When everything interacts, we need mathematics. We need a Variational Theory of Electrolytes

(1)

Life

is in a

Complex Mixture of Electrolytes

mostly Na

+

, K

+

, and Ca

++

Cl

-Everything Interacts Through the Eelctric Field

Ions Come ‘in pairs’

i.e., electrically balanced neutral combinations

Cl

-Na+

(2)

2

Life Occurs in a Complex Fluid

~200 mM salt solutions

mostly Sodium Na+, Potassium K+, and Calcium Ca++ Chloride Cl

-Chemistry and Biology

are about

Chemically Specific Properties

are the same thing as their

DEVIATION

(3)

When everything interacts, we need mathematics.

Hünenberger & Reif (2011) Single-Ion Solvation

Variational

Mathematics:

‘Everything’

Interacts

with

(4)

Under physiologically appropriate conditions,

it is

Almost Never Valid to use

Debye-Hückel Theory

it is important to take proper account of

Ion Size

(5)

Mathematics

describes only a little of

Daily Life

But

Mathematics* Creates

our

Standard of Living





u

(6)

6

Mathematics Creates

our

Standard of Living

Mathematics replaces

Trial and Error

with Computation

*e.g.,

Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, …..





u

(7)

Mathematics

increases the

Efficiency of Experimentation

and

Efficiency of design

by orders of magnitude

We can do more with less





u

(8)

8

What mathematics?

What is most helpful?





u

(9)

I believe

Variational Approach

has a

Special Value

1

2

0

u









(10)

10

Variational Approach

is

Always self-consistent

Allows adding components

with minimal parameters

1

2

0

u









(11)

Scientific Discussion

Converges

Rapidly

when

1

2

0

u









(12)

12

catalyzes

Science as a Social Process

1

2

0

u









(13)

(14)

Complex Schemes

produce

Unresolved Discussion

and

(15)

Complex Schemes

produce

(16)

Complex Schemes

need to be replaced by a

Variational Field Theory

in my opinion

Here we consider

Electrolyte Solutions

in general,

not just infinitely dilute NaCl

(17)

Poisson Boltzmann

does not fit

Solutions of Divalent Ions

Torrie and Valleau

exact quotation, emphasis Bob Eisenberg:

When the counter-ions are doubly charged … the

Classical Theory Fails

Altogether

even for

(18)

Good Data

(19)

1.

>139,175 Data Points

[Sept 2011] on-line

IVC-SEP Tech Univ of Denmark

http://www.cere.dtu.dk/Expertise/Data_Bank.aspx

2. Kontogeorgis, G. and G. Folas, 2009:

Models for Electrolyte Systems. Thermodynamic John Wiley & Sons, Ltd. 461-523.

3. Zemaitis, J.F., Jr., D.M. Clark, M. Rafal, and N.C. Scrivner, 1986,

Handbook of Aqueous Electrolyte Thermodynamics. American Institute of Chemical Engineers

4. Pytkowicz, R.M., 1979,

Activity Coefficients in Electrolyte Solutions. Vol. 1.

Good Data

(20)

Bad Theory

even without flow

2 0

“It is still a fact that over the last decades,

it was easier to fly to the moon

than to describe the

free energy of even the simplest

salt solutions

beyond a concentration of 0.1M or so.”

(21)

Everything

Interacts

with

Everything

Ions in Water are the Liquid of Life

They are not ideal solutions

For Modelers and Mathematicians

(22)

22

Mathematics of Chemistry

must deal

Naturally

with

Interactions

Everything Interacts

‘Law of Mass Action’ assumes nothing interacts

(23)

Mathematics of Chemistry

must deal

Naturally

with

Interactions

Everything Interacts

‘Law of Mass Action’

assumes

Nothing Interacts

(24)

Page 24

‘

Law

’

of Mass Action

including

Interactions

From Bob Eisenberg p. 1-6, in this issue

Variational Approach

EnVarA

Conservative Dissipative

‘New’

Mathematics

of

Interactions

1

2

-

0

(25)

Where to start?

