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Enzyme reactions at the surface of living cells
J. theor. Biol. (1984) 109, 555-569
Enzyme Reactions at the Surface of Living Cells. I. Electric Repulsion of Charged Ligands and Recognition of Signals from the External Milieu JACQUES RICARD AND GEORGES NOAT
Centre de Biochimie et de Biologie Moldculaire du CNRS, BP 71, 13402 Marseille Cedex 9, France (Received 12 December 1983, and in revised form 7 March 1984) The dynamic behaviour of a polyelectrolyte-bound enzyme is studied when diffusion of substrate or diffusion of product is coupled to electric repulsion and to Michaelis-Menten enzyme reaction. The definition of the classical concepts of electric partition coefficients and Donnan potential of a polyelectrolyte membrane has been extended under global non-equilibrium conditions. This extension is permissible when a strong repulsion exists of substrate and product by the fixed negative charges of the membrane. Coupling between product diffusion, electric repulsion and enzyme reaction at constant advancement may result in a hysteresis loop of the partition coefficient as the product concentration is increased in the reservoir. This hysteresis loop vanishes as the rate of product diffusion increases. No hysteresis loop may occur when electric repulsion effects are coupled to substrate diffusion and reaction. The existence of multiple values of the partition coefficient for a fixed concentration of product implies that the membrane may store short-term memory of the former product concentration present in the external milieu. The occurrence of hysteresis generated by coupling enzyme reaction, product diffusion, electric partition effects at constant advancement of the reaction may be viewed as a sensing device of product concentration in the external milieu. Surprisingly, non-linearities required to generate this sensing device come from electrostatic effects and not from enzyme kinetics. M a n y enzymes are b o u n d at the outer surface of living cells. This is the case, in particular, for hydrolytic enzymes located in plant cell walls, in close contact with the external milieu. W h e n an enzyme is e m b e d d e d in a biological m e m b r a n e or in a cell wall diffusional resistances of substrate and product coupled to enzyme reaction m a y generate apparently complex kinetic b e h a v i o u r of the b o u n d enzyme. M o r e o v e r if the substrate and the product of the reaction are bearing charges, electrostatic partition o f these mobile ions m a y give rise to an a p p a r e n t co-operativity o f the enzyme, even if this enzyme were displaying classical M i c h a e l i s - M e n t e n kinetics (Engasser & Horvath, 1975; Rieard eta/., 1981). 555 0022--5193/84/160555 + 15 $03.00/0
© 1984 Academic Press Inc. (London) Ltd.
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These effects, which have been found to occur experimentally, are highly controlled by the ionic strength of the solution. Therefore an important question to be answered is to determine which kind of kinetic behaviour may occur when diffusion and electrostatic effects are coupled to an enzyme reaction, for biological membranes are polyanions and most enzymes have ions as substrates or products. In order to answer this question it is obviously of major importance to know whether substrate or product repulsion effect is explainable, under non-equilibrium conditions, on the basis of the classical Donnan theory. The classical concepts of Donnan potential and partition coefficient have been defined under equilibrium conditions and one may wonder whether these concepts may still be applied when the enzyme reaction is coupled to substrate and product diffusion. It has been shown that, under certain conditions, coupling diffusion with enzyme reaction may create multiple steady states of the system (Engasser & Horvath, 1974, 1976; Thomas et al., 1977). It is therefore important to know whether coupling the diffusion of a charged ligand with enzyme reaction in a charged membrane may generate multiple electrical states of this membrane. The possible existence of these multiple steady states at the surface of the cell may be considered as a process which allows sensing the external milieu by the cell. The aims of the present paper are twofold: firstly to determine whether the concepts of Donnan potential and electrostatic partition coefficient apply under steady state conditions of the system; and secondly to determine whether electric repulsion of substrate and product, when coupled to diffusion and reaction, may generate multiple steady state values of electric partition coefficient at the surface of the cell.
Theory THE
MODEL
Let us consider an enzyme which catalyses the following reaction: E-
kl[S]
" (ES, EPQ) ~
k2
k_ I
EQ-
k3
"E
k-3[Q]
P where S, P and Q are the substrate and the two products, respectively. In this scheme it is postulated that the first step of product release is nearly irreversible, for the appearance of P is determined under steady state conditions and (or) P has a poor affinity for the E Q complex. As previously mentioned this situation is precisely the one known to occur for a number of hydrolytic enzymes located in the outer membrane of plant cells (Crasnier,
ENZYME
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Ricard & Noat, 1982). Clearly any enzyme which follows the reaction scheme above must display in free solution classical Michaelis-Menten behaviour. However, when bound to a polyanionic membrane, the same enzyme may acquire novel kinetic properties. The simplest situation is the one where the bound enzyme molecules are arranged as a layer located close to the surface of the polyanion. The change of kinetic properties may be due to diffusional resistances of substrate and product as well as to electrostatic interactions between these ligands and the fixed charges of the membrane. This situation is schematized in Fig. 1.
f
,
J So
o,. "
Oo
F I G . I. S i m p l e m o d e l o f a n e n z y m e r e a c t i o n o c c u r r i n g in a m e m b r a n e .
If the external medium is unstirred, there exists a coupling between enzyme reaction and substrate transport. The flux of substrate in free solution, J~, is
oS°(x)
Js=-D--
z:~ o~,°(x)
DS°(x) RT
Ox
Ox
(1)
where D is the diffusion coefficient, z the valence of the substrate, S ° ( x ) its concentration which varies with the distance x and @°(x) the diffusion potential. 37 is the Faraday, R and T have their usual significance. If the system is in steady state and is such that the transport of electric potential does not vary in time,
o@°(x)
- - = 0
(2)
Ot
This leads to
o2~,°(x)
o-=
0
(3)
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G. N O A T
Equation (3) is formally analogous to the second Fick's law as applied to a potential and may be derived from equation (2) exactly as the second Fick's law is derived from the definition of a steady state (see for instance Castellan, 1973). Expression (3) may then be reexpressed as
-u
aO°(x) Ox
= ~ {~,°(o)- ~,°(m)} =
h~aq, °
m
(4)
where m is the distance between the bulk phase and the surface, ha the transport coefficient and A~ ° the diffusion potential difference between the bulk phase (distance o) and the membrane (distance m). Under these conditions, the diffusion potential d/°(x) varies linearly with the distance x (x ¢ [0, m]). The flux of substrate is then
_DOS°(X) +haSO(x) z~;
o
This equation may be easily integrated in the interval [0, m] and one finds
.I~ = ha{S°(o) - S°(m)}
(6)
where/~a is an apparent transport coefficient equal to
ha = ha
z~;AqJ° "/exp ~
(7)
)
and S°(o) an apparent bulk concentration of substrate, namely
S°(o)=S°(o) exp \
3
RT ]
(8)
Equation (6) is similar to the Goldman equation (Heinz, 1978) and implies that O°(x) vary linearly with the distance in the interval [0, m]. This in turn means that the variation of S°(x) cannot be a linear function of the distance x. A similar reasoning may be applied to product diffusion and need not be presented here.
