short course mathematical molecular biology - sjtuins.sjtu.edu.cn/files/common/20160623095502_sjtu...

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Short Course

Mathematical Molecular BiologyBob Eisenberg

Shanghai Jiao Tong University

Sponsor Zhenli Xu

u

2

How can we use mathematics to describe biological systems?

I believe some biology isPhysics ‘as usual’‘Guess and Check’

But you have to know which biology!

u

3

All of Biology occurs in Salt Solutions of definite composition and concentration

and that matters!

Salt Water is the Liquid of LifePure H2O is toxic to cells and molecules!

Salt Water is a Complex FluidMain Ions are Hard Spheres, close enough

Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl-

3 Å

K+Na+ Ca++ Cl-

Page 4

Trajectories in Condensed Phases are Noisy

Note: Brownian noise looks the same on all scales! Function has unbounded variation, crossing any line an infinite number of times in any interval no matter how small.

Page 5

Instruction to BobShow Videos!!!

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Trajectories Over Barriers in Condensed Phases are Noisy

Barcilon, Chen, Ratner, Eisenberg

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From Trajectories to Probabilitiesin Diffusion Processes

‘Life Work’ of Ze’ev Schuss Department of Mathematics, Tel Aviv University

Theory and Applications of Stochastic Differential Equations. 1980 John Wiley

Theory And Applications Of Stochastic Processes: An Analytical Approach 2009 Springer

Singular perturbation methods for stochastic differential equations of mathematical physics.

SIAM Review, 1980 22: p. 116-155.

Schuss, Nadler, Singer, Eisenberg

10

Here I start from

Stochastic PDEand

Field Theory

Other methodsgive nearly identical results

MSA (Mean Spherical Approximation)SPM (Primitive Solvent Model)

Non-equil MMC (Boda, Gillespie) several forms

DFT (Density Functional Theory of fluids, not electrons)DFT-PNP (Poisson Nernst Planck)

EnVarA (Energy Variational Approach)Steric PNP (simplified EnVarA)

Poisson Fermi

MATHField

Theory

ChemistryModels

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Solved with PNP including Correlations

Other methodsgive nearly identical results

MMC Metropolis Monte Carlo (equilibrium only)DFT (Density Functional Theory of fluids, not electrons)

DFT-PNP (Poisson Nernst Planck)MSA (Mean Spherical Approximation)

SPM (Primitive Solvent Model)EnVarA (Energy Variational Approach)

Non-equil MMC (Boda, Gillespie) several formsSteric PNP (simplified EnVarA)

Poisson Fermi

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Always start with Trajectoriesbecause

Always start with TrajectoriesZe’ev Schuss

Department of Mathematics, Tel Aviv University

1) Trajectories are the equivalent of SAMPLES in probability theory

2) Trajectories satisfy PHYSICAL boundary conditions

3) Trajectories satisfy classical PHYSICAL ordinary differential equations(we hope)

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From Trajectories to Probabilitiesin Diffusion Processes

‘Life Work’ of Ze’ev Schuss Department of Mathematics, Tel Aviv University

Theory and Applications of Stochastic Differential Equations1980

Theory and Applications Of Stochastic Processes: An Analytical Approach 2009

Singular perturbation methods for stochastic differential equations of mathematical physics

SIAM Review, 1980 22: 116-155

Schuss, Nadler, Singer, Eisenberg

Page 14

Trajectories in Condensed Phases are Noisy

Note: Brownian noise looks the same on all scales! Function has unbounded variation, crossing any line an infinite number of times in any interval no matter how small.