(26)

26

Multi-Scale Issues

Journal of Physical Chemistry C (2010 )114:20719, invited review

Biological Scales Occur Together

so must be

Computed Together

This may be impossible in simulations

Computational

Scale

Biological

Scale

Ratio

Time 10

-15

sec

10

-4

sec

10

11

Space 10

-11

m

10

-5

m

10

6

Spatial Resolution

10

12

Solute Concentration

10

-11

to 10

1

M

10

12

(27)

Force Fields are Calibrated

Ignoring Interactions with ions

but

come from

Interactions

(28)

28 Molecular Dynamics Simulations

almost always

Assume No Interactions

Real Solutions Always Have Interactions

Electric Field

Every ion interacts with every other ion

through the

Ionic Atmosphere

(29)

Molecular Dynamics Force Fields are Calibrated

assuming no interactions with concentrations

Force Fields must be REcalibrated

in each Biological Solution

Just ask the author(s) of CHARMM

Chemically Specific Properties

of Ionic Solutions come from

(30)

30

Calibration is Hard Work

Force Fields must be RE-calibrated

in each Biological Solution

to verify equilibrium potentials

(chemical potentials)

Fitting Real Experiments

requires Accurate Chemical Potentials in mixtures

Channels are Identified by Equilibrium Potentials

If simulations are uncalibrated,

(31)

Uncalibrated Simulations

will not make devices that

(32)

32

Biological Theory

and

Molecular Dynamics Simulations

almost always assume ideal solutions

In my opinion

‘New’ Mathematics

is needed to deal with the

INTERACTIONS

that make ionic solutions non-ideal

and create the

CHEMICAL SPECIFICITY

(33)

No theory

is available for

Mixtures of Ions

In my opinion

is needed to deal

with the INTERACTIONS

that make ionic solutions non ideal and that can create the

(34)

34

No theory

is available for

Flow

of any kind.

In my opinion

is needed to deal

that make ionic solutions non ideal and that can create the

(35)

No theory

is available for

Brownian Motion of Ions

Brownian Motion theory is for UNcharged particles.

Brownian Motion theory ignores Interactions

In my opinion

is needed to deal

(36)

36

Where to start?

Mathematically ?

(37)

ompF porin

3 Å

K+

Na+

Ca++

Channels are Selective

Different Ions Carry Different Signals through Different Channels

0.7 nm = Channel Diameter

(38)

38

Different Types of Channels

use

Different Types of Ions

for

Different Information

(39)

Natural nano-valves* for atomic control of biological function

Ion channels

coordinate contraction of cardiac muscle, allowing the heart to function as a pump

Ion channels

coordinate contraction in skeletal muscle

Ion channels

control all electrical activity in cells

Ion channels

produce signals of the nervous system

Ion channels

are involved in secretion and absorption in all cells: kidney, intestine, liver, adrenal glands, etc.

Ion channels

are involved in thousands of diseases and many drugs act on channels

Ion channels

are proteins whose genes (blueprints) can be manipulated by molecular genetics

Ion channels

have structures shown by x-ray

Ion Channels are Biological Devices

(40)

Thousands

of

Molecular Biologists

Study Channels

every day,

One protein molecule at a time

This number is not an exaggeration. We have sold >10,000 AxoPatch amplifiers

40

Ion Channel Monthly

AxoPatch 200B

Femto-amps

(41)

Where to start?

‘Law of Mass Action’

must be

Replaced

by a

(42)

42

Working Hypothesis

Biological Adaptation is

Crowded Ions

and

Side Chains

Comparison with Experiments shows

Potassium K

+

Sodium Na

+

Must include Biological

Adaptation!

(43)

Active Sites

of Proteins are

Very

Charged

7 charges ~

20

M net charge

= 1.2×10

22

cm

-3

-+ -+ -+ -+

+

-

-K

+ Na+ Ca2+ Hard Spheres

liquid Water is 55 M solid NaCl is 37 M

OmpF Porin

Physical basis of function

(44)

Ionizable Residues

Density =

22 M

#AA MS_A^3 CD_MS+ CD_MS- CD_MSt

EC1:Oxidoreductases Average 47.2 1,664.74 7.58 2.82 10.41

Median 45.0 1,445.26 6.12 2.49 8.70 EC2:Transferases Average 33.8 990.42 13.20 6.63 19.83

Median 32.0 842.43 8.18 6.71 14.91 EC3:Hydrolases Average 24.3 682.88 13.14 13.48 26.62

Median 20.0 404.48 11.59 12.78 23.64 EC4:Lyases Average 38.2 1,301.89 13.16 6.60 19.76

Median 28.0 822.73 10.81 4.88 16.56 EC5:Isomerases Average 31.6 1,027.15 24.03 11.30 35.33

Median 34.0 989.98 9.05 7.76 16.82 EC6:Ligases Average 44.4 1,310.03 9.25 9.93 19.18