Electrostatic partition and Donnan potential under global non-equilibrium conditions If substrate and product are anions bearing z and ;tz (z is a positive integer and A is such that Az is a positive integer as well) charges, respectively, one may define, under equilibrium conditions, that is in absence of
ENZYME
R E A C T I O N S AT C E L L S U R F A C E S
559
chemical reaction, an electric partition coefficient, ~, equal to
= ~
\~'i]
=exp\ RT ]
(9)
In this expression S and () are equilibrium concentrations, the superscripts o and i refer to the outside and the inside of the matrix. A~o is the electrostatic potential difference between the Donnan phase and the bulk phase, namely A~o ----~ o _ ~ . ~: is the Faraday, R and T have their usual significance.
V,°, O°(O)
~ 5"'(m)
FIG. 2. Coupling between diffusion, electric partition of reactants and enzyme reaction in a membrane. The enzyme molecules are located on an impermeable surface and are embedded in a polyanionic matrix. The concentration of substrate and product in the bulk phase are S°(o) and Q°(o). The corresponding concentrations, in the immediate vicinity of the surface are S°(m) and Q°(m). Inside the matrix the concentrations of substrate and product are S~(m) and Q~(m). (a) Coupling between substrate diffusion, electric partition and enzyme reaction. (b) Coupling between product diffusion, electric partition and enzyme reaction.
If this system is in a steady state and if there exists a coupling between diffusion of ions S and Q and an enzyme reaction, a steady state between these processes may occur. Even under these non-equilibrium conditions a partition coefficient may still be defined if substrate S and product Q are submitted to a strong electrostatic repulsion at the surface of the membrane, when entering the Donnan phase. The transport of the substrate from the bulk phase (distance o) to the membrane and the array of enzyme molecules (distance m) may be viewed as shown in Fig. 2 and in the following kinetic diagram Input-*
~
S ° ( o ) ~ . S°(m) . ~ S~(m)-* Output hd
gd~
(10)
where /~a is the apparent transport coefficient, ~ and /3 electrostatic coefficients which express the electrostatic repulsion of the substrate by the fixed negative charges of the matrix. These coefficients may be found to be
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J. R 1 C A R D A N D G. N O A T
of the form (Hill &Chen, 1970, 1971 ; Hill, 1977) a = exp (-~;A~b/2RT) /3 = exp (~A~b/2RT)
(11)
A¢ is the potential difference between the external milieu, at the surface of the matrix and the inside. In general it is different from a Donnan potential since the system is under non-equilibrium conditions, and the point is to discover which conditions allow ACt to become nearly identical to a Donnan potential, A0o. The steady state rate of transport of substrate in the Donnan phase may be expressed as v=
1 +ct+fl
S°(o)
(12)
If it is assumed now that S°(m) and Si(m) equilibrate rapidly during the steady state transport of S, diagram (10) may be reduced by eliminating the fast relaxing variables to (Cha, 1968; Reich & Selk'ov, 1982) Input~ S°(o) - . r,. ~ X ~ Output
(13)
X = g°(m) +.~'(m)
(14)
hdf
where and 1
f=~z+l
(15)
Under steady state conditions the rate of transport assumes the form V = l + /~a ~ Z gO(o )
(16)
Equations (12) and (16) are indeed different. However if exp (3:AO/2RT) >> 1 +exp (-~A4,/2RT)
(17)
equation (12) reduces to
v-
g°(o)
(18)
This situation obviously occurs when Aqs is large. Then the partition coefficient is much greater than one and equation (16) becomes
h"7 g°(o) v=~--
(19)
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Expressions (18) and (19) are indeed equivalent and A¢=AeD
(20)
Therefore if the electric potential difference between the matrix and the free solution in contact with that matrix is large enough, the second step of diagram (10) equilibrates rapidly. Then the electric potential difference is equivalent to a Donnan potential and the concept of partition coefficient may still be applied when the whole system is far from equilibrium. Coupling between diffusion, reaction and electric partition of ligands Taking account of the definition of the partition coefficient, the flux of substrate which reaches the enzyme molecules is J. = h~{g°(o)- ~zS'(m)}
(21)
Alternately the flux of product which leaves the array of enzyme molecules is, under steady state Jq = / ~ { ~ Z Q i ( m ) - (~°(o)}
(22)
The enzyme reaction rate may be expressed as VmS'(m)/ Ks Ve - 1 + S'(m)/ Ks + Q'(m)/ K o
(23)
where V,., Ks and KQ are the maximum reaction rate and Michaelis constants for substrate and product, respectively. It is convenient to rewrite equations (21), (22) and (23) in dimensionless form by setting S o. =
S'(m) Ks
So =
Qi(m) q~=
Ko
h* = haKs
vm
S°(o) Ks (~°(0)
q°=
Ko
(24)
h *' = h'a Ko
vm
If substrate diffusion is a rather "slow" process whereas product diffusion is a "'fast" one, there may be a coupling between substrate diffusion, partitioning effects and enzyme reaction, namely h*(so- ~Zs~)
s¢ =0 1 +s¢ +q~,
(25)
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RICARD
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Alternately if it is product diffusion which is "slow", whereas substrate diffusion is not, the relevant coupling equation between product diffusion and enzyme reaction assumes the form
s,.
h*'(~=q~. - qo) = 0
I +s~ +q,.
(26)
Under quasi-equilibrium conditions, at the surface of the membrane, the electric partition coefficient may be expressed as ~=
_ S°(m)/Ks_ s" Si(m)/Ks s~
(27)
and
~A~_ Q°(m)/Ko Q'(m)/Ko
-
q" q,~
(28)
The values of s" and q" represent scaled substrate and product concentration in the bulk phase at the surface of the membrane. Then the coupling equation (25) rewrites as
h*(so-s'~)
I
s.l~>, +s'l@ z + q . / @ x : - o
(29)
and the other coupling equation assumes the form s'~/@= 1 +s'l@" +q'l@~=
l,*,l~, ,,d ~ -
q°) = o
(3o)
One may easily show that for a monovalent anion X - , the electrostatic partition coefficient under equilibrium conditions is (Ricard et al., 1981) -
A-
Ix-]
(31)
where A- is the "concentration" or the density of fixed negative charges. Therefore under equilibrium conditions ~ monotonically declines as [X-] is increased. If z = A = 1, the expression of the electric partition coefficient at the surface of the membrane is A-
~'-SO(m)+QO(ra)
(32)
and one may wonder about the variation of ~ when the concentration of substrate or product is varied in the reservoir. In order to study the effect of variation of substrate or product concentration in the reservoir on the
ENZYME
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electric partition coefficient, one has to maintain at a constant value the degree of advancement of the reaction. Therefore for studying the effect of substrate concentration on ~ under global non-equilibrium conditions, one has to change simultaneously the product concentration in such a way that the ratio
Q°(m) P-so(m)
(33)
is held constant. Similarly, in order to study the effect of product concentration in the reservoir on ~, one has to change simultaneously the substrate concentration in that reservoir as to maintain constant the p value. It is possible to calculate how the concentrations of substrate and product should be varied in the reservoir in order to maintain constant this p value (see Appendix). When it is so the expression of the electric partition coefficient ~ (32) assumes the form = 8-z~= 8-s(34) s" q~ where 6s and
8q are
expressed as Z~t~s =
8q
1
Ks l + p
(35)
A- p KQ l + p
where A- is still the density or "concentration" of fixed negative charges. Taking account of equation (34) the equation of coupling between substrate diffusion and enzyme reaction assumes the form
8~ 8°\ h * ( s o - ~ ) ( 1 +~-~ + ~ ) -
8,
~-i = 0
(36,
which can be rearranged to
h*so~3-h*8,~2+{h*so(8,+6q)-6~}~-h'~8,(6~+6q)=O
(37)
Similarly if the product diffusion is coupled with enzyme reaction, the equation of coupling is
8~ ~2
h*'(
q 0-)-( l + ~8~ 5 + ~ -8., 5)=0
(38)
which can be rewritten as
h*'qo~ 3- h*'Sq~ 2 +{h*'qo(8, + 8.) + 6,}~ - h*'Sq(8~ + 8q) = 0 (39)
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J. R I C A R D A N D G. N O A T
Although both equations (37) and (39) represent coupling processes at constant advancement of the reaction, these equations are indeed incompatible. Equation (37) expresses the coupling between substrate diffusion and reaction when the product is assumed to diffuse at a very high rate in such a way that qo = q" (Fig. 3a). Alternately equation (39) describes the coupling between product diffusion and reaction when substrate diffusion is fast and such that So= s'~ (Fig. 3b). When coupling equations (37) and (39) both hold, the whole system is in steady state and its behaviour will be considered in the next paper (Ricard & Noat, 1984). (o)
qo
s~
q~
(b)
so
q~
%
FIG. 3. Coupling between substrate or product diffusion, electric repulsion and enzyme reaction. (a) Coupling between substrate diffusion, electric partition and enzyme reaction. The situation in this scheme is the one described by equation (37). Diffusion of product (right hand side) is assumed to be very fast with respect to that of the substrate in such a way that qo = q~ (see text). (b) Coupling between product diffusion, electric repulsion and enzyme reaction. The situation in this scheme is the one described by equation (39). Diffusion of the substrate is assumed to be very fast and such that s o = s~.