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We start with Langevin equations of charged particlesSimplest stochastic trajectories

areBrownian Motion of Charged Particles

Einstein, Smoluchowski, and Langevin ignored chargeand therefore

do not describe Brownian motion of ions in solutions

Once we learn to count Trajectories of Brownian Motion of Charge,we can count trajectories of Molecular Dynamics

Opportunity and Need

We useTheory of Stochastic Processes

to go

from Trajectories to Probabilities

Schuss, Nadler, Singer, Eisenberg

16Schuss, Nadler, Singer, Eisenberg

Langevin Equations Bulk Solution

; 2kp

k x q pp pkk k

f kTm mx x w Positive cat ion,

e.g., p = Na+

;

Newton's Law Friction & Noise

2kn

k x q nn nkk k

f kTm mx x w

Negative an ion, e.g., n = Cl¯

Global Electric Forcefrom all charges including

Permanent charge of Protein,Dielectric Boundary charges,Boundary condition charge

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00

0( ) ( ) ( ) ( )div ( )xs k ik

i

px x ix xef q xe Ef ze

ρΡ

Electric Force in Ion Channelsnot assumed

Excess ‘Chemical’

Force

‘All Spheres” Implicit Solvent

‘Primitive’ Model

GLOBAL Electric Forcefrom all charges including

Permanent charge of Protein,Dielectric Boundary charges,Boundary condition charge,

MOBILE IONS

Schuss, Nadler, Singer, Eisenberg

Total Force

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From Trajectories to Probabilities

Joint probability density of position and velocity

Coordinates are positions and velocities of N particles in 12N dimensional phase space

Schuss, Nadler, Singer, Eisenberg

c cj j

c c c c c

x vj j j j j

cj c

j

kTmp v p v v f m p p L

with Fokker Planck Operator

,, x vp x v Pr2N

j = 1

Main Result of Theory of Stochastic ProcessesSum the trajectories

Sum satisfies Fokker-Planck equation

More MathMany papers

• We actually performed the sum and showed it was the same as a MARGINAL PROBABILITY estimator of SINGLET CONCENTRATION defined in chemistry

• We actually did a nonequilibrium BBGKY expansion with electrostatic & steric correlations

• We like everyone else had to assume a closure

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1) Nadler, B., T. Naeh and Z. Schuss (2001). SIAM J Appl Math 62: 443-447.

2) Nadler, B., T. Naeh and Z. Schuss (2003). "SIAM J Appl Math 63: 850-873.

3) Nadler, B., Z. Schuss and A. Singer (2005). "" Physical Review Letters 94(21): 218101.

4) Nadler, B., Z. Schuss, A. Singer and B. Eisenberg (2003). Nanotechnology 3: 439.

5) Nadler, B., Z. Schuss, A. Singer and R. Eisenberg (2004). Journal of Physics: Condensed Matter 16: S2153-S2165.

6) Schuss, Z., B. Nadler and R. S. Eisenberg (2001). Physical Review E 64: 036116 1-14.

7) Schuss, Z., B. Nadler and R. S. Eisenberg (2001). "Phys Rev E Stat Nonlin Soft Matter Phys 64(3 Pt 2): 036116.

8) Schuss, Z, B Nadler, A Singer, R Eisenberg (2002) Unsolved Problems Noise & Fluctuations, UPoN 2002, Washington, DC AIP

9) Singer, A, Z Schuss, B Nadler, R. Eisenberg (2004) Physical Review E Statistical Nonlinear Soft Matter Physics 70 061106.

10) Singer, A, Z Schuss, B Nadler, R Eisenberg (2004). Fluctuations & Noise in Biological Systems II V. 5467. D. Abbot, S. M.

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Finite OPEN System Fokker Planck EquationBoltzmann Distribution

TrajectoriesConfigurations

NonequilibriumEquilibriumThermodynamics Schuss, Nadler, Singer &

Eisenberg

Thermodynamics Device Equation

lim ,N V

StatisticalMechanics

Theory of Stochastic Processes

Conditional PNP

0 |

| |

yy ye y x

y y

y

x x

P

|1 0y y xy y

xxx e

m

Other Forces

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Electric Force depends on Conditional Density of Charge

Nernst-Planck gives UNconditional Density of Charge

Schuss, Nadler, Singer, Eisenberg

Closure Needed: CORRELATIONS‘Guess and Check’

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Probability and Conditional Probability are

