Paraphrase of Chapters 2223: Ions in solution

1
Paraphrase of Chapters 22-23Ions in solution

• BME 531
• Nathan Baker
• baker_at_biochem.wustl.edu

2
How solvent interacts with biomolecules

• Water properties
• Dipolar solvent (1.8 D)
• Hydrogen bond donor and acceptor
• Polarizable
• Functional behavior
• Bulk polarization
• Site binding or specific solvation
• Preferential hydration
• Acid/base chemistry

Carbonic anhydrase reaction mechanism (Stryer,
Biochemistry)
Spine of hydration in DNA minor groove (Kollman
et al)
3
How ions interact with biomolecules

• Non-specific screening effects
• Depends only on ionic strength (not species)
• Results of damped electrostatic potential
• Described by Debye-Hückel and Poisson-Boltzmann
  theory for low concentration
• Influences
• Described throughout these lectures
• Binding constants
• Rates??

Electrostatic potential of AChE at 0 mM and 150
mM NaCl. Rate and binding affinity decrease with
NaCl has been attributed to screening effects
although species-dependent influences have been
observed. Radic Z, Kirchhoff PD, Quinn DM,
McCammon JA, Taylor P. 1997. J Biol Chem 272
(37) 23265-23277.
4
How ions interact with biomolecules

• Site-specific binding
• Ion-specific
• Site geometry, electrostatics, coordination, etc.
  enables favorable binding
• Influences co-factors, allosteric activation,
  folding (RNA)

Site of sodium-specific binding in thrombin.
Sodium binding converts thrombin to a
procoagulant form by allosterically enhancing the
rate and changing substrate specificity. Pineda
AO, Carrell CJ, Bush LA, Prasad S, Caccia S, Chen
ZW, Mathews FS, Di Cera E. 2004. J Biol Chem 279
(30) 31842-53.
5
How ions interact with biomolecules

• Non-specific screening effects
• Depends only on ionic strength (not species)
• Results of damped electrostatic potential
• Described by Debye-Hückel and Poisson-Boltzmann
  theory for low concentration
• Site-specific binding
• Ionic specific (concentration of specific ion,
  not necessarily ionic strength)
• Site geometry, electrostatics, coordination, etc.
  enables favorable binding
• Influences
• Co-factors
• Allosteric activation
• Folding (RNA)
• Hofmeister effects (preferential hydration)
• Partitioning of ions between water and
  non-specific sites on biomolecule
• Dependent on ion type (solvation energy, etc.)
• Dominate at high salt concentrations
• Influences
• Protein stability
• Membrane structure and surface potentials
• Protein-protein interactions

6
MODELS OF SOLVENT

• Explicit
• Quantum mechanical
• Polarizable
• Monte Carlo and molecular dynamics
• Integral equations
• RISM
• 3D HNC
• DFT
• Implicit solvent
• Coulombs law
• Poisson equation
• Phenomenological
• Generalized Born
• Modified Coulomb

Level of detail or Computational cost
7
Explicit solvent simulations

• Sample the configuration space of the system
• Ions
• Atomically-detailed water
• Solute
• Simulation performed under particular macroscopic
  conditions
• Constant number/chemical potential
• Constant pressure/volume
• Constant temperature/energy
• Algorithms
• Molecular dynamics integrate equations of
  motion
• Monte Carlo randomly sample coordinates

8
Electrostatics in explicit solvent simulations

• Charge descriptions
• Fixed monopoles (dipoles, quadrupoles, etc.)
• Polarization induced charge
• Observables
• Average potential, energy, etc. over simulation
• Potentials of mean force interaction averaged
  over a reaction coordinate

9
Boundary conditions in explicit solvent
simulations

• Electrostatic interactions have infinite range
• Simulations can only include a finite number of
  degrees of freedom
• What happens outside the simulation box?
• Periodic boundaries
• Pro Infinite system
• Con Artificial order
• Finite simulation domain
• Pro No artificial order
• Con Boundary effects

10
Explicit solvent simulations pros and cons

• Advantages
• High levels of detail
• Easy inclusion of additional degrees of freedom
• All interactions considered explicitly
• Disadvantages
• No simple theories available
• Slow (and uncertain) convergence
• Time-consuming
• Boundary effects
• Poor scaling to larger systems
• Some effects still not considered in many force
  fields

11
Distribution theories

• Provide average information what is the
  probability of finding a molecule at a specific
  position with a given configuration?
• Example what is the probability of finding
  argon a certain distance from a hard sphere?
• Example what is the probability of finding a
  (rigid) water molecule 3 Å from a spherical ion
  with its dipole pointing toward the ion?