Algebraic treatment of equation (37) shows that this equation may have only one positive real root, whereas equation (39) may have three (see Appendix). This implies that when substrate diffusion is coupled to enzyme reaction, the electric partition coefficient monotonically declines as substrate concentration in the reservoir is varied at constant degree of advancement of the reaction (Fig. 4a). Alternately if coupling occurs between enzyme reaction and product diffusion, the partition coefficient exhibits a region of
ENZYME REACTIONS AT CELL SURFACES @
565
@
60
30
40
20
20
0
0"2
0"4
0'6
0"8
So
I'0
0
0'1
0"2
0'3
0.4
qo
0"5
FIG. 4. Variation of the electric partition coefficient, under non-equilibrium conditions, as a function of substrate or product concentration in the reservoir and at constant advancement of the reaction. (a) Variation of the electric partition coefficient as a function of dimensionless substrate concentration in the reservoir. This is the situation depicted by equation (37) and Fig. 3(a), that is a coupling between substrate diffusion, electric partition and reaction. Product diffusion is assumed to be very fast. Simulation of equation (37) is ettected with the following values: 8s=3 , Bq=6, ha*=0.06 (curve 1), 0.12 (curve 2), 0.5 (curve 3) and 5.0 (curve 4). (b) Variation of the electric partition coefficient as a function of dimensionless product concentration in the reservoir. The curves correspond to equation (39) and describe a coupling between product diffusion, electric repulsion and reaction. Substrate diffusion is assumed to be very fast. This is the situation shown in Fig. 3(b). Simulation of equation (39) is effected with the following values: 8s = 3, 8~ = 6, ha*'= 0.09 (curve 3), 0" 10 (curve 2), 0" 11 (curve 3), 0.12 (curve 4). Dotted arrows represent jumps of partition coefficient values when qo concentration is changed in the reservoir. instability (Fig. 4b). This implies that w h e n qo is varied at c o n s t a n t degree o f a d v a n c e m e n t o f the reaction, the c o r r e s p o n d i n g value o f the partition coefficient describes a hysteresis loop. S t u d y i n g the effect o f substrate or p r o d u c t c o n c e n t r a t i o n in the reservoir o n the electric partition coefficient at c o n s t a n t a d v a n c e m e n t o f the reaction implies that b o t h ligands are s i m u l t a n e o u s l y varied a c c o r d i n g to either e q u a t i o n (2') or (3') o f the A p p e n d i x .
Discussion T h e classical c o n c e p t s o f D o n n a n potential a n d electrostatic partition coefficient, which express quantitatively the strength o f electrostatic interaction b e t w e e n c h a r g e d ligands a n d the fixed negative charges o f a p o l y a n i o n , have b e e n defined u n d e r equilibrium conditions. I f an e n z y m e reaction involving c h a r g e d substrate a n d p r o d u c t o c c u r s in the polyelectrolyte matrix and if there exists a c o u p l i n g between this e n z y m e reaction, electric partition a n d diffusion o f substrate a n d p r o d u c t , the system m a y be in steady state. T h e n the reactions w h i c h describe this c o u p l i n g m a y b e c o m e extremely c o m p l e x (Shuler, A d s & Tsuchize, 1972; H a m i l t o n , S t o c k m e y e r & C o l t r o n ,
566
J. RICARD AND G. NOAT
1973). However it may be shown that during the steady state and far from equilibrium conditions, the concept of Donnan potential may still be applied at the surface of the polyelectrolyte matrix, if there is a strong repulsion of the mobile charges by the fixed charges of the matrix. It has already b e e r shown, in different experimental and theoretical studies, that diffusional resistances coupled to enzyme reaction may generate multiple steady states of substrate and product concentration inside the membrane. However this effect occurs if the enzyme reaction is inhibited by an excess substrate or in a more general way if this enzyme does not follow classical Michaelis-Menten kinetics (Engasser & Horvath, 1974, 1976; Thomas et al., 1977). The results presented in this paper show that if substrate and product are anions, their electric repulsion by the fixed negative charges of the membrane may generate a new type of behaviour when coupled to diffusion and reaction. Even if the enzyme does follow classical Michaelis-Menten kinetics, there may be multiple values of the partition coefficient and therefore of the electric potential, for fixed values of substrate and product concentration. If reaction is coupled to product, but not to substrate diffusion, the partition coefficient declines, at constant advancement of the reaction, as the product concentration increases in the reservoir and may even exhibit multiple values in a given range of product concentration. The existence of the hysteresis loop (Everett & Whiton, 1952; Everett & Smith, 1954; Everett, 1954, 1955; Enderby, 1955, 1956; Friboulet & Thomas, 1982) depends on the rate of product diffusion. The hysteresis loop disappears as the rate of product diffusion is increased. Multiple values of the partition coefficient imply that there exists multiple steady states of substrate and product at the surface of the polyelectrolyte membrane. The situation generated by electric effects is thus quite different from that originating from diffusion coupled to reaction in absence of electric effects. Hysteresis phenomena do not seem to exist when reaction is coupled to substrate diffusion. At least with the simple model investigated, the partition coefficient monotonically declines as the substrate concentration is raised in the reservoir. The existence of these multiple values of the partition coefficient, for fixed values of substrate and product concentrations, implies that the membrane may store short-term memory of the former concentrations of charged ligand in the bulk phase. The hysteresis loop generated by coupling electric effects, diffusion and reaction may therefore be considered as a sensing device of the ligand concentration in the external milieu. Owing to these coupling effects the surface of a cell may "recognize" whether the concentration of a ligand is increasing or decreasing in the external milieu and the electric properties of this surface must vary accordingly.