Median 49.0 1,637.98 8.32 7.95 17.89

#AA MS_A^3 CD_MS+ CD_MS- CD_MSt

Total n= 150

Average 36.6 1,162.85 13.39 8.46 21.86

Median 33.0 916.21 8.69 7.23 16.69

EC#: Enzyme Commission Number based on chemical reaction catalyzed #AA: Number of residues in the functional pocket

MS_A^3: Molecular Surface Area of the Functional Pocket (Units Angstrom^3) CD_MS+: Base Density (probably positive)

CD_MS-: Acid Density (probably negative) CD_MSt: Total Ionizable density

EC#: Enzyme Commission Number based on chemical reaction catalyzed #AA: Number of residues in the functional pocket

MS_A^3: Molecular Surface Area of the Functional Pocket (Units Angstrom^3)

CD_MS+: Base Density (probably positive)

CD_MS-: Acid Density (probably negative)

CD_MSt: Total Ionizable density

(45)

EC2: TRANSFERASES

Average Ionizable Density: 19.8 Molar

Example:

UDP-N-ACETYLGLUCOSAMINE ENOLPYRUVYL TRANSFERASE

(PDB:1UAE)

Functional Pocket Molecular Surface Volume: 1462.40 A3

Density : 19.3 Molar (11.3 M+. 8 M-)

Green: Functional pocket residues

Blue: Basic = Probably Positive = R+K+H

(46)

EC3: HYDROLASES

Average Ionizable Density: 26.6 Molar

Example:

ALPHA-GALACTOSIDASE (PDB:1UAS)

Functional Pocket Molecular Surface Volume:

286.58 A3

Density : 52.2 Molar (11.6 M+. 40.6 M-)

Green: Functional pocket residues

Blue: Basic = Probably Positive = R+K+H

Red: Acid = Probably Negative = E + Q

Brown ALPHA D-GALACTOSE

Jimenez-Morales, Liang, Eisenberg

(47)

RyR

Receptor

Gillespie, Meissner, Le Xu, et al,

not

Bob Eisenberg



DFT/PNP

vs

Monte Carlo Simulations

Concentration Profiles

(50)

50

Divalents

KCl

CaCl

₂

CsCl

CaCl

₂

NaCl

CaCl

₂

KCl

MgCl

₂

Misfit

Misfit

Error < 0.1 kT/e

2 kT/e

(51)

KCl

Misfit

Error < 0.1 kT/e 4 kT/e

(52)

Theory fits Mutation with Zero Charge

No parameters adjusted

Gillespie et al

J Phys Chem 109 15598 (2005)

Protein charge density

wild type*

13

M

Water is 55

M

*some wild type curves not shown, ‘off the graph’



0

M

in

D4899

Theory Fits Mutant

in K + Ca

Theory Fits Mutant

in K

Error < 0.1 kT/e

1 kT/e

(53)

Replacement of

“Law of Mass Action”

is

Feasible for

(54)

54 Mutants of ompF Porin

Atomic Scale Macro Scale 30 60 -30 30 60 0

pA

mV

LECE (-7e)

LECE-MTSES

-

(-8e)

LECE-GLUT

-

(-8e)

E

_Ca

E

_Cl WT (-1e)

Calcium selective

Experiments have built

Two Synthetic Calcium Channels

As density of permanent charge increases, channel becomes calcium selective

E_rev  E_Ca in 0.1M 1.0 M CaCl₂

Unselective Wild Type

built by Henk Miedema, Wim Meijberg of BioMade Corp.,Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004)

MUTANT ─ Compound

Glutathione derivatives

Designed by Theory

(55)

Variational Principles Deal with Interactions

Consistently and Automatically

New Component

(or Scale)

implies

New Field Equations (Euler Lagrange)

by

EnVarA

Chun Liu,

with YunKyong Hyon, and Bob Eisenberg

(56)

Page 56

Energetic Variational Approach

EnVarA

Chun Liu, Rolf Ryham, Yunkyong Hyon, and Bob Eisenberg

Mathematicians and Modelers: two different ‘partial’ variations written in one framework, using a ‘pullback’ of the action integral

Action Integral, after pullback Rayleigh Dissipation Function

Field Theory of Ionic Solutions

that allows boundary conditions and flow and deals with Interactions of Components self-consistently

Composite

Variational Principle

Euler Lagrange Equations



1

2

0

E











'

' Dissipative 'Force' Conservative Force

(57)

Field Equations with Lennard Jones Spheres

Nernst Planck Diffusion Equation

for number density c_n of negative n ions; positive ions are analogous

Non-equilibrium variational field theory EnVarA

Coupling Parameters

Ion Radii

Poisson Equation

Number Densities

Diffusion Coefficient Dielectric Coefficient Thermal Energy 12 , 14 12 , 14

12

(

) (

)

