Notizen 127
The R ole of E lectrical and V olum e Transport in H eterogeneous R eaction System s
E. Kahrig, H. Beßerdich, E. Brecht, and D. K irstein
Z entralinstitut für M olekularbiologie der Akadem ie der Wissenschaften der DDR, Selbständige A bteilung Bio
katalyse, Berlin
(Z. Naturforsch. 30 c, 127 — 128 [1975] ; received Septem ber 13, 1974)
T ransport Reaction Coupling, Periodic C oncentration Profiles For autocatalysis coupled w ith diffusion the local periodic concentration profiles are influenced essentially by electrical transport and convection.
W hen diffusion is negligible for M ichaelis-M enten-kinetics equations are given to determ ine the kinetic param eters
Fmax a° d K m from transport m easurem ents.
Concentration profiles of substrate and product in enzyme membranes have been computed for dif
fusion and chemical reaction of zero or first order as well as Michaelis-Menten kinetics 1-6. A m athe
matical treatment considering other transport p ro cesses (electrical transport, convection) is not known till now, but the consideration of these phenomena can lead to additional param eters valuable for in fluencing and optimizing the systems.
If the substrate is transported by diffusion, elec
trical current and volume flow and the following assumptions are m ade: steady state; electrical field intensity E, electrical mobility u, diffusion coef
ficient D and convective velocity v cony_ are indepen
dent of the local coordinate; coupling of the tran s
port processes negligible, the mass balance gives
d2c _ u E + tyonv. dc _ r
d r 2 I) ~ dx D
( c = substrate concentration, x — local coordinate, r = reaction ra te ).
W ith the boundary conditions c= c0 at x = 0 c = c t at x = 1
the concentration profiles of the substrate are com
puted i n 7 for reactions of zero and first order.
They are complicated functions of the param eters U E + Veonv. 7 £() i ^1
« - ---^ D > “ l “ “o ’ ( K 0 , K x = rate constants).
Requests for rep rin ts should be sent to Dr. E. K ahrig, Z entralinstitut für M olekularbiologie der Akadem ie der Wissenschaften der DDR, Selbständige A bteilung Bio
katalyse, X-1058 B erlin, W olliner Str. 71.
Assuming that the reaction includes an auto- catalytic step, for instance
A + C — 2 C
with a second substrate A of constant concentration a, the reaction rate is given by
r = — K ' a c = — K c .
It can be shown that now — depending on the sign of the discrim inant — various solutions are possible. It must be emphasized that a negative discrim inant leads to periodic solutions
a
c (a:) = e-x (A cos b' x + B sin b' x) where
The onset — b < 0 j and the wave length 2 7 l \
/ = “jjT") °f the spatial periodic concentration distributions are now dependent on diffusion and reaction as well as on charge and volume transport.
It is im portant that the latter phenomena inhibit the occurence of the periodicity, because if a = 0 there is always a periodical profile 8 - n . When dif
fusion is negligible compared to the other transport processes e. U^ ^ and - - - - - 1 ) ,E q. (1) can be simplified.
Integration gives linear profiles for zero order reaction and exponential profiles for first order reaction 1.
The shape of the concentration profiles depends above all on the param eters
K 0 K t
«0 — 77 5 — 77
U E -f- U c o n v . U E -\- V conVm
For Michaelis-Menten kinetics it follows:
(u E + uconv.) -{-K.^Eq— — = 0 (2) ax &m + c
( K 2 = turnover number, E0 = enzyme concentration, K m = M ichaelis-constant).
Eq. (2) gives the following implicite relation for the concentration profile:
K m In c + c = — a mx + K m In c0 + c0 (3) where
k2e0
— r>
U E + U g o n v .
Using Eq. (3) it is possible to determine the im portant kinetic param eters ümax = K 2 E0 and K m
This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution-NoDerivs 3.0 Germany License.
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128 Notizen by m easuring the substrate concentrations c0 and cx in the compartments on both sides of the membrane.
Provided that c0 , cx Km, K.2 E0 is given by K SE , - - C , ~ ± - ( u E + vconr.) and provided that cx ^ c0 , K m is given by
K 2E0 l
In i L fo
An experimental verification of these equations is 1 R. Goldm an, 0 . Kedem, and E. Katchalsky, Biochemistry
7, 4518 [1968].
2 E. Selegny, S. Avrameas, G. Broun, and D. Thomas, C.
R. Acad. Sei., P aris 266, 1431 [1968].
3 E. Selegny, G. Broun, J. Geffroy, and D. Thomas, J.
Chim. Phys. et Physico. Chim. Biol. 66, 391 [1969].
4 E. Selegny, G. Broun, and D. Thomas, Physiol. Veg. 9, 25 [1971].
5 E. Selegny, J.-P. Kernevez, G. Broun, and D. Thomas, Physiol. Veg. 9, 51 [1971].
not yet done. This procedure is sim ilar to that of Selegny et al. 2’ 3 for reaction and diffusion alone.
Sum m arizing it can be stated that
1. the spatial periodic concentration profiles for autocatalysis are influenced essentially by elec
trical and volume transport. If charge and volume transport occurs, the dynamic patterns in m embranes — first observed by de Simone, Beil and S c riv en 11 — should show characteristical changes e. g. of wave length and onset.
2. It is possible to determine the kinetic param eters Umax and K m if diffusion is negligible and a superposition of electrical and/or volume trans
port with Michaelis-Menten kinetics occurs.
6 D. Thomas, G. Broun, and E. Selegny, Biochimie 54, 229 [1972],
7 E. K ahrig, H. Besserdich, E. Brecht, and D. K irstein, A cta Biol. Med. Germ. 34, H eft 4 [1975].
8 H. G. Busse, J. Phys. Chem. 73, 750 [1969].
9 A. N. Z haikin and A. M. Z habotinsky, N ature 225, 535 [1970].
10 A. T. W infree, Science 175, 634 [1972].
11 J. A. De Simone, D. L. Beil, and L. E. Scriven, Science 180, 946 [1973].
