short course mathematical molecular biology · in diffusion processes ‘life work’ of ze’ev...
TRANSCRIPT
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Short Course
Mathematical Molecular BiologyBob Eisenberg
Shanghai Jiao Tong University
Sponsor Zhenli Xu
u
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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
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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-
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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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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
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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
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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
Page
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-
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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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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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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
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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
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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.
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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.
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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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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
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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
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Inside GramicidinWater Density
Dielectric Functionan OUTPUT of
model
Liu & EisenbergJ Chem Phys 141: 22D532
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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
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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?