=

( )

|

6

(

) (

)

( )

,

|

n n n n

n n

n n n n

B

n p n p

p

a

x

y

c

D

c

z e

c y dy

t

k T

x

y

a

x

y

c y d y

x

y













 

 

 































 





 

(

) =

or

N

z ec

i

n

p







  











(58)

Page 58

EnVarA across biological scales: molecules, cells, tissues

developed by Chun Liu

with

(1) Hyon, Eisenberg

Ions in

Channels (2) Horng, Lin, Lee

Ions in

Channels

(3) Bezanilla, Hyon, Eisenberg

Conformation Change of

Voltage Sensor

(4) Ryham, Cohen

Virus fusion to

Cells

(5) Mori, Eisenberg

Water flow in

Tissues

Multiple

Scales

creates a new

Multiscale

Field Theory of Interacting Components

that allows boundary conditions and flow

and deals with

(59)

developed

by Chun Liu

Preliminary Results

demonstrate

Feasibility

for

(60)

60

Eisenberg, Hyon, and Liu

Layering: Classical Interaction Effect

Comparison between PNP-DFT and MC

Anion PNP-PNP-DFTDFT Cation

Anion MC Cation MC

C

ha

rg

e

D

en

si

ty

Position

(61)

Binding Curves Current Voltage

Time

Curves

Nonequilibrium Computations

with Variational Field Theory

(62)

62

The End

(63)

(64)

(65)

Energetic Variational

Approach

EnVarA

*

if they define an energy and its variation

Energy defined by simulations or theories or experiments is OK

Full micro/macro treatment is needed for an Atomic Model, with closure, as in liquid crystals

New mechanisms*

(

e.g.

, active transport)

(66)

Page 66

Energetic Variational Analysis

EnVarA

Chun Liu, Yunkyong Hyon and Bob Eisenberg

New Interpretations

likely to be

Controversial

but

(67)

is needed to

Add Components

and

Mechanisms

with

Minimal Confusion

(68)

68 .

EnVarA here deals with Reduced Models

Reduced Models are Needed

Reduced Models are Device Equations

like Input Output Relations of Engineering Systems.

The device equation is the mathematical statement of how the system works.

Device Equations describe ‘Slow Variables’

(69)

.

Finding the reduced model,

checking its validity,

estimating its parameters

,

and their effects,

are all part of the

Inverse Problem

(70)

70

Biology is Easier than Physics

Reduced Models Exist*

for important biological functions

or the

Animal would not survive

to reproduce

(71)

Biology is Easier than Physics

Biology Says a Simple Model Exists

Existence of Life

(72)

72

Existence of Life

implies the

Existence of Robust Multiscale Models

Biology Says

there is a

Simple Model

of

Specificity

(73)

Reduced models are the adaptation

created by evolution

to perform the biological function of selectivity

Inverse Methods

are needed to

Establish the Reduced Model

(74)

Inverse Problems

Badly posed,

simultaneously over and under determined.

Exact choice of question and data are crucial

74 Underlying Math Problem

(with DFT, noise and systematic error) has actually been solved using Tikhonov Regularization as in the

Inverse Problem of a Blast Furnace

(75)

Inverse Problems

Many answers are possible

Central Issue

Which answer is right?

Modelers and Mathematicians, Bioengineers: this is reverse engineering

Underlying Math Problem

(76)

How does the

Channel control Selectivity?

Inverse Problems: many answers possible

Central Issue

Which answer is right?

Key is

ALWAYS

Large Amount of Data

from

Many Different Conditions

76 Almost too much data was available for Burger, Eisenberg and Engl (2007)

(77)

Solving Inverse Problems

depends on

Fitting Large Amounts of Data

from many

(78)

78

Dealing with

Different Experimental Traditions

is an unsolved social problem

What was measured?

With what setup?

(79)

Ion Channels are a good test case

because I know the experimental tradition

Channels are also Biologically Very Important

Help!

