillespie, Eisenberg We have extended the functional to Charged Inhomogeneous Systems with a bootstrap perturbation method that fits MC simulations nearly perfectly.

139. 139 Density Functional Theory Energy density depends on “non-local densities”: Nonner, Gillespie, Eisenberg

140. 140 Nonner, Gillespie, Eisenberg The non-local densities (  = 0, 1, 2, 3, V1, V2) are averages of the local densities: where  is the Dirac delta function, is the Heaviside step function, and R i is the radius of species i.

141. 141 We use Rosenfeld’s perturbation approach to compute the electrostatic component. Specifically, we assume that the local density  i (x) is a perturbation of a reference density  i ref (x): The ES Excess Chemical Potential Density Functional Theory

142. 142 The Reference Fluid In previous implementations, the reference fluid was chosen to be a bulk fluid. This was both appropriate for the problem being solved and made computing its ES excess chemical potential straight-forward. However, for channels a bulk reference fluid is not sufficient. The channel interior can be highly-charged and so 20+ molar ion concentrations can result. That is, the ion concentrations inside the channel can be several orders of magnitude larger than the bath concentrations. For this reason we developed a formulation of the ES functional that could account for such large concentration differences.

143. 143 Test of ES Functional To test our ES functional, we considered an equilibrium problem designed to mimic a calcium channel. two compartments were equilibrated edge effects fully computed 24 M O -1/2 CaCl 2 NaCl or KCl 0.1 M The dielectric constant was 78.4 throughout the system.

144. 144

145. 145

146. 146 ‘New’ Mathematics is Needed: Analysis of Simulations

147. 147 Can Simulations serve as “Fundamental Physical Laws”? Direct Simulations are Problematic Even today

148. 148 Can simulations serve as fundamental physical laws? Direct Simulations are Problematic Even today Simulations so far cannot reproduce macroscopic variables and phenomena known to dominate biology

149. 149 Simulations so far often do not reproduce Concentration (i.e., number density) (or activity coefficient) Energy of Electric Field Ohm’s ‘law’ (in simple situations) Fick’s ‘law’ (in simple situations) Conservation Laws (e.g., of energy) Fluctuations in number density

150. 150 The larger the calculation, the more work done, the greater the error First Principle of Numerical Integration First Principle of Experimentation The more work done, the less the error Simulations as fundamental physical laws (?)

151. 151 How do we include Macroscopic Variables in Atomic Detail Calculations? Another viable approach is Hierarchy of Symplectic Simulations

152. 152 Analysis of Simulations e.g., How do we include Macroscopic Variables Conservation laws in Atomic Detail Calculations? Because mathematical answer is unknown, I use an Engineering Approach Hierarchy of Low Resolution Models

153. 153 Simulations produce too many numbers 10 6 trajectories each 10 -6 sec long, with 10 9 samples in each trajectory, in background of 10 22 atoms Why not simulate?

154. 154 Simulations need a theory that Estimates Parameters (e.g., averages) or Ignores Variables Theories and Models are Unavoidable! (in my opinion)

155. 155 Symplectic integrators are precise in ‘one’ variable at a time! It is not clear (at least to me) that symplectic integrators can be precise in all relevant variables at one time