Measures on DIFFERENT Sets

that may be VERY DIFFERENT

Considerall trajectories that end on the right

vs.all trajectories that end on the left

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Conditioning and Correlations are VERY strong and GLOBAL

when Electric Fields are Involved, as in

Ionic Solutions and Channels

so cannot do the probability theorywithout variational methods

We had to guessthe conditioned sets

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Main Biological Ions are Hard Spheres, close enough

Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl-

General Theory of Hard Spheresis now available

Thanks to Chun Liu, more than anyone else

Took a long time, because dissipation, multiple fields, and multiple ion types had to be included

VARIATIONAL APPROACH IS NEEDED

3 Å

K+Na+ Ca++ Cl-

Page 25

EnVarA

212 ( log log )B n n p pk T c c c c E dx

Microscopic

Finite Size EffectElectrostatic Entropy

(atomic)

Solid Spheres

212 ( )IPE t u w

Macroscopic

Hydrodynamc Potential EnergyHydrodynamcEquation of StateKinetic Energy

(hydrodynamic)

Primitive Phase;

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Energetic Variational ApproachEnVarA

Chun Liu, Rolf Ryham, and Yunkyong Hyon

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

12 0 E

'' Dissipative 'Force'Conservative Force

x u

Action Integral, after pullback Rayleigh Dissipation Function

Field Theory of Ionic Solutions: Liu, Ryham, Hyon, Eisenberg

Allows boundary conditions and flowDeals Consistently with Interactions of Components

Composite

Variational Principle

Euler Lagrange Equations

Shorthand for Euler Lagrange processwith respect to

x

Shorthand for Euler Lagrange processwith respect to

u

2

,

= , = ,

i i iB i i j j

B ii n p j n p

D c ck T z e c d y dx

k T c

=

Dissipative

,

= = , ,

0

, , =

1log

2 2i

B i i i i i j j

i n p i n p i j n p

ck T c c z ec c d y dx

ddt

Conservative

Hard Sphere Terms

Permanent Charge of proteintime

ci number density; thermal energy; Di diffusion coefficient; n negative; p positive; zi valence; ε dielectric constantBk T

Number Density

Thermal Energyvalence

proton charge

Dissipation Principle Conservative Energy dissipates into Friction

= ,

0

2122 i i

i n p

z ec

Note that with suitable boundary conditions

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28

12 0 E

'' Dissipative 'Force'Conservative Force

x u

is defined by the Euler Lagrange Process,as I understand the pure math from Craig Evans

which gives Equations like PNP

BUTI leave it to you (all)

to argue/discuss with Craig about the purity of the process

when two variations are involved

Energetic Variational ApproachEnVarA

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0ff ft

u 0f u

2 ( )ff f ffff f

pu kMt

Pressure ViscosityAcceleration Coupling DragConvective GradientAcceleration

uu u uu

( ) 0t

u

Ionic SolutionPrimitive Model Part 1

Solvent Water Phase treated as incompressible conductive dielectric

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12

x yx y,(| | ) =

| |i j

i j

a ac e

骣 + ÷ç ÷ç- ÷ç ÷-ç ÷桫

r rr r

Ionic SolutionPrimitive Model Part 2

Macroscopic and atomic scale combined.

Ions in incompressible conductive dielectric

2

12

12

( ) ( ( ) ( ))

( )

( )

f n p

n n p

p n p

pt

M k c c

c c c d

c c c d

Coupling Drag Coulomb Force

u u u

u u u x x

x x y y y y

x y yx y yLennard Jones Solid Sphere

Convective PressureAcceleration Acceleration Gradient

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PNP (Poisson Nernst Planck) for Spheres

Eisenberg, Hyon, and Liu

12,

14

12,

14

12 ( ) ( )= ( )

| |

6 ( ) ( )( ) ,

| |

n n n nn nn n n n

B

n p n pp

a a x yc cD c z e c y dy

t k T x y

a a x yc y d y

x y

Nernst Planck Diffusion Equationfor number density cn of negative n ions; positive ions are analogous