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REFERENCES CASTELLAN, G. W. (1973). Physical Chemistry. San Francisco: Addison-Wesley Publishing
Company. CHA, S. (1968). J. bioL Chem. 243, 820. CRASNIER, M., RICARD, J. & NOAT, G. (1982). FEBS Lett. 144, 309 ENDERaY, J. A. (1955). Trans. Farad. Soc. 51, 835. ENDERBY, J. A. (1956). Trans. Farad. Soc. 52, 106. ENGASSER, J. M. & HORVATH, C. (1974). Biochemistry 13, 3855. ENGASSER, J. M. & HORVATH, C. (1975). Biochem. J. 145, 431 ENGASSER, J. M. & HORVATH, C. (1976). In: Immobilized enzyme. Principle L Wingard, L. B., Katchalski-Katzir, E. & Gotdstein L. eds. pp. 127-220. New York: Academic Press. EVERETT, D. H. & WHITTON, W. I. (1952). Trans. Farad. Soc. 48, 749. EVERE'rr, D. H. & SMITH, F. W. (1954). Trans. Farad. Soe. 50, 187. EVERETT, D. H. (1954). Trans. Farad. So~ 50, 1077. EVERETT, D. H. (1955). Trans. Farad. Soc. 51, 155. FRIBOULET, A. & THOMAS, D. (1982). Biophys. Chem. 16, 153. HAMILTON, B. K., STOCKMEYER, L. J. & COLTON, C. K. (1973). J. theor. Biol. 41, 547. HEINZ, E. (1978). Mechanics and Energetics of Biophysical Transport. Berlin: Springer. HILL, T. L. & CHEN, Y. (1970). Proc. natn. Acad. Sci. U.S.A. 66, 607. HILL, T. L. & CHEN, Y. (1971). Bh~phys. J. 11, 685. HILL, T. L. (1977). Free Energy Transduction in Biology. New York: Academic Press. REICH, J. G. & SELK'OV, E. E. (1982). Energy Metabolism of the Cell. A Theoretical Treatise. London: Academic Press. RICARD, J., NOAT, G., CRASNIER, M. & JOB, D. (1981). Biochem. J. 195, 357. RICARD, J. & NOAT, G. (1984). J. theor. Biol. 109, 571. SCHULER, M. L., ARIS, R. & TSUCHIZE, H. M. (1972). J. theor. Biol. 35, 67. THOMAS, D., BARBOTIN, J. N., DAVID, H., HERVAGAULT,F. F. & ROME'I'FE,J. L. (1977). Proc. natn. Acad. ScL U.S.A. 12, 5314. APPENDIX
Variation of Substrate and Product Concentration in the Reservoir Required to M a i n t a i n the A d v a n c e m e n t o f the R e a c t i o n Constant I f t h e r e is a c o u p l i n g b e t w e e n s u b s t r a t e d i f f u s i o n a n d r e a c t i o n , b u t n o c o u p l i n g b e t w e e n p r o d u c t d i f f u s i o n a n d r e a c t i o n , q " = qo, t h e n
K o qo S~- K s p
(1')
a n d e q u a t i o n (36) c a n b e r e a r r a n g e d to
h* K o l l ( l + K O l ) q ~ d Ks ~ p \ Ks / . Ko 1 1 1 -~h~S°Ks p-~+h*s°-~-h*a
Ko 1 Ks p
K o 1 1) Ks p ~-,qo-h*so=O
(2')
W h e n p is m a i n t a i n e d c o n s t a n t , this e q u a t i o n s h o w s h o w t h e s c a l e d c o n c e n t r a t i o n s so a n d qo s h o u l d b e s i m u l t a n e o u s l y v a r i e d .
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Alternately if there exists a coupling between product diffusion and reaction but no coupling between substrate diffusion and reaction, s" = so and equation (38) rewrites as
Ks
Ks \ h*' Ks p__+l_h d,, _K_~Qp)so_ a qo = 0 KQ (3')
Again when p is constant this equation shows how So and qo should be simultaneously varied in the reservoir.
Real Positive Roots of Equations (37) and (39) If a third degree equation in ~ has three positive real roots, one must have ~3 -- (h i + A2 -t"A3)~2 -{-(A 1A2 -{"Al,,')t3"{-A2A3)~) -- A1A2A3= 0
(4')
where the As are the positive roots. One may therefore expect that equation (37) and (39) have three positive real roots (the so-called Descartes rule of signs). However there may be constraints between the coefficients of the equations and the values of So, qo, z., "d, t.,, "d, ~ ~'~ and 8q are of necessity positive. This may imply that for realistic values of the above parameters and of So and qo, equations (37) and (39) in the main text may have only one real positive and two imaginary roots. Equation (37) of the main text may be rewritten as
~'3-~'~+ 8,+Sq-h-~s° ~'-~(8~+8q)=0 S0 SO
(5')
Comparison with equation (3') shows that 8s so
- - = A 1+ A 2 + A 3
AIA2A2
a, + aq =
(6')
At + A 2 '+A 3
1
A)A2A3
AIA2+AIA3 +A2A 3
h*
(A1 + A 2 + A 3 ) 2
A~+A2+A 3
Clearly the expression of l/h*, and therefore of h*, is of necessity negative which is a physical impossibility. Therefore equations (2') and (37) may have only one positive real and two imaginary roots. Similarly equation (39) of the main text may be reexpressed as
(
~3-~--q~2+ 8~+8q qo ha qo]
qo
(8, +qo) = 0
(7')
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Comparison with equation (4') shows that 8q=AI+A2+A 3 qo
8, + 8q
h 1A2A3 A! + A 2 + A 3
(8')
6s 1
AIA2A3 qo h*'=AIA2+AIA3+A:Aa AI+A2+A3
The expressions of 8q/q0, 8s + 8q and 6J(h*'qo) are all positive and therefore equations (7') or (39) may have three positive real roots. It is of interest to study the effect of the magnitude of the sealed diffusion constant h*' on the hysteresis loop. If the value of h*' is very large, equation (7') reduces to ~3 _6q ~2 + (6s + 6q)~ _6q (8, + 8q) = 0 qo qo
(9')
and one must have A1A2+AIA3 +A2A3 -
A1A2A3 A! + A 2 + A 3
(10')
which is clearly impossible. Therefore as the rate of product diffusion is increased, equation (39) has only one real positive root and cannot generate a hysteresis loop. Alternately if the rate of product diffusion is made slower, equation (4') reduces to ~03_ 8q ~2
qo
6,
8q (6~ +6q) = 0
(11')
+ h*'q'-'----oo~-q'--oo
and one must have 8....q 3"1 + A 2 + A 3
qo
A!A2A3 t$s +tSq -- AI + A 2 +A3
(12')
8~ 1
AIA2+A!A3+AzA3 qo h*d' Ai +A2+A3
These expressions are indeed possible and therefore equation (11') may have three real positive roots. Decreasing the rate of product diffusion h*' results in the appearance of a hysteresis loop.