12
Distribution theories

• Offer static information about molecular
  distributions
• Solvent density and polarization
• Ion density
• Average forces, free energies, potentials of mean
  force
• Very complicated formulations
• Integral equations (hard to solve)
• Statistical mechanics
• Approximate
• Difficult to include dynamic information

13
MODELS OF IONS

• Primitive solvent models
• Ions included explicitly
• Ions included as distribution functions
• Continuum models
• Debye-Hückel
• Manning-Oosawa
• Possion-Boltzmann

14
Strong electrolytes

• Strong electrolytes have a dissociation constant
  above 0.1
• Strong electrolytes have a (relatively)
  straightforward theoretical framework for low
  concentrations
• Debye-Hückel theory
• Poisson-Boltzmann theory

15
Debye-Hückel theory

• Suppose we have several species of ions with
  different charges in an external field or
  potential
• Assume a Boltzmann-like probability for finding
  an ion at a specific position
• Given this probability, what is the charge
  density due to all the ions?
• What if the charge-potential interaction energy
  is very small?

16
Debye-Hückel theory

• Now assume the system of ions is electroneutral
• Consider these system of ions in the presence of
• homogeneous dielectric
• fixed charge distribution one that does not
  respond to potential changes
• Use the Poisson equation from above
• Factor out the Debye parameter

17
Debye-Hückel theory

• The resulting equation is a Helmholtz equation
• For point charges, the solution is the
  superposition of Helmholtz Green functions
• The solutions decay much more rapidly due to
  ionic screening

18
Debye-Hückel theory

• The energies are calculated in the usual way
• We can also infer ion number and charge
  distributions

19
Debye length fundamental length scale for
Debye-Hückel theory
Avogadros (6.021023 mol-1)
Electron charge (1.6010-19 C)
Species charge (e)
Species concentration (M)
Ionic strength (M)
For water (e80) at 298 K

• Debye parameter represents effective screening of
  potential
• Inverse parameter is Debye length

20
Debye length fundamental length scale for
Debye-Hückel theory

• Introduces a characteristic length
• Also gives the maximum for the charge density
  distribution

21
Chemical potential a review

• Measures the change in free energy upon a change
  in particle number
• Given with respect to a standard state
• Ideal chemical potential
• Non-interacting components
• Uses concentrations or mole fractions
• Excess chemical potential
• Measures non-ideality
• Uses activities
• Activity coefficient measures deviation from
  ideality

22
DH theory and thermodynamics

• DH theory describes activity coefficients for
• Simple electrolytes
• At low ionic strengths (I lt10-3 M)
• When only non-specific electrostatics matters
• Need to solve for the electrostatic potential for
  a finite size ion (radius a) in electrolyte
  solution
• Write solution as
• Contribution from central ion
• Contribution from electrolyte solution

Surface potential due to electrolyte solution
23
DH theory and thermodynamics

• What is the energy of inserting an ion into
  solution? Chemical potential
• Guntelberg charging calculate the energy due to
  mobile counterions for
• Creating the cavity (zero in this model)
• Charging the cavity
• Result single-ion activity coefficients

24
DH theory and thermodynamics

• Unfortunately, single ion activities cant be
  measured
• However, we can use the property of chemical
  potentials at equilibrium to relate individual
  species activities to salt
• This defines a mean activity coefficient for the
  salt

25
Debye-Hückel activity coefficients

• Mean activity coefficient can be determined by
• Approximating the single-ion coefficients by DH
  theory
• Approximately equal ionic radii
• Enforcing electroneutrality

Monovalents
Divalents
26
Limitations of Debye-Hückel

• Remember this works best for low ionic
  strengths where ion-ion and ion-solvent
  interactions dont matter

Meissner HP, Tester JW. Activity coefficients of
strong electrolytes in aqueous solutions. Ind.
Eng. Chem. Process Des. Develop. 11 (1) 128-133,
1972.
27
APPLICATION TO BIOLOGY