Non-equilibrium variational field theory EnVarA

Coupling Parameters

Ion Radii

=1

or( ) =

N

i i

i

z ec i n p 0ρ

Poisson Equation

Permanent Charge of Protein

Number Densities

Diffusion Coefficient

Dielectric Coefficient

valenceproton charge

Thermal Energy

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Semiconductor PNP EquationsFor Point Charges

ii i i

dJ D x A x xdx

Poisson’s Equation

Drift-diffusion & Continuity Equation

0

i ii

d dx A x e x e z xA x dx dx

0idJdx

Chemical Potential

ex*x

x x ln xii i iz e kT

Finite Size

Special Chemistry

Thermal Energy

ValenceProton charge

Permanent Charge of Protein

Cross sectional Area

Flux Diffusion Coefficient

Number Densities

( )i x

Dielectric Coefficient

valenceproton charge

Page 33

Layering Against Charged WallClassical Interaction Effect

Coupling Parameter λ= 0.5Lagrange Multiplier

Coupling Parameter λ= 0.8Lagrange Multiplier

Motivation and Assumption for Fermi-Poisson

Largest Effect of Crowded Chargeis

Saturation

Saturation cannot be described at all by classical Poisson Boltzmann approach

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Simulating saturation by interatomic repulsion(Lennard Jones)

is a singular mathematical challengeto be side-stepped if possible, particularly in three dimensions,

Eisenberg, Hyon and Liu (2010) JChemPhys 133: 104104

A Nonlocal Poisson-Fermi Model for Electrolyte Solutions

Jinn Liang Liu

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劉晉良

Jinn-Liang is first author on our papers

J Comp Phys (2013) 247:88J Phys Chem B (2013) 117:12051J Chem Phys (2014) 141: 075102J Chem Phys, (2014) 141: 22D532Physical Review E (2015) 92: 012711Chem Phys Letters (2015) 637: 1J Phys Chem B (2016) 120: 2658

MotivationNatural Description of Crowded Charge

is a Fermi Distribution

because it describes Saturationin a simple way

used throughout Physics

and Biophysics, where it has a different name!

Simulating saturation by interatomic repulsion (Lennard Jones)is a significant mathematical challenge

to be side-stepped if possibleEisenberg, Hyon and Liu (2010). JChemPhys 133: 104104

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( ) exp ( ) ( )ibath teric

i iC C S r r r

Boltzmann distribution in PhysiologyBezanilla and Villalba‐Galea J. Gen. Physiol. (2013) 142: 575–578

Saturates!

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Does not Saturate

Fermi Description usesEntropy of Mixture of Spheres

from Combinatoric Analysis

W is the mixing entropy of UNEQUAL spheres with N available NON-UNIFORM sites

1 1 1

1

2 2

1

2

! ( !( - )!) = .

N==

andk

k

W N N N NN N

WWW

combinations for species in all vacant sites

combinations for species, and so on, ? through

combinations for

combinations of to fill space an

water voids d compute robustly & efficiently

2 2

11

2

1 ! !

!K K

j j

jj

K

jj N N N

NW W

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Connection to volumes of spheres and voids, and other details are published in 5 papers

J Comp Phys (2013) 247:88 J Phys Chem B (2013) 117:12051J Chem Phys (2014) 141: 075102 J Chem Phys, (2014) 141: 22D532

Physical Review E (2015) 92:012711

Expressions in other literature are not consistent with this entropy

Fermi Description uses

Energy of Mixture of Spheres

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i

i

free energyElectrostatic + Entropy of Spheres and Voids

mole

(Electro)Chemical Potential and Voids i

Voids are Needed

It is impossible to treat all ions and water molecules as

hard spheres and

at the same time haveZero Volume of interstitial Voids

between all particles

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*Previous treatments

Bazant, Storey & Kornyshev,. Physical Review Letters, 2011. 106(4): p. 046102.Borukhov, Andelman & Orland, Physical Review Letters, 1997. 79(3): p. 435.

Li, B. SIAM Journal on Mathematical Analysis, 2009. 40(6): p. 2536-2566.Liu, J.-L., Journal of Computational Physics 2013. 247(0): p. 88-99.