Enzyme Reactions at the Surface of Living Cells. I. Electric Repulsion of Charged Ligands and Recognition of Signals from the External Milieu JACQUES RICARD AND GEORGES NOAT
Centre de Biochimie et de Biologie Moldculaire du CNRS, BP 71, 13402 Marseille Cedex 9, France (Received 12 December 1983, and in revised form 7 March 1984) The dynamic behaviour of a polyelectrolyte-bound enzyme is studied when diffusion of substrate or diffusion of product is coupled to electric repulsion and to Michaelis-Menten enzyme reaction. The definition of the classical concepts of electric partition coefficients and Donnan potential of a polyelectrolyte membrane has been extended under global non-equilibrium conditions. This extension is permissible when a strong repulsion exists of substrate and product by the fixed negative charges of the membrane. Coupling between product diffusion, electric repulsion and enzyme reaction at constant advancement may result in a hysteresis loop of the partition coefficient as the product concentration is increased in the reservoir. This hysteresis loop vanishes as the rate of product diffusion increases. No hysteresis loop may occur when electric repulsion effects are coupled to substrate diffusion and reaction. The existence of multiple values of the partition coefficient for a fixed concentration of product implies that the membrane may store short-term memory of the former product concentration present in the external milieu. The occurrence of hysteresis generated by coupling enzyme reaction, product diffusion, electric partition effects at constant advancement of the reaction may be viewed as a sensing device of product concentration in the external milieu. Surprisingly, non-linearities required to generate this sensing device come from electrostatic effects and not from enzyme kinetics. M a n y enzymes are b o u n d at the outer surface of living cells. This is the case, in particular, for hydrolytic enzymes located in plant cell walls, in close contact with the external milieu. W h e n an enzyme is e m b e d d e d in a biological m e m b r a n e or in a cell wall diffusional resistances of substrate and product coupled to enzyme reaction m a y generate apparently complex kinetic b e h a v i o u r of the b o u n d enzyme. M o r e o v e r if the substrate and the product of the reaction are bearing charges, electrostatic partition o f these mobile ions m a y give rise to an a p p a r e n t co-operativity o f the enzyme, even if this enzyme were displaying classical M i c h a e l i s - M e n t e n kinetics (Engasser & Horvath, 1975; Rieard eta/., 1981). 555 0022--5193/84/160555 + 15 $03.00/0
© 1984 Academic Press Inc. (London) Ltd.
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These effects, which have been found to occur experimentally, are highly controlled by the ionic strength of the solution. Therefore an important question to be answered is to determine which kind of kinetic behaviour may occur when diffusion and electrostatic effects are coupled to an enzyme reaction, for biological membranes are polyanions and most enzymes have ions as substrates or products. In order to answer this question it is obviously of major importance to know whether substrate or product repulsion effect is explainable, under non-equilibrium conditions, on the basis of the classical Donnan theory. The classical concepts of Donnan potential and partition coefficient have been defined under equilibrium conditions and one may wonder whether these concepts may still be applied when the enzyme reaction is coupled to substrate and product diffusion. It has been shown that, under certain conditions, coupling diffusion with enzyme reaction may create multiple steady states of the system (Engasser & Horvath, 1974, 1976; Thomas et al., 1977). It is therefore important to know whether coupling the diffusion of a charged ligand with enzyme reaction in a charged membrane may generate multiple electrical states of this membrane. The possible existence of these multiple steady states at the surface of the cell may be considered as a process which allows sensing the external milieu by the cell. The aims of the present paper are twofold: firstly to determine whether the concepts of Donnan potential and electrostatic partition coefficient apply under steady state conditions of the system; and secondly to determine whether electric repulsion of substrate and product, when coupled to diffusion and reaction, may generate multiple steady state values of electric partition coefficient at the surface of the cell.
Theory THE
MODEL
Let us consider an enzyme which catalyses the following reaction: E-
kl[S]
" (ES, EPQ) ~
k2
k_ I
EQ-
k3
"E
k-3[Q]
P where S, P and Q are the substrate and the two products, respectively. In this scheme it is postulated that the first step of product release is nearly irreversible, for the appearance of P is determined under steady state conditions and (or) P has a poor affinity for the E Q complex. As previously mentioned this situation is precisely the one known to occur for a number of hydrolytic enzymes located in the outer membrane of plant cells (Crasnier,
ENZYME
REACTIONS
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557
Ricard & Noat, 1982). Clearly any enzyme which follows the reaction scheme above must display in free solution classical Michaelis-Menten behaviour. However, when bound to a polyanionic membrane, the same enzyme may acquire novel kinetic properties. The simplest situation is the one where the bound enzyme molecules are arranged as a layer located close to the surface of the polyanion. The change of kinetic properties may be due to diffusional resistances of substrate and product as well as to electrostatic interactions between these ligands and the fixed charges of the membrane. This situation is schematized in Fig. 1.
f
,
J So
o,. "
Oo
F I G . I. S i m p l e m o d e l o f a n e n z y m e r e a c t i o n o c c u r r i n g in a m e m b r a n e .
If the external medium is unstirred, there exists a coupling between enzyme reaction and substrate transport. The flux of substrate in free solution, J~, is
oS°(x)
Js=-D--
z:~ o~,°(x)
DS°(x) RT
Ox
Ox
(1)
where D is the diffusion coefficient, z the valence of the substrate, S ° ( x ) its concentration which varies with the distance x and @°(x) the diffusion potential. 37 is the Faraday, R and T have their usual significance. If the system is in steady state and is such that the transport of electric potential does not vary in time,
o@°(x)
- - = 0
(2)
Ot
This leads to
o2~,°(x)
o-=
0
(3)
558
J. R I C A R D
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G. N O A T
Equation (3) is formally analogous to the second Fick's law as applied to a potential and may be derived from equation (2) exactly as the second Fick's law is derived from the definition of a steady state (see for instance Castellan, 1973). Expression (3) may then be reexpressed as
-u
aO°(x) Ox
= ~ {~,°(o)- ~,°(m)} =
h~aq, °
m
(4)
where m is the distance between the bulk phase and the surface, ha the transport coefficient and A~ ° the diffusion potential difference between the bulk phase (distance o) and the membrane (distance m). Under these conditions, the diffusion potential d/°(x) varies linearly with the distance x (x ¢ [0, m]). The flux of substrate is then
_DOS°(X) +haSO(x) z~;
o
This equation may be easily integrated in the interval [0, m] and one finds
.I~ = ha{S°(o) - S°(m)}
(6)
where/~a is an apparent transport coefficient equal to
ha = ha
z~;AqJ° "/exp ~
(7)
)
and S°(o) an apparent bulk concentration of substrate, namely
S°(o)=S°(o) exp \
3
RT ]
(8)
Equation (6) is similar to the Goldman equation (Heinz, 1978) and implies that O°(x) vary linearly with the distance in the interval [0, m]. This in turn means that the variation of S°(x) cannot be a linear function of the distance x. A similar reasoning may be applied to product diffusion and need not be presented here.