(80)

80

Ion Channels are a good Test Case

Simple Physics

(Electrodiffusion)

Single Structure

(once open)

Simple Theory is Possible

and Reasonably Robust

because Channels are Devices

with well defined

Inputs, Outputs

and

Power Supplies

(81)

6 4 4 0 0 6 4 3 0 0

6 4 2 0 0 6 4 1 0 0

6 4 0 0 0 6 3 9 0 0

6 3 8 0 0

T im e ( m s )

Ip at ch (p A ) 6 4 2 0 - 2 - 4 - 6 - 8 - 1 0 - 1 2 - 1 4 - 1 6

6 5 1 0 0 6 5 0 0 0

6 4 9 0 0 6 4 8 0 0

6 4 7 0 0 6 4 6 0 0

6 4 5 0 0

T i m e ( m s )

Ip at ch (p A ) 6 4 2 0 - 2 - 4 - 6 - 8 - 1 0 - 1 2 - 1 4 - 1 6

7 2 7 0 0 7 2 6 0 0

7 2 5 0 0 7 2 4 0 0

7 2 3 0 0 7 2 2 0 0

T im e ( m s )

Ip at ch (p A ) 6 4 2 0 - 2 - 4 - 6 - 8 - 1 0 - 1 2 - 1 4 - 1 6

Open Duration /ms

O p e n A m p lit u d e, p A

Lowpass Filter = 1 kHz Sample Rate = 20 kHz

Current vs. time

Amplitude vs. Duration

Channel Structure Does Not Change

once the channel is open

(82)

82

Ideal Ions are Identical

if they have the same charge

Channels are Selective

because

Diameter Matters

Ions are NOT Ideal

Potassium K

+

= Na

/

+

Sodium

3 Å

K+ Na+

In ideal solutions K+ = Na+

(83)

Goal:

Understand Selectivity

well enough to

Fit Large Amounts of Data

from many solutions and concentrations

and to

(84)

84

Everything

Interacts

with

Everything

Ions in Water are the Liquid of Life

They are not ideal solutions

For Modelers and Mathematicians

Tremendous Opportunity for Applied Mathematics Chun Liu’s Energetic Variational Principle

(85)

Working Hypothesis

Biological Adaptation is

Crowded Ions

and

Side Chains

(86)

Page 86

Energetic Variational Analysis

EnVarA

being developed by

Chun Liu

Yunkyong Hyon and Bob Eisenberg

creates a

Field Theory of Ionic Solutions

that allows boundary conditions and flow

and deals with

(87)

‘

Law

’

of Mass Action

including

Interactions

EnVarA

Conservative Dissipative

Great Opportunity

for

New Mathematics

and

Its Applications

1

2

-

0

(88)

Page 88

Variational Analysis of Ionic Solution

EnVarA

Generalization of Chemical Free

Energy

Eisenberg, Hyon, and Liu from Poisson Eq. Number Densities Lagrange Multiplier Dielectric Coefficient





2 1

2

k T c

B

( log

n

c

n

c

p

log

c

p

)

E

d x

  







 















 















 









Microscopic

Finit

Electrostatic Entropy e Size Eff

( e atom ct ic)

Solid Spheres





₁ 2

2 IP

( )

E

t



u

w



















Hydrodynamc Potential Energy Hydrodynamic _{Equation of State} Kinetic Energy

(hydrodynamic)

Primitive Phase;











Macroscopic

(89)

EnVarA

Dissipation Principle for Ions

Hard Sphere Terms Permanent Charge of protein

time

Number Density

Thermal Energy

valence

proton charge 2

,

= , = ,

i i i

B i i j j

B i

i n p j n p

D c

c

k T

z e

c d y

dx

k T

c







 

 







 













=

Conservative







, = , = , , = 0 ,

1

log

2

i

B i i i i i j j

i n p i n p i j n p

c

d

k T

c

z ec

c d y dx

dt













(90)

90

Replacement of “Law of Mass Action”

is

Feasible for Electrolyte Solutions

because

come from

Interactions

mostly from

(91)

of ions (e.g. activity = free energy per mole)

are known to come from interactions of their

Diameter and Charge

and dielectric ‘constant’ of ionic solution

(92)

92

Learned from Douglas Henderson, J.-P. Hansen, and Stuart Rice…Thanks!

Electrolytes

in a solution are a

Highly Compressible Plasma

of Interacting Spherical Particles

Central Result of Physical Chemistry

although the

Liquid

itself is

Incompressible

Debye-Hückel and Poisson-Nernst-Planck PNP

(93)

Ion Channels

are the

Valves of Cells

Ion Channels are Devices

*

that Control Biological Function

Chemical Bonds are lines Surface is Electrical Potential

Redis negative (acid)

Blue is positive (basic)

Selectivity

Different Ions

carry

Different Signals

Figure of ompF porin by Raimund Dutzler

~30 Å

0.7 nm = Channel Diameter

+

Ions in Water

are the

Liquid of Life

3 Å

K+

Na+