Biochem. J. (1990) 269, 115-122 (Printed in Great Britain)
Pattern formation in
an
115
immobilized bienzyme system
A morphogenetic model Sonia CORTASSA,*$ Hao SUN,t J. Pierre KERNEVEZt and Daniel THOMAS* Laboratoire de Technologie Enzymatique, U.R.A. N° 41 du Centre National de la Recherche Scientifique and t Departement de Genie Informatique, Universite de Technologie de Compiegne, Compiegne, France *
(CNRS),
Experimental and theoretical studies of a reaction-diffusion model of two immobilized enzymes participating in the cellular acid-base metabolism, namely glutaminase and urease, are presented. The system shows an unstable steady state at pH 6.0, where any perturbation will drive the system towards a more alkaline or more acidic pH, owing to the autocatalytic behaviour with respect to pH exhibited by both enzymes. When diffusion is coupled to reaction by means of immobilization, different patterns of the internal pH profile appear across the membrane. If the bienzymic membrane is subjected to a perturbation at its boundaries, of the same amplitude but in opposite directions, the internal pH evolves through an asymmetric pattern to attain a nearly symmetric distribution of pH. The pH value at the final steady state is more acidic or more alkaline than the initial state according to the initial and boundary conditions. The final nearly symmetric state is attained more rapidly when less enzyme is immobilized (1.8 x 10-4 M -s- as against 3.3 x 10-4 M -S-1 of total enzyme activity in the membrane volume). The experimental results agree rather well qualitatively with numerical predictions of the model equations.
INTRODUCTION One of the major challenges in biochemistry and biology today is the problem of the manner in which the activities of billions of molecules are co-ordinated in space and time to produce cellular functions. In spite of our everyday deeper knowledge of the structure and function of isolated biological molecules, understanding of the functional organization of the cell is rather less developed [1,2]. Many hypotheses have proposed rates ofchemical change (reactions) coupled to transport of matter (diffusion) as a plausible mechanism to explain how biological organization arises. These reaction-diffusion models have been extensively elaborated, although their experimental use lags far behind their theoretical development [3]. About 37 years ago, Turing [4] demonstrated mathematically that reaction-diffusion models are able to generate form, i.e. morphogenesis. The concept of a morphogen was introduced in the pioneering paper of Turing [4] as a name for a chemical substance able to generate form through a reaction-diffusion mechanism. Autocatalysis, or reaction activation by its product, or inhibition by its substrate, under far-from-equilibrium conditions, is a necessary requirement in Turing's [4] model for pattern formation and in all models derived from it [5-7]. Many examples of pattern formation have been reported in chemical and physical systems, such as chemical waves in the Belousov-Zhabotinsky reaction, and photochemical structures for a large number of photochromic and chromogenic compounds [8-10]. However, most examples of photochemical pattern formation or stable patterns in Belousov-Zhabotinsky reactions do not arise via a Turing instability, i.e. a diffusiondriven instability, but contain convective or surface effects [1 1,12]. In biochemical systems only a few experimental studies support the ability of reaction-diffusion mechanisms to produce patterns. The recent work of Aon et al. [13] shows a phenomenon of
pattern formation in a photobiochemical system. Those authors suggested that the non-linear reduction of an electron acceptor by photosynthetic activity, when coupled to diffusion, can trigger a symmetry change that redistributes a monotonic gradient of the electron acceptor in a banding pattern. The coupling of autocatalysis with diffusion has also been postulated to account for periodic precipitation processes, namely Liesegang rings, where the threshold for crystal growth would be the non-linear mechanism participating in the phenomenon [14]. The immobilization of enzyme systems gives rise to diffusional limitations (mass-transfer constraints) on the supply of substrates or removal of products in the immediate environment of the enzymes. In consequence, local metabolite-concentration profiles may be established across the membrane width, which will, in turn, determine the kinetic behaviour of the enzymes. In addition, immobilized enzymes constitute a well-defined system from the point of view of composition, thereby allowing knowledge and control of all variables. Thus modelling of these systems with reaction-diffusion equations is straightforward [15]. Although concentration profiles can be calculated from reaction-diffusion equations, little attention has been paid to their experimental visualization [16]. In the present work, a theoretical and experimental approach to a reaction-diffusion model of two immobilized enzymes participating in acid-base metabolism is developed. The enzymes involved are glutaminase and urease, which show autocatalytic behaviour with respect to pH. When diffusion is coupled to the autocatalytic reaction by means of immobilization, different patterns of the internal pH profile of the membrane appear. Theory predicts, and experiments confirm, that the evolution of the internal pH of the membrane towards a nearly symmetric steady state, through a transient asymmetric pattern of pH, is triggered by slight perturbations of the pH in an asymmetric fashion at the boundaries.