Lu & Zhou, Biophysical Journal, 2011. 100(10): p. 2475-2485.Qiao, Tu & Lu, J Chem Phys, 2014. 140(17):174102

Silalahi, Boschitsch, Harris & Fenley, JCCT 2010. 6(12): p. 3631-3639.Zhou, Wang & Li Physical Review E, 2011. 84(2): p. 021901.

Consistent Fermi Approach is NovelConsistent Fermi approach has not been previously applied to ionic solutions

as far as we, colleagues, referees, and editors know

Previous treatments* have inconsistent treatment of particle size They do not reduce to Boltzmann functionals in the appropriate limitPrevious treatments often do not include non-uniform particle size

Previous treatments* are inconsistent with electrodynamics and nonequilibrium flows including convection

DetailsPrevious treatments do not include discrete water or voids.

They cannot deal with volume changes of channels, or pressure/volume in general Previous treatments do not include polarizable water

with polarization as an output

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42

Challenge

Can Simplest Fermi Approach

• Describe ion channel selectivity and permeation?

• Describe non-ideal properties of bulk solutions?

There are no shortage of chemical complexities to include, if needed!

Classical Treatments of Chemical Complexities

Evidence (start)

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*Topic in Lecture Courses at Jiaotong, thanks toZhenli Xuand at Politechnico Milano, thanks to RiccardoSacco

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Poisson Fermi Approach to

Bulk Solutions

Same Fermi Poisson Equations, different model of nearby atoms in

Hydration Shells

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Bulk SolutionHow well does the Poisson Fermi Approach

for Bulk Solutions?Same equations, different model of nearby atoms

Occupancy is 6 + 12 Waters*held Constant in

Model of Bulk Solutionin this oversimplified Poisson Fermi Model

Liu & Eisenberg (2015) Chem Phys Ltr 10.1016/j.cplett.2015.06.079

*in two shells: experimental Data on OccupancyRudolph & Irmer, Dalton Trans. (2013) 42, 3919 Mähler & Persson, Inorg. Chem. (2011) 51, 425

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ParametersOne adjustable

Chem Phys Ltrs (2015) 637 1

47

Activity CoefficientsNa+ Cl-

‘normalized’ free energy per mole

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Activity CoefficientsCa2+ Cl2¯

‘normalized’ free energy per mole

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Gramicidin AUnusual SMALL Bacterial Channel

often simulated and studiedMargaret Thatcher,

student of Nobelist Dorothy HodgkinBonnie Wallace leading worker

Validation of PNP Solvers with Exact Solution

following the lead of Zheng, Chen & Wei

J. Comp. Phys. (2011) 230: 5239.

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Three Dimensional TheoryComparison with Experiments

Gramicidin A

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Steric Effect is Large in (crowded) GramicidinPNPF spheres vs PNP points

Water Occupancy

K+ Occupancy

Currentvs

Voltage

Three Dimensional Calculation Starting with Actual Structure

Points

Points

Spheres

Spheres

Points

Spheres

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Cardiac Calcium Channel CaV.n

Binding Curve

Liu & Eisenberg J Chem Phys 141(22): 22D532

Lipkind-Fozzard Model

Na Channel

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Signature of Cardiac Calcium Channel CaV1.nAnomalous* Mole Fraction (non-equilibrium)

Liu & Eisenberg (2015) Physical Review E 92: 012711

*Anomalous because CALCIUM CHANNEL IS A SODIUM CHANNEL at [CaCl2] 10-3.4

Ca2+ is conducted for [Ca2+] > 10-3.4, but Na+ is conducted for [Ca2+] <10-3.