• Provides simple model for interpreting
  non-specific salt effects on
• Solubility
• Binding constants
• Rate constants
• Wide range of other applications
• Virial coefficients
• Solution osmotic pressure
• Ionic conductances (Debye-Hückel-Onsager)
• Keep theorists busy

Be careful is the effect really due to ionic
strength independent of valency or species?
Chemical reaction rates of ions as a function of
their combined charge (zAzB).
28
Solubility

• Solubility products can be written in terms of
  mean activities (activity of solid 1)
• Inert strong electrolytes
• Decrease mean activities according to DH theory
• Leave solubility coefficient unchanged
• Therefore aqueous concentrations increase with
  inert electrolyte concentration (until DH
  limits) salting in

29
Solubility
Solubilities of NaCl ? and glycine ? in various
solvents.
30
Solubility

• However, activity coefficients dont necessary
  decrease for all concentrations of inert
  electrolyte
• DH range of concentrations predict salting in
• Larger concentrations may cause salting out

Salting in
Salting out
Reference solubility
31
Binding constants

• Similar arguments as solubility
• Care must be taken to identify an inert salt
• Only works for very simple, non-specific binding
  models
• Extensions possible with more complicated models
  (Poisson-Boltzmann, etc.)

32
MORE COMPLICATED MODELS

• Primitive solvent, explicit ion simulations
• Integral equation theories
• Manning-Oosawa models of counterion condensation
• Explict ion-ion correlations
• Anything from plasma physics

33
COMPUTATIONAL APPROACHES
Computation
Theory
34
Why is theory useful?

• Interpretation of data models
• Map observations onto physical principles
• Generate models of system behavior
• Better understanding
• Prediction of response to change
• Engineering
• Ask why instead of what
• The best theories are simple
• Theories do not require theoreticians

35
Why is computation necessary?

• Although the model might be simple and contain
  few parameters, solving the model for
  observations can be difficult
• Examples
• Schrödinger equation
• Classical force fields and partition functions
• Differential equations
• Polymer models of folding

36
Explicit particle simulation recap

• All atoms treated explicitly
• Electrostatic interactions handled at Coulombs
  Law level
• Lots of detail
• All observables from continuum simulations
• plus atomic-level information
• Issues with finite system size
• Very time-consuming
• Limits system size
• Limits accuracy (convergence)

37
CONTINUUM SIMULATIONS

• The remainder of these lectures applies exactly
  the same principles discussed earlier to systems
  where analytical results are not available
• We will consider two types of models
• Simple phenomenological models (modified Coulomb,
  etc.) where there is no obvious way to predict
  the error in the model
• Poisson-Boltzmann models which rely on similar
  principles to Debye-Hückel and therefore have the
  same limitations
• These models include detailed structural
  information into the calculation of electrostatic
  interaction energies and forces

38
Simple analytic models

• Usually phenomenological generalization of
  Poisson or Debye-Hückel models
• Include
• Generalized Born (coming up)
• Distance-modified dielectrics (not a good idea)
• Surface-based models
• Coulombs law (often in conjuction with surface
  terms)
• Most models for apolar contributions are
  phenomenological and/or heuristic

39
Apolar solvation
Figures from Dill KA, Truskett TM, Vlachy V,
Hribar-Lee B. 2005. Modeling water, the
hydrophobic effect, and ion solvation. Annu Rev
Biophys Biomol Struct. Experiments Ar gas ?
water transfer.

• All of the models we have discussed so far only
  describe polar solvation
• Interaction of solute with water dipoles
• Interaction of solute with ions
• Experimental observations
• Solubility has temperature minimum
• Large positive heat capacity
• Temperature-sensitive (negative) entropy
  structuring of water

40
Apolar solvation

• What should be included in a model
• Cavity terms the probability of finding a
  hole (or inert species of a given size) in the
  solvent
• Weak interactions dispersive and repulsive
  interactions between solute and solvent
• Careful explicit solvent simulations will include
  these details

Figure from Levy RM, Zhang LY, Gallichio E,
Felts AK. 2003. J Am Chem Soc 125 (31) 9523-9530.
41
Apolar solvation

• What do we do for implicit solvent models?
• Cavity terms
• Scaled particle theory
• Apolar free energy surface area times surface
  tension coefficient
• Surface tension coefficient between 25 to 50 cal
  mol-1 (model-dependent)
• Spolar RS, Ha JH, Record MT, Jr. 1989. Proc Natl
  Acad Sci USA 86 (21) 8382-8385 Sitkoff D, Sharp
  KA, Honig B. 1994. Biophys Chem 51 (2-3)
  397-409.
• Dispersion/repulsion terms
• Hard solute
• Integral of attractive potential over accessible
  space
• Levy RM, Zhang LY, Gallichio E, Felts AK. 2003. J
  Am Chem Soc 125 (31) 9523-9530.
• Not widely used!!

