Lambert w

Lambert W-FunctionProblem: W.eW = x, find W(x) - ?

Solution: the Lambert W-Function

Lambert W-FunctionRef. :

Lambert, J. H. "Observationes variae in Mathes in Puram." Acta Helvitica, physico-mathematico-anatomico-botanico-medica 3, 128-168, 1758.

Euler, L. "De serie Lambertina plurimisque eius insignibusproprietatibus." Acta Acad. Scient. Petropol. 2, 29-51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350-369, 1921.

Lambert W-FunctionThe Lambert W -function, also called the omega function or the product logfunction , is the inverse function of discovered by:

Johann Lambert,

Zurich/Berlin, 1758

Leonhard Euler,

St.-Petersburg Academy Academyof Science, 1783

and

Lambert W-FunctionW(1) = 0.56714 is called the omegaconstant and can be considered a sortof "golden ratio" of exponents.

The Lambert W -function has the series expansion!

Lambert W-Function

The real (left) and imaginary (right) parts of the analytic continuation of over the complex plane are illustrated above.

Euler, L. "De serie Lambertina plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29-51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350-369, 1921.

(x,y) Re: (x,y) Im:

Lambert W-FunctionThe General Problem :

The General Solution :

Lambert W-Function has numerousapplications:

1) Banwell and Jayakumar (2000) showed that a W-function describes therelation between voltage, current and resistance in a diode

2) Packel and Yuen (2004) applied the W -function to a ballistic projectile in thepresence of air resistance.

3) Other applications have been discovered in:

statistical mechanics, quantum chemistry, combinatorics, enzyme kinetics,

physiology of vision, engineering of thin films, hydrology,

analysis of algorithms (Hayes 2005) , and solar wind.

Lambert W-FunctionThe Isothermal Solar Wind Problem :

where v is the outflow velocity of the wind, which isthe quantity we wish to solve for, r is the distance (measured here from the center of the Sun), a is thespeed of sound in the outer solar atmosphere, whichis proportional to the temperature of the gas, and which we assume to be constant. Also, rc is the so-called ``Parker critical-point distance'' where the wind accelerates past the sound speed:

Steven R. Cranmer, New views of the solar wind with the Lambert W function,Am. J. Phys., 2005, Vol. 72, No. 11, 1397-1403.

Some applications of the Lambert W Function to Physical Chemistry

1) The kinetics of the electromechanical vesicle elongation

Kakorin, S. and Neumann, E. (1998) Kinetics of the electroporativedeformation of lipid vesicles. Ber. Bunsenges. Phys. Chem.102: 670-675.

Kakorin, S., Redeker, E. and Neumann, E. (1998) Electroporativedeformation of salt filled vesicles. Eur. Biophys. J. 27: 43-53.

Cell volume V0 > VVV0 E

efflux

Some applications of the Lambert W Function to Physical Chemistry

1) The kinetics of the electromechanical vesicle elongation

V t a mn

LambertWm

nm

C t( ) exp[ ( ) ]= +

2 2 2

11

1

mE N r

d aw p=

3320

02 4

nN r c

d ap=

310

1 64 05

( / )

_

2C (m / n ) ln | m |=

4p p 2

0 w 9

3 r Nd V VE 32dt 160 d a

V(t 0) 0

= = =

Cell volume V0 > VVV0 Eefflux

Some applications of the Lambert W Function to Physical Chemistry

,

2) Conductivity of electroporated lipid bilayer membranes

Kakorin, S. and Neumann, E. (2002) Ionic conductivity of electroporatedlipid bilayer membranes, Bioelectrochem., 56: 163-166.

Griese, T., Kakorin, S. and Neumann, E. (2002) Conductometric and electrooptic relaxation spectrometry of lipid vesicle electroporation at high fields, Phys. Chem. Chem. Phys. 4: 1217-1227.

E

d

( )2 0p im0

im

F D c(0) c(d) RT h Fexp (1 ) | |RT F d RT

+ =

Some applications of the Lambert W Function to Physical Chemistry

,

0 0im

p exex

F ( 3 a E n / 2 )LambertW expRT

= )hEa3F/(RTd4 2 =

n = h / d

0 0p 0 p p im

ex

a Fexp n | | (1 f )2d RT

=

Integrated Nernst-Planck equation for the membrane conductivity:

( ) RT/)d(c)0(cDF20 += ))F/(RT1( 0im=

2) Conductivity of electroporated lipid bilayer membranes

Solution:

Lambert W-Function

A.R. Tzafriri, E.R. Edelman, The total quasi-steady-state approximationis valid for reversible enzyme kinetics,Journal of Theoretical Biology 226 (2004) 303313.

A.R. Tzafriri, MichaelisMenten Kinetics at High EnzymeConcentrations,Bulletin of Mathematical Biology (2003) 65, 11111129.

S. Schnell and C. Mendoza,Enzyme kinetics of multiple alternative substrates,Journal of Mathematical Chemistry 27 (2000) 155170.

2) Enzyme Kinetics:

Lambert W-FunctionDas Michaelis-Menten-Modell:(Enzym - Reaktion mit einem Fliegleichgewicht)

1 cat

1

k kk

E S (ES) E P

+ +ZZZXYZZZEnzym - Substrat Komplex

Die Nhrung des fluss-stationren Zustandes:

1 1 catd[(ES)] k [E] [S] k [(ES)] k [(ES)] 0

dtBildung Zerfall Zerfall in P

aus E und S in E und S

= =

1 1 catk [E] [S] (k k ) [(ES)] = +

Michaelis-Menten-Modell

1 1 catk [E] [S] (k k ) [(ES)] = + Die Gesamtkonzentration an Enzym ist konstant:

0 0[E] [E] [(ES)]; [E] [E] [(ES)] const.= + = =Nach der Umformung: 1 0 1 catk ([E] [(ES)]) [S] (k k ) [(ES)] = +

1 0 1 cat 1 1 cat 1k [E] [S] (k k ) [(ES)] k [S] [(ES)] (k k k [S]) [(ES)] = + + = + +

cat 1M

1

k kKk

+ (M = mol/L, Michaelis Konstante)

1 0 0 0

1 cat1 cat 1 M

1

k [E] [S] [E] [S] [E] [S][(ES)] k kk k k [S] K [S][S]k

= = =++ + ++

1 cat

1

k kk

E S (ES) E P

+ +ZZZXYZZZ

Michaelis-Menten-ModellDie Bildungsgeschwindigkeit des Produktes:

0cat cat 0 cat 0

M M 0

00 cat 0

M 0

d[P] [S] [S ] [P]v k [(ES)] k [E] k [E]dt K [S] K [S ] [P]

[S ]v k [E] ; [P(t 0)] 0 ???K [S ]

= = = = + + = = =+