Ca Channel

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More Detail

COMPUTING FLOW

551PhysRev E (2006) 73:041512 2PhysRev Ltrs (2011) 106:046102 3JCompPhys (2013) 247:88 4J PhysChem B (2013) 117:12051

approximates dielectric of entire bulk solution including correlated motions of ions, following Santangelo 20061 with Liu’s correctedand consistent Fermi treatment of spheres.2,3,4

We introduce3,4 two second order equations and boundary conditionsThat give the polarization charge density water pol

Three Dimensional computation is facilitated by using 2nd order equations

0

( / ) terici i i i b i iD C z k T C C S

JJ e

2

2 2 1 ( )water cl

r

What is PNPF? PNPF = Poisson-Nernst-Planck-Fermi

Implemented fully in 3D Code to accommodate 3D Protein Structures

Flow

Force

2 2 1water cl

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Flows are Essential in Devices & BiologyStructure is Essential in Devices & Biology

Implemented fully in 3D Code to accommodate 3D Protein Structures

1) PNPF uses treatment by Santangelo 20061 used by Kornyshev 20112

of near/far fields crudely separated by fixed correlation length

2) PNPF introduces steric potential3,4 so unequal spheres are dealt with consistently

3) PNPF force equation reduces3,4 to pair of 2nd order PDE’s andAppropriate boundary conditions

that are consistent and allowRobust and Efficient Numerical Evaluation

4) PNPF combines Force Equation and Nernst-Planck Description of Flow

1PhysRev E (2006) 73:041512 2PhysRev Ltrs (2011) 106:046102 3JCompPhys (2013) 247:88 4J PhysChem B (2013) 117:12051

cl

Poisson-Fermi Analysis is NON-Equilibrium

Flows cease only at death

Computational Problems Aboundand are Limiting

if goal is to fit real data

It is very easy to get results that only seem to converge, and are in fact Not Adequate approximations to the converged solutions

Jerome, J. (1995) Analysis of Charge Transport. Mathematical Theory and Approximation of Semiconductor Models. New York, Springer-Verlag.

Markowich, P. A., C. A. Ringhofer and C. Schmeiser (1990). Semiconductor Equations. New York, Springer-Verlag.

Bank, R. E., D. J. Rose and W. Fichtner (1983). Numerical Methods for Semiconductor Device Simulation IEEE Trans. on Electron Devices ED-30(9): 1031-1041.

Bank, R, J Burgler, W Coughran, Jr., W Fichtner, R Smith (1990) Recent Progress Algorithms for Semiconductor Device SimulationIntl Ser Num Math 93: 125-140.

Kerkhoven, T. (1988) On the effectiveness of Gummel's method SIAM J. Sci. & Stat. Comp. 9: 48-60.Kerkhoven, T and J Jerome (1990). "L(infinity) stability of finite element approximations to elliptic gradient equations."

Numer. Math. 57: 561-575.

Scientists must grasp, …….not just reach, if we want devices to work and models to be transferrable

57

Computational Electronics has solved these problems over the last 40 years in thousands

of papers used to design our digital devicesDevices and calculations work

Models are transferrable

Vasileska, D, S Goodnick, G Klimeck (2010) Computational Electronics: Semiclassical and Quantum Device Modeling and Simulation. NY, CRC Press.

Selberherr, S. (1984). Analysis and Simulation of Semiconductor Devices. New York, Springer-Verlag.Jacoboni, C. and P. Lugli (1989). The Monte Carlo Method for Semiconductor Device Simulation. New York, Springer Verlag.

Hess, K. (1991). Monte Carlo Device Simulation: Full Band and Beyond. Boston, MA USA, Kluwer.Hess, K., J. Leburton, U.Ravaioli (1991). Computational Electronics: Semiconductor Transport and Device Simulation. Boston, Kluwer.

Ferry, D. K. (2000). Semiconductor Transport. New York, Taylor and Francis.Hess, K. (2000). Advanced Theory of Semiconductor Devices. New York, IEEE Press.

Ferry, D. K., S. M. Goodnick and J. Bird (2009). Transport in Nanostructures. New York, Cambridge University Press.

It is very easy to get results that only seem to converge, but are in fact not adequate approximations to the converged solutions.

Jerome, J. W. (1995). Analysis of Charge Transport. Mathematical Theory and Approximation of Semiconductor Models. New York, Springer-Verlag.