42
Generalized Born

• Used to calculate solvation energies (forces)
• Modification of Born ion solvation energy
• Adjust effective radii of atoms based on
  environment
• Differences between buried and exposed atoms
• Fast to evaluate
• Lots of variations
• Very sensitive to parmeterziation
• Good parmeterization can give results comparable
  to Poisson-Boltzmann
• Parameterization should change with conformation

Figure from Onufriev A, Bashford D, Case DA.
2000. J Phys Chem B 104 (15) 3712-3720.
43
Poisson-Boltzmann

• Same basic principles as Debye-Hückel theory
• Continuum dielectric (Poisson equation)
• Non-interacting mobile ions (mean field
  approximation)
• Same limitations
• Low ion concentration
• Low ion valency
• No specific interactions solute-solvent,
  solute-ion, ion-solvent, ion-ion

44
Poisson-Boltzmann derivation step 1

• Start with Poisson equation to describe solvation
• Supplement biomolecular charge distribution with
  mobile ion term

Dielectric function
Biomolecular charge distribution
Mobile charge distribution
45
Poisson-Boltzmann derivation step 2

• Choose mobile ion charge distribution form
• Boltzmann distribution ? no explicit ion-ion
  interaction
• No detailed structure for atom (de)solvation

Ion charges
Ion bulk densities
Ion-protein steric interactions
46
Poisson-Boltzmann derivation step 3

• Substitute mobile charge distribution back into
  Poisson equation
• Result Nonlinear partial differential equation

47
Poisson-Boltzmann special cases

• 11 electrolyte (NaCl)
• Assume similar steric interactions for each
  species with protein
• Simplify two-term series to hyperbolic sine

Modified screening coefficient zero inside
biomolecule
11 electrolyte charge distribution
48
Poisson-Boltzmann special cases

• 11 electrolyte (NaCl)
• Assume similar steric interactions for each
  species with protein
• Simplify two-term series to hyperbolic sine
• Small charge-potential interaction
• Linearized Poisson-Boltzmann
• Homogeneous dielectric PB ? Debye-Hückel

49
Poisson-Boltzmann energies

• Similar to Poisson equation
• Functional integral over solution domain
• Solution extremizes free energy
• Basis for calculating forces
• Charge-field
• Dielectric boundary
• Osmotic pressure

Fixed charge- potential interactions
Dielectric polarization
Mobile charge energy
50
Solving the PE or PBE

• Determine the coefficients based on the
  biomolecular structure
• Discretize the problem
• Solve the resulting linear or nonlinear algebraic
  equations

51
Equation coefficients charge distribution

• Charges are delta functions hard to model
• Often discretized as splines to smooth the
  problem
• What about higher-order charge distributions?

52
Equation coefficients mobile ion distribution

• Provides
• Bulk ionic strength
• Ion accessibility
• Usually constructed based on inflated van der
  Waals radii

53
Equation coefficients dielectric function

• Describes change in dielectric response
• Low dielectric interior (2-20)
• High dielectric solvent (80)
• Many definitions
• Molecular (solid line)
• Solvent-accessible (dotted line)
• van der Waals (gray circles)
• Inflated van der Waals (previous slide)
• Smoothed definitions (spline-based and Gaussian)
• Results can be very sensitive to the choice of
  surface!!!