Electrostatic partition and Donnan potential under global non-equilibrium conditions If substrate and product are anions bearing z and ;tz (z is a positive integer and A is such that Az is a positive integer as well) charges, respectively, one may define, under equilibrium conditions, that is in absence of
ENZYME
R E A C T I O N S AT C E L L S U R F A C E S
559
chemical reaction, an electric partition coefficient, ~, equal to
= ~
\~'i]
=exp\ RT ]
(9)
In this expression S and () are equilibrium concentrations, the superscripts o and i refer to the outside and the inside of the matrix. A~o is the electrostatic potential difference between the Donnan phase and the bulk phase, namely A~o ----~ o _ ~ . ~: is the Faraday, R and T have their usual significance.
V,°, O°(O)
~ 5"'(m)
FIG. 2. Coupling between diffusion, electric partition of reactants and enzyme reaction in a membrane. The enzyme molecules are located on an impermeable surface and are embedded in a polyanionic matrix. The concentration of substrate and product in the bulk phase are S°(o) and Q°(o). The corresponding concentrations, in the immediate vicinity of the surface are S°(m) and Q°(m). Inside the matrix the concentrations of substrate and product are S~(m) and Q~(m). (a) Coupling between substrate diffusion, electric partition and enzyme reaction. (b) Coupling between product diffusion, electric partition and enzyme reaction.
If this system is in a steady state and if there exists a coupling between diffusion of ions S and Q and an enzyme reaction, a steady state between these processes may occur. Even under these non-equilibrium conditions a partition coefficient may still be defined if substrate S and product Q are submitted to a strong electrostatic repulsion at the surface of the membrane, when entering the Donnan phase. The transport of the substrate from the bulk phase (distance o) to the membrane and the array of enzyme molecules (distance m) may be viewed as shown in Fig. 2 and in the following kinetic diagram Input-*
~
S ° ( o ) ~ . S°(m) . ~ S~(m)-* Output hd
gd~
(10)
where /~a is the apparent transport coefficient, ~ and /3 electrostatic coefficients which express the electrostatic repulsion of the substrate by the fixed negative charges of the matrix. These coefficients may be found to be
560
J. R 1 C A R D A N D G. N O A T
of the form (Hill &Chen, 1970, 1971 ; Hill, 1977) a = exp (-~;A~b/2RT) /3 = exp (~A~b/2RT)
(11)
A¢ is the potential difference between the external milieu, at the surface of the matrix and the inside. In general it is different from a Donnan potential since the system is under non-equilibrium conditions, and the point is to discover which conditions allow ACt to become nearly identical to a Donnan potential, A0o. The steady state rate of transport of substrate in the Donnan phase may be expressed as v=
1 +ct+fl
S°(o)
(12)
If it is assumed now that S°(m) and Si(m) equilibrate rapidly during the steady state transport of S, diagram (10) may be reduced by eliminating the fast relaxing variables to (Cha, 1968; Reich & Selk'ov, 1982) Input~ S°(o) - . r,. ~ X ~ Output
(13)
X = g°(m) +.~'(m)
(14)
hdf
where and 1
f=~z+l
(15)
Under steady state conditions the rate of transport assumes the form V = l + /~a ~ Z gO(o )
(16)
Equations (12) and (16) are indeed different. However if exp (3:AO/2RT) >> 1 +exp (-~A4,/2RT)
(17)
equation (12) reduces to
v-
g°(o)
(18)
This situation obviously occurs when Aqs is large. Then the partition coefficient is much greater than one and equation (16) becomes
h"7 g°(o) v=~--
(19)
ENZYME
REACTIONS
AT CELL
SURFACES
561
Expressions (18) and (19) are indeed equivalent and A¢=AeD
(20)
Therefore if the electric potential difference between the matrix and the free solution in contact with that matrix is large enough, the second step of diagram (10) equilibrates rapidly. Then the electric potential difference is equivalent to a Donnan potential and the concept of partition coefficient may still be applied when the whole system is far from equilibrium. Coupling between diffusion, reaction and electric partition of ligands Taking account of the definition of the partition coefficient, the flux of substrate which reaches the enzyme molecules is J. = h~{g°(o)- ~zS'(m)}
(21)
Alternately the flux of product which leaves the array of enzyme molecules is, under steady state Jq = / ~ { ~ Z Q i ( m ) - (~°(o)}
(22)
The enzyme reaction rate may be expressed as VmS'(m)/ Ks Ve - 1 + S'(m)/ Ks + Q'(m)/ K o
(23)
where V,., Ks and KQ are the maximum reaction rate and Michaelis constants for substrate and product, respectively. It is convenient to rewrite equations (21), (22) and (23) in dimensionless form by setting S o. =
S'(m) Ks
So =
Qi(m) q~=
Ko
h* = haKs
vm
S°(o) Ks (~°(0)
q°=
Ko
(24)
h *' = h'a Ko
vm
If substrate diffusion is a rather "slow" process whereas product diffusion is a "'fast" one, there may be a coupling between substrate diffusion, partitioning effects and enzyme reaction, namely h*(so- ~Zs~)
s¢ =0 1 +s¢ +q~,
(25)
562
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AND
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Alternately if it is product diffusion which is "slow", whereas substrate diffusion is not, the relevant coupling equation between product diffusion and enzyme reaction assumes the form
s,.
h*'(~=q~. - qo) = 0
I +s~ +q,.
(26)
Under quasi-equilibrium conditions, at the surface of the membrane, the electric partition coefficient may be expressed as ~=
_ S°(m)/Ks_ s" Si(m)/Ks s~
(27)
and
~A~_ Q°(m)/Ko Q'(m)/Ko
-
q" q,~
(28)
The values of s" and q" represent scaled substrate and product concentration in the bulk phase at the surface of the membrane. Then the coupling equation (25) rewrites as
h*(so-s'~)
I
s.l~>, +s'l@ z + q . / @ x : - o
(29)
and the other coupling equation assumes the form s'~/@= 1 +s'l@" +q'l@~=
l,*,l~, ,,d ~ -
q°) = o
(3o)
One may easily show that for a monovalent anion X - , the electrostatic partition coefficient under equilibrium conditions is (Ricard et al., 1981) -
A-
Ix-]
(31)
where A- is the "concentration" or the density of fixed negative charges. Therefore under equilibrium conditions ~ monotonically declines as [X-] is increased. If z = A = 1, the expression of the electric partition coefficient at the surface of the membrane is A-
~'-SO(m)+QO(ra)
(32)
and one may wonder about the variation of ~ when the concentration of substrate or product is varied in the reservoir. In order to study the effect of variation of substrate or product concentration in the reservoir on the
ENZYME
REACTIONS
AT
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SURFACES
563
electric partition coefficient, one has to maintain at a constant value the degree of advancement of the reaction. Therefore for studying the effect of substrate concentration on ~ under global non-equilibrium conditions, one has to change simultaneously the product concentration in such a way that the ratio
Q°(m) P-so(m)
(33)
is held constant. Similarly, in order to study the effect of product concentration in the reservoir on ~, one has to change simultaneously the substrate concentration in that reservoir as to maintain constant the p value. It is possible to calculate how the concentrations of substrate and product should be varied in the reservoir in order to maintain constant this p value (see Appendix). When it is so the expression of the electric partition coefficient ~ (32) assumes the form = 8-z~= 8-s(34) s" q~ where 6s and
8q are
expressed as Z~t~s =
8q
1
Ks l + p
(35)
A- p KQ l + p
where A- is still the density or "concentration" of fixed negative charges. Taking account of equation (34) the equation of coupling between substrate diffusion and enzyme reaction assumes the form
8~ 8°\ h * ( s o - ~ ) ( 1 +~-~ + ~ ) -
8,
~-i = 0
(36,
which can be rearranged to
h*so~3-h*8,~2+{h*so(8,+6q)-6~}~-h'~8,(6~+6q)=O
(37)
Similarly if the product diffusion is coupled with enzyme reaction, the equation of coupling is
8~ ~2
h*'(
q 0-)-( l + ~8~ 5 + ~ -8., 5)=0
(38)
which can be rewritten as
h*'qo~ 3- h*'Sq~ 2 +{h*'qo(8, + 8.) + 6,}~ - h*'Sq(8~ + 8q) = 0 (39)
564
J. R I C A R D A N D G. N O A T
Although both equations (37) and (39) represent coupling processes at constant advancement of the reaction, these equations are indeed incompatible. Equation (37) expresses the coupling between substrate diffusion and reaction when the product is assumed to diffuse at a very high rate in such a way that qo = q" (Fig. 3a). Alternately equation (39) describes the coupling between product diffusion and reaction when substrate diffusion is fast and such that So= s'~ (Fig. 3b). When coupling equations (37) and (39) both hold, the whole system is in steady state and its behaviour will be considered in the next paper (Ricard & Noat, 1984). (o)
qo
s~
q~
(b)
so
q~
%
FIG. 3. Coupling between substrate or product diffusion, electric repulsion and enzyme reaction. (a) Coupling between substrate diffusion, electric partition and enzyme reaction. The situation in this scheme is the one described by equation (37). Diffusion of product (right hand side) is assumed to be very fast with respect to that of the substrate in such a way that qo = q~ (see text). (b) Coupling between product diffusion, electric repulsion and enzyme reaction. The situation in this scheme is the one described by equation (39). Diffusion of the substrate is assumed to be very fast and such that s o = s~.