$ Present address and address for correspondence and reprint requests: Laboratory of Biochemistry, E. C. Slater Institute, University of Amsterdam, P.O. Box 20151, IOOOHD, Amsterdam, The Netherlands. Vol. 269
116
S. Cortassa and others
MODEL DESCRIPTION Nomenclature used Parameter Description Units Enzyme of the reaction i Ei Substrate and Product of M S, and Pi the reaction i Maximal rate of enzyme i V1 at a given pH Vi M(opt) Maximal rate of enzyme i at the optimal pH Ratio of maximal rate at a Nongiven pH to maximal rate dimensional at the optimal pH of the enzyme i Ionization constants of the M Kies,i Kie 2 enzyme i pH of the unstable steady NonpH* state dimensional M-1 ,3 Normalizing constant cm e Membrane width in the direction parallel with diffusion Diffusion coefficient for H+ cm2/s DH + M O = Vi me2/DH+ Kinetic parameter for enzyme i Characteristic time constant S d= e2/DH+ of the system NonNormalized factor of the dimensional system
The model considers an artificial protein membrane in which two enzymes, E1 and E2 are homogeneously immobilized. The product of the reaction catalysed by E1 is an acid, whereas that produced by E2 is a base. For the present experimental system, El and E2are glutaminase and urease respectively. The reactions involved may be schematically represented as follows:
Si
Pi
i=
following equation [17]:
viM(pH) -Kies 10p
+
I0-pH
Kies 10-PH + Ki K
i= 1,2 (1)
where Kiesl and K'es2 are the ionization constants of enzyme i, and V1M(opt) and VtM(pH) are the maximal rates at the optimum pH or at a given pH respectively. This expression is valid only if the relation between the ionization constants of the enzyme Kiesl/Kies2 > 102 holds [17]. The optimum pH for EB activity will occur at:
pHi
PKiesl + pKes2
(opt)
[PJ]/[P2] < 1,
and may be verified experimentally close to this pH. Although the expression may look rather ad hoc, it may be obtained by expanding the function pH = pH* + log [(/3 [P2] + 1)/(3 [P1]+ 1)] as a Taylor series and neglecting the quadratic and higher-order terms. This function was derived from the following reasoning: since P1 is an acid, an increase in its concentration will lead to a decrease in the pH with respect to pH*; the converse is true for an increase in the concentration of the alkaline product, P2. Unity is added to the [Pi] values in order to account for their positive contributions to the change of pH (since for an argument of the logarithm lower than unity the logarithmic function will be negative); then only the non-dimensional ratio (/3 [P2] + 1)/(, [P1] + 1) is allowed to reach values over or under unity. From the experiments with reaction products of glutaminase and urease the order of magnitude of ,3 was estimated as 1 M-1. Under these conditions, the temporal change of pH at any point in the membrane width (the spatial co-ordinate actually taken into account) (OpH/Ot) is accounted for by a diffusional term (J2pH/ax2) and a reaction term [o-F(pH)]. Thus the kinetic equation describing the pH dynamics is a reactiondiffusion equation:
apH a2pH
OF(pH) = ° Boundary conditions: pH(O,t) = pHo and pH(l,t) =pH
1,2
The model assumes that both enzymes obey Michaelis-Menten kinetics and that the substrates are present in excess to assure zero-order kinetics for both reactions. Moreover, the conditions of the reaction were assumed to allow the substrate concentrations to remain constant inside the membrane; that is, substrate consumption was assumed to be negligible compared with initial concentrations. The pH-dependence of the enzymic activity may be treated using the classical assumption that the enzymes behave as dibasic acids, exhibiting two ionization equilibria. Of the three forms in which each enzyme may exist, only the monoprotonated species shows activity. The pH-dependence may be expressed by the
V iM(opt)
(pK'es2 + pK2esi) This expression is strictly valid only around pH*, where
a,
Ei
-
The pH optima for the activities of E, and E2 should be different, with that of E, lower than that of E2 (Fig. 1) in order to obtain an asymmetric pattern (see below). Another condition required by the model is that the immobilized enzymes are homogeneously distributed and that only substrates and products can diffuse in the membrane. The instantaneous pH is assumed to be described by the following phenomenological expression: pH = pH* +,/ ([P2/[P1]) (3) with pH*, the pH of the unstable steady state, given by
2
i = 1, 2
(2)
-
-
(4)