54
Discretization

• Choose your problem domain finite or infinite?
• Usually finite domain
• Requires relatively large domain
• Uses asymptotically-correct boundary condition
  (e.g., Debye-Hückel, Coulomb, etc.)
• Infinite domain requires appropriate basis
  functions
• Choose your basis functions global or local?
• Usually local map problem onto some sort of
  grid or mesh
• Global basis functions (e.g. spherical harmonics)
  can cause numerical difficulties

55
Discretization local methods

• Polynomial basis functions (defined on interval)
• Locally supported on a few grid points
• Only overlap with nearest-neighbors ? sparse
  matrices

Boundary element (Surface discretization)
Finite element (Volume discretization)
Finite difference (Volume discretization)
56
Discretization pros cons

• Finite difference
• Sparse numerical systems and efficient solvers
• Handles linear and nonlinear PBE
• Easy to setup and analyze
• Non-adaptive representation of problem
• Finite element
• Sparse numerical systems
• Handles linear and nonlinear PBE
• Adaptive representation of problem
• Not easy to setup and analyze
• Less efficient solvers
• Boundary element
• Very adaptive representation of problem
• Surface discretization instead of volume
• Not easy to setup and analyze
• Less efficient solvers
• Dense numerical system
• Only handles linear PBE

57
Basic numerical solution

• Iteratively solve matrix equations obtained by
  discretization
• Linear multigrid
• Nonlinear Newtons method and multigrid
• Multigrid solvers offer optimal solution
• Accelerate convergence
• Fine ? coarse projection
• Coarse problems converge more quickly
• Big systems are still difficult
• High memory usage
• Long run-times
• Need parallel solvers

58
Errors in numerical solutions

• Electrostatic potentials are very sensitive to
  discretization
• Grid spacings lt 0.5 Å
• Smooth surface discretizations
• Errors most pronounced next to biomolecule
• Large potential and gradients
• High multipole order
• Errors decay with distance
• Approximately follow multipole expansion behavior
• Coarse grid spacings will correctly resolve
  electrostatics far away from molecule

59
Sequential focusing

• Finite difference
• What should we do if were only interested in a
  part of the protein?
• Use a coarse mesh for the whole domain
• Use a fine mesh for the region of interest
• Coarse mesh solution sets fine mesh boundary
  conditions
• Sequential focusing algorithm
• Useful for site-specific calculations (binding,
  pKa, etc.)
• Uses multipole behavior of error

60
Parallel focusing

• Finite difference
• Extension of sequential focusing to large systems
• Parallel runs of sequential focusing
• Focus to overlapping finer grids
• Provide coverage for all or subset of domain
• Energies obtained from disjoint partitions
• Details
• Good parallel scaling
• Easy implementation
• Amenable to dynamics

61
Electrostatics Software
62
APPLICATIONS OF CONTINUUM ELECTROSTATICS

• Thermodynamics
• Basic concepts
• Solvation energies
• Binding energies
• Acid dissociation constants
• Equilibrium ion distributions
• Kinetics
• Rate constant calculations
• Molecular dynamics simulation
• Other
• Structural analysis
• Classification

63
PB energy calculations

• Beware self energies!
• Energies calculated with PB equation contain
  self-interaction terms
• These terms tend toward infinity for small grid
  spacings
• These terms are very sensitive to the system
  setup artifacts
• Solution
• Perform reference calculation with the same
  grid setup, etc.
• This removes the self-energy artifacts by
  cancellation
• The most natural reference calculation is a
  solvation energy

64
Solvation energy

• Physical model transfer solute from dielectric
  of 1 (vacuum) to 80 (water)
• Computational model transfer solute from
  homogeneous dielectric to inhomogeneous
  dielectric eliminate self-interaction terms
• Two models can be reconciled through free energy
  cycle set the reference state

65
Solvation energy two Born ions

• Water dielectric
• Two ions
• 3 Å radii
• Internal dielectric of 1
• Opposite charges of 1 e
• Basic calculation
• Calculate solvation energies of isolated ions
• Calculate solvation energy of complex
• Subtract solvation energies
• Add in Coulombs law attraction

66
Binding energy

• This calculation assumes no conformational
  change!
• Separate calculation into two steps
• Calculate electrostatic interaction in
  homogeneous dielectric (Coulombs law)
• Calculate solvation energy change upon binding
  (Poisson or Poisson-Boltzmann equation)
• Self-interactions are removed in solvation energy
  calculation

67
Binding energy example

• Protein kinase A inhibition by balanol
• Wong CF, Hunenberger PH, Akamine P, Narayana N,
  Diller T, McCammon JA, Taylor S, Xuong NH. 2001.
  J Med Chem 44 (10) 1530-1539.
• Continuum electrostatics predicts binding
  affinities of several inhibitors for 400 kinases