Algebraic treatment of equation (37) shows that this equation may have only one positive real root, whereas equation (39) may have three (see Appendix). This implies that when substrate diffusion is coupled to enzyme reaction, the electric partition coefficient monotonically declines as substrate concentration in the reservoir is varied at constant degree of advancement of the reaction (Fig. 4a). Alternately if coupling occurs between enzyme reaction and product diffusion, the partition coefficient exhibits a region of
ENZYME REACTIONS AT CELL SURFACES @
565
@
60
30
40
20
20
0
0"2
0"4
0'6
0"8
So
I'0
0
0'1
0"2
0'3
0.4
qo
0"5
FIG. 4. Variation of the electric partition coefficient, under non-equilibrium conditions, as a function of substrate or product concentration in the reservoir and at constant advancement of the reaction. (a) Variation of the electric partition coefficient as a function of dimensionless substrate concentration in the reservoir. This is the situation depicted by equation (37) and Fig. 3(a), that is a coupling between substrate diffusion, electric partition and reaction. Product diffusion is assumed to be very fast. Simulation of equation (37) is ettected with the following values: 8s=3 , Bq=6, ha*=0.06 (curve 1), 0.12 (curve 2), 0.5 (curve 3) and 5.0 (curve 4). (b) Variation of the electric partition coefficient as a function of dimensionless product concentration in the reservoir. The curves correspond to equation (39) and describe a coupling between product diffusion, electric repulsion and reaction. Substrate diffusion is assumed to be very fast. This is the situation shown in Fig. 3(b). Simulation of equation (39) is effected with the following values: 8s = 3, 8~ = 6, ha*'= 0.09 (curve 3), 0" 10 (curve 2), 0" 11 (curve 3), 0.12 (curve 4). Dotted arrows represent jumps of partition coefficient values when qo concentration is changed in the reservoir. instability (Fig. 4b). This implies that w h e n qo is varied at c o n s t a n t degree o f a d v a n c e m e n t o f the reaction, the c o r r e s p o n d i n g value o f the partition coefficient describes a hysteresis loop. S t u d y i n g the effect o f substrate or p r o d u c t c o n c e n t r a t i o n in the reservoir o n the electric partition coefficient at c o n s t a n t a d v a n c e m e n t o f the reaction implies that b o t h ligands are s i m u l t a n e o u s l y varied a c c o r d i n g to either e q u a t i o n (2') or (3') o f the A p p e n d i x .
Discussion T h e classical c o n c e p t s o f D o n n a n potential a n d electrostatic partition coefficient, which express quantitatively the strength o f electrostatic interaction b e t w e e n c h a r g e d ligands a n d the fixed negative charges o f a p o l y a n i o n , have b e e n defined u n d e r equilibrium conditions. I f an e n z y m e reaction involving c h a r g e d substrate a n d p r o d u c t o c c u r s in the polyelectrolyte matrix and if there exists a c o u p l i n g between this e n z y m e reaction, electric partition a n d diffusion o f substrate a n d p r o d u c t , the system m a y be in steady state. T h e n the reactions w h i c h describe this c o u p l i n g m a y b e c o m e extremely c o m p l e x (Shuler, A d s & Tsuchize, 1972; H a m i l t o n , S t o c k m e y e r & C o l t r o n ,
566
J. RICARD AND G. NOAT
1973). However it may be shown that during the steady state and far from equilibrium conditions, the concept of Donnan potential may still be applied at the surface of the polyelectrolyte matrix, if there is a strong repulsion of the mobile charges by the fixed charges of the matrix. It has already b e e r shown, in different experimental and theoretical studies, that diffusional resistances coupled to enzyme reaction may generate multiple steady states of substrate and product concentration inside the membrane. However this effect occurs if the enzyme reaction is inhibited by an excess substrate or in a more general way if this enzyme does not follow classical Michaelis-Menten kinetics (Engasser & Horvath, 1974, 1976; Thomas et al., 1977). The results presented in this paper show that if substrate and product are anions, their electric repulsion by the fixed negative charges of the membrane may generate a new type of behaviour when coupled to diffusion and reaction. Even if the enzyme does follow classical Michaelis-Menten kinetics, there may be multiple values of the partition coefficient and therefore of the electric potential, for fixed values of substrate and product concentration. If reaction is coupled to product, but not to substrate diffusion, the partition coefficient declines, at constant advancement of the reaction, as the product concentration increases in the reservoir and may even exhibit multiple values in a given range of product concentration. The existence of the hysteresis loop (Everett & Whiton, 1952; Everett & Smith, 1954; Everett, 1954, 1955; Enderby, 1955, 1956; Friboulet & Thomas, 1982) depends on the rate of product diffusion. The hysteresis loop disappears as the rate of product diffusion is increased. Multiple values of the partition coefficient imply that there exists multiple steady states of substrate and product at the surface of the polyelectrolyte membrane. The situation generated by electric effects is thus quite different from that originating from diffusion coupled to reaction in absence of electric effects. Hysteresis phenomena do not seem to exist when reaction is coupled to substrate diffusion. At least with the simple model investigated, the partition coefficient monotonically declines as the substrate concentration is raised in the reservoir. The existence of these multiple values of the partition coefficient, for fixed values of substrate and product concentrations, implies that the membrane may store short-term memory of the former concentrations of charged ligand in the bulk phase. The hysteresis loop generated by coupling electric effects, diffusion and reaction may therefore be considered as a sensing device of the ligand concentration in the external milieu. Owing to these coupling effects the surface of a cell may "recognize" whether the concentration of a ligand is increasing or decreasing in the external milieu and the electric properties of this surface must vary accordingly.