58

Keys to Successful Computation

1) Avoid errors by checking against analytical solutions of Guowei and collaborators

2) Avoid singularities (i.e., acid/base charges) on protein boundaries that wreck convergence

3) Use a simplified Matched Interface Boundary sMIB method of Guowei and collaborators modified to embed Scharfetter Gummel SG criteria of computational electronics (extended to include steric effects).

Scharfetter Gummel is REQUIREDto ENSURE CONTINUITY OF CURRENT

Charge Conservation is not enough

Scharfetter and Gummel, IEEE Trans. Elec. Dev.16, 64 (1969)P. Markowich, et al, IEEE Trans. Elec. Dev. 30, 1165 (1983).

Zheng, Chen, and G.-W. Wei, J. Comp. Phys. 230, 5239 (2011).Geng, S. Yu, and G.-W. Wei, J. Chem. Phys. 127, 114106 (2007).

S. M. Hou and X.-D. Liu, J. Comput. Phys. 202, 411 (2005).J.-L. Liu, J. Comp. Phys. 247, 88 (2013).

4) Modified Successive Over-relaxation SOR for fourth order PNPF 59

Poisson FermiStatus Report

Nonequilibrium implemented fully in 3D Codeto accommodate

3D Protein Structures

Only partially compared to experimentsIn Bulk or Channels, so far.

60

Poisson FermiStatus Report

• Gramicidin tested with real three dimensional structure, including flowPhysical Review E, 2015. 92:012711

• CaV1.n EEEE, i.e., L-type Calcium Channel, tested with homology model J Phys Chem B, 2013 117:12051 (nonequilibrium data is scarce)

• PNPF Poisson-Nernst-Planck-Fermi for systems with volume saturation

General PDE, Cahn-Hilliard Type, Four Order, Pair of 2nd order PDE’sNot yet tested by comparison to bulk data

J Chem Phys, 2014. 141:075102; J Chem Phys,141:22D532

Numerical Procedures tailored to PNPF have been testedJ Comp Phys, 2013 247:88; Phys Rev E, 2015. 92:012711

NCX Cardiac Ca2+/Na+ exchanger branched Y shape KNOWNstructure. Physical analysis of a transporter using consistent mathematicsand known crystallographic structure This is an all atom calculation withpolarizable water molecules as outputs J Phys Chem B 120: 2658

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62

NCX Sodium Calcium Transporter Crucial* to Cardiac Function

strongly implicated in short term memory and learning

*More than 1,000 experimental references in Blaustein & Lederer Physiological Reviews,1999

Green is Sodium

Blue is Calcium

Liu, J.-L., H.-j. Hsieh and B. Eisenberg (2016) J Phys Chem B 120: 2658-2669

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More Detail

INSIDE CHANNELS

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Steric Effect is Significant

Gramicidin is CrowdedShielding is Substantial

Electric Potential

Steric PotentialShielding

Shielding

Shielding has been ignored in many papers, whereResults are often at one concentration or unspecified concentration,as in most molecular dynamics

Channel is often described as a potential profileThis is inconsistent with electrodynamicsas in classical rate models

65

GramicidinTwo K+ Binding Sites

OUTPUTS of our calculations

Binding sites are prominent in NMR measurements & MD calculationsBUT they VARY

with conditions in any consistent model and socannot be assumed to be of fixed size or location

66

Inside GramicidinWater Density

Dielectric Functionan OUTPUT of

model

Liu & EisenbergJ Chem Phys 141: 22D532

67

Inside the Cardiac Calcium Channel

CaV1.n

Water Density

Dielectric FunctionAn Output of this Model

Liu & Eisenberg (2015) Phys Rev E 92: 012711 Liu & Eisenberg J Chem Phys 141(22): 22D532

68

Steric Potential Estimator of Crowding

Electric Potential

Inside the Cardiac Calcium Channel

CaV1.n

Liu & Eisenberg (2015) Phys Rev E 92: 012711

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The End

Any Questions?