68
Binding energy example

• Zero (low) ionic strength, dielectric
  coefficients of 2 and/or 80, molecular surface
  definition (sensitivity!)
• No structural rearrangement (based on X-ray
  structure)
• Procedure
• Calculate solvation and Coulombic energy of
  inhibitor
• Calculate solvation and Coulombic energy of PKA
• Calculate solvation and Coulombic energy of
  complex
• Results show good trend with experimental
  measurements
• Used to parameterize QSAR model

69
Conformational changes

• Same concepts as binding energy calculation
• Calculate electrostatic energy due to
  configuration change in homogeneous dielectric
  (Coulombs law)
• Calculate electrostatic energy due to change in
  solvation between configurations (Poisson or
  Poisson-Boltzmann)

70
pKa calculations
71
pKa calculations

• Want acid dissociation constant for residues in a
  particular structural context
• Use model pKas for amino acids
• Calculate pKa from two binding calculations
• Binding of unprotonated residue
• Binding of protonated residue
• Consider two conformational distributions for
  protein?

72
Force calculations

• Energy gradients can be obtained from PB
  calculations
• Useful for
• Structure minimization
• Docking
• Dynamics
• Obtained by functional differentiation of PB
  free energy with respect to atom positions

Animation from Dave Sept.
Charge-field interaction
Dielectric boundary force
Osmotic pressure
73
OTHER APPLICATIONS

• Things we probably wont get to cover

74
Application to microtubules

• Important cytoskeletal components structure,
  transport, motility, division
• Typically 250-300 Å in diameter and up to
  millimeters in length
• Computationally difficult due to size (1,500
  atoms/Å ) and charge (-4.5 e/Å)
• Solved LPBE at 150 mM ionic strength on 686
  processors for 600 Å-long, 1.2-million-atom
  microtubule
• Resolution to 0.54 Å for largest calculation
  quantitative accuracy

75
Application to microtubules
76
Application to microtubules
77
Microtubule stability and assembly

• Performed series of calculations on tubulin
  dimers and protofilament pairs
• Poisson-Boltzmann electrostatics and SASA apolar
  energies
• Observed 7 kcal/mol stronger interactions between
  protofilaments than within
• Determined energetics for helix properties
  predict correct minimum for experimentally-observe
  d A (52 Å) and B (8-9 Å) lattices

78
Microtubule stability and assembly
79
Microtubule stability and assembly
80
Quantitative analysis of electrostatic potentials

• Do electrostatic potentials tell us anything
  about biomolecular function?
• Ligand binding sites?
• Biomolecular binding sites?
• Active sites or shifted pKas?
• Structural destabilization?
• Are we learning anything that couldnt be learned
  from structure or sequence analysis methods?

81
Breakdown of implicit solvent models

• Is the water near a zwitterionic bilayer a
  featureless dielectric continuum?
• Examine behavior of TIP3P water around POPC
  bilayer using molecular dynamics
• Systematically replace each layer of explicit
  solvent with dielectric continuum and calculate
  potential
• Average results over MD trajectory
• Determine
• Nature of water at membrane surface
• Discrepancies between implicit and explicit
  electrostatics

82
Water around membrane systems results

• 4 layers of water significantly different than
  bulk
• Dielectric response (orientational fluctuations)
  of water near membrane much lower
• Membrane potential dramatically affected by water
  polarization
• Conclusion Water near membrane has
  significantly different structure and response
  properties from bulk
• Future work non-neutral membranes and mobile
  ions

83
Breakdown of the implicit ion model

• Canonical electrostatics test case
  non-polarizable hard sphere with point charge
• Mild test of continuum electrostatics (no
  condensation, etc.)
• Good agreement with mean-field results at low
  mobile ion concentrations
• Agreement deteriorates with
• Increasing macroion charge
• Increasing mobile ion concentration

84
Breakdown of the implicit ion model
85
Ions around DNA

• NPBE simulations predict 2000 M Mg near DNA
  surface for 20 mM MgCl2, 150 mM NaCl solution
• Ran GCMC calculations for similar conditions
  around 20 bp D-DNA
• Observe max density of 10 M Mg in GCMC
  simulations
• Much lower than PBE results
• Implicit solvent model questionable
• Reproduce expected condensation behavior
• 88 charge compensation at 17 Å radius
• Divalent cations affect charge screening

86
Ions around DNA
87
Ions around DNA