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REFERENCES CASTELLAN, G. W. (1973). Physical Chemistry. San Francisco: Addison-Wesley Publishing
Company. CHA, S. (1968). J. bioL Chem. 243, 820. CRASNIER, M., RICARD, J. & NOAT, G. (1982). FEBS Lett. 144, 309 ENDERaY, J. A. (1955). Trans. Farad. Soc. 51, 835. ENDERBY, J. A. (1956). Trans. Farad. Soc. 52, 106. ENGASSER, J. M. & HORVATH, C. (1974). Biochemistry 13, 3855. ENGASSER, J. M. & HORVATH, C. (1975). Biochem. J. 145, 431 ENGASSER, J. M. & HORVATH, C. (1976). In: Immobilized enzyme. Principle L Wingard, L. B., Katchalski-Katzir, E. & Gotdstein L. eds. pp. 127-220. New York: Academic Press. EVERETT, D. H. & WHITTON, W. I. (1952). Trans. Farad. Soc. 48, 749. EVERE'rr, D. H. & SMITH, F. W. (1954). Trans. Farad. Soe. 50, 187. EVERETT, D. H. (1954). Trans. Farad. So~ 50, 1077. EVERETT, D. H. (1955). Trans. Farad. Soc. 51, 155. FRIBOULET, A. & THOMAS, D. (1982). Biophys. Chem. 16, 153. HAMILTON, B. K., STOCKMEYER, L. J. & COLTON, C. K. (1973). J. theor. Biol. 41, 547. HEINZ, E. (1978). Mechanics and Energetics of Biophysical Transport. Berlin: Springer. HILL, T. L. & CHEN, Y. (1970). Proc. natn. Acad. Sci. U.S.A. 66, 607. HILL, T. L. & CHEN, Y. (1971). Bh~phys. J. 11, 685. HILL, T. L. (1977). Free Energy Transduction in Biology. New York: Academic Press. REICH, J. G. & SELK'OV, E. E. (1982). Energy Metabolism of the Cell. A Theoretical Treatise. London: Academic Press. RICARD, J., NOAT, G., CRASNIER, M. & JOB, D. (1981). Biochem. J. 195, 357. RICARD, J. & NOAT, G. (1984). J. theor. Biol. 109, 571. SCHULER, M. L., ARIS, R. & TSUCHIZE, H. M. (1972). J. theor. Biol. 35, 67. THOMAS, D., BARBOTIN, J. N., DAVID, H., HERVAGAULT,F. F. & ROME'I'FE,J. L. (1977). Proc. natn. Acad. ScL U.S.A. 12, 5314. APPENDIX
Variation of Substrate and Product Concentration in the Reservoir Required to M a i n t a i n the A d v a n c e m e n t o f the R e a c t i o n Constant I f t h e r e is a c o u p l i n g b e t w e e n s u b s t r a t e d i f f u s i o n a n d r e a c t i o n , b u t n o c o u p l i n g b e t w e e n p r o d u c t d i f f u s i o n a n d r e a c t i o n , q " = qo, t h e n
K o qo S~- K s p
(1')
a n d e q u a t i o n (36) c a n b e r e a r r a n g e d to
h* K o l l ( l + K O l ) q ~ d Ks ~ p \ Ks / . Ko 1 1 1 -~h~S°Ks p-~+h*s°-~-h*a
Ko 1 Ks p
K o 1 1) Ks p ~-,qo-h*so=O
(2')
W h e n p is m a i n t a i n e d c o n s t a n t , this e q u a t i o n s h o w s h o w t h e s c a l e d c o n c e n t r a t i o n s so a n d qo s h o u l d b e s i m u l t a n e o u s l y v a r i e d .
568
J. R I C A R D
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Alternately if there exists a coupling between product diffusion and reaction but no coupling between substrate diffusion and reaction, s" = so and equation (38) rewrites as
Ks
Ks \ h*' Ks p__+l_h d,, _K_~Qp)so_ a qo = 0 KQ (3')
Again when p is constant this equation shows how So and qo should be simultaneously varied in the reservoir.
Real Positive Roots of Equations (37) and (39) If a third degree equation in ~ has three positive real roots, one must have ~3 -- (h i + A2 -t"A3)~2 -{-(A 1A2 -{"Al,,')t3"{-A2A3)~) -- A1A2A3= 0
(4')
where the As are the positive roots. One may therefore expect that equation (37) and (39) have three positive real roots (the so-called Descartes rule of signs). However there may be constraints between the coefficients of the equations and the values of So, qo, z., "d, t.,, "d, ~ ~'~ and 8q are of necessity positive. This may imply that for realistic values of the above parameters and of So and qo, equations (37) and (39) in the main text may have only one real positive and two imaginary roots. Equation (37) of the main text may be rewritten as
~'3-~'~+ 8,+Sq-h-~s° ~'-~(8~+8q)=0 S0 SO
(5')
Comparison with equation (3') shows that 8s so
- - = A 1+ A 2 + A 3
AIA2A2
a, + aq =
(6')
At + A 2 '+A 3
1
A)A2A3
AIA2+AIA3 +A2A 3
h*
(A1 + A 2 + A 3 ) 2
A~+A2+A 3
Clearly the expression of l/h*, and therefore of h*, is of necessity negative which is a physical impossibility. Therefore equations (2') and (37) may have only one positive real and two imaginary roots. Similarly equation (39) of the main text may be reexpressed as
(
~3-~--q~2+ 8~+8q qo ha qo]
qo
(8, +qo) = 0
(7')
E N Z Y M E R E A C T I O N S AT C E L L S U R F A C E S
569
Comparison with equation (4') shows that 8q=AI+A2+A 3 qo
8, + 8q
h 1A2A3 A! + A 2 + A 3
(8')
6s 1
AIA2A3 qo h*'=AIA2+AIA3+A:Aa AI+A2+A3
The expressions of 8q/q0, 8s + 8q and 6J(h*'qo) are all positive and therefore equations (7') or (39) may have three positive real roots. It is of interest to study the effect of the magnitude of the sealed diffusion constant h*' on the hysteresis loop. If the value of h*' is very large, equation (7') reduces to ~3 _6q ~2 + (6s + 6q)~ _6q (8, + 8q) = 0 qo qo
(9')
and one must have A1A2+AIA3 +A2A3 -
A1A2A3 A! + A 2 + A 3
(10')
which is clearly impossible. Therefore as the rate of product diffusion is increased, equation (39) has only one real positive root and cannot generate a hysteresis loop. Alternately if the rate of product diffusion is made slower, equation (4') reduces to ~03_ 8q ~2
qo
6,
8q (6~ +6q) = 0
(11')
+ h*'q'-'----oo~-q'--oo
and one must have 8....q 3"1 + A 2 + A 3
qo
A!A2A3 t$s +tSq -- AI + A 2 +A3
(12')
8~ 1
AIA2+A!A3+AzA3 qo h*d' Ai +A2+A3
These expressions are indeed possible and therefore equation (11') may have three real positive roots. Decreasing the rate of product diffusion h*' results in the appearance of a hysteresis loop.