Initial conditions: pH(x,O) = g(x) where F(pH) =a2-al VIM (opt) e2 DH+
T2
e VM(opt) DH+
C2; 0. = /13 = AXT2 where pH(O,t) and pH(l,t) indicate the pH values at any time, t, at the boundaries of the membrane width (spatial co-ordinates 0 and 1). The initial condition is accounted for by the function g(x) describing the spatial distribution of pH at time 0. In this case g(x) is the equation of a straight line with slope (pHo-pHj)/e and y-intercept pH1. In the model we have chosen o- = q2 in order to reduce the number of parameters to be investigated. The experimental conditions were selected so that V'M(Opt)= V2M(Opt) to fulfil alT
OI1
=
=
(r
Eqn. (4) has been made non-dimensional by introduction of the factors e2/DH+ and e, the membrane width, as characteristic time and space units respectively. This characteristic time is proportional to the time constant of the system, equal to 0 = e2/DH+. Thus the parameter o- contains kinetic and spatial information about the system. Mathematically, the existence and uniqueness of a solution to the system of eqns. (4) can be proved by the Leray-Schauder-Tychonov theorem [18]. The numerical
1990
Pattern formation in an immobilized bienzyme system
117
simulation is done with an explicit-finite-difference method. pH n is an approximation of pH(iAx,nAt), where Ax and At are the space and the time steps respectively, and i and n the number of the spatial and temporal steps being considered. The approximation is defined by the following scheme:
pHin+ -pHi5 pHi+l + pHiAt
2pHi
oF(pH
n)
=
Glutaminase
0;
Ax2
0 1I
Pattern formation in
an
115
immobilized bienzyme system
A morphogenetic model Sonia CORTASSA,*$ Hao SUN,t J. Pierre KERNEVEZt and Daniel THOMAS* Laboratoire de Technologie Enzymatique, U.R.A. N° 41 du Centre National de la Recherche Scientifique and t Departement de Genie Informatique, Universite de Technologie de Compiegne, Compiegne, France *
(CNRS),
Experimental and theoretical studies of a reaction-diffusion model of two immobilized enzymes participating in the cellular acid-base metabolism, namely glutaminase and urease, are presented. The system shows an unstable steady state at pH 6.0, where any perturbation will drive the system towards a more alkaline or more acidic pH, owing to the autocatalytic behaviour with respect to pH exhibited by both enzymes. When diffusion is coupled to reaction by means of immobilization, different patterns of the internal pH profile appear across the membrane. If the bienzymic membrane is subjected to a perturbation at its boundaries, of the same amplitude but in opposite directions, the internal pH evolves through an asymmetric pattern to attain a nearly symmetric distribution of pH. The pH value at the final steady state is more acidic or more alkaline than the initial state according to the initial and boundary conditions. The final nearly symmetric state is attained more rapidly when less enzyme is immobilized (1.8 x 10-4 M -s- as against 3.3 x 10-4 M -S-1 of total enzyme activity in the membrane volume). The experimental results agree rather well qualitatively with numerical predictions of the model equations.
INTRODUCTION One of the major challenges in biochemistry and biology today is the problem of the manner in which the activities of billions of molecules are co-ordinated in space and time to produce cellular functions. In spite of our everyday deeper knowledge of the structure and function of isolated biological molecules, understanding of the functional organization of the cell is rather less developed [1,2]. Many hypotheses have proposed rates ofchemical change (reactions) coupled to transport of matter (diffusion) as a plausible mechanism to explain how biological organization arises. These reaction-diffusion models have been extensively elaborated, although their experimental use lags far behind their theoretical development [3]. About 37 years ago, Turing [4] demonstrated mathematically that reaction-diffusion models are able to generate form, i.e. morphogenesis. The concept of a morphogen was introduced in the pioneering paper of Turing [4] as a name for a chemical substance able to generate form through a reaction-diffusion mechanism. Autocatalysis, or reaction activation by its product, or inhibition by its substrate, under far-from-equilibrium conditions, is a necessary requirement in Turing's [4] model for pattern formation and in all models derived from it [5-7]. Many examples of pattern formation have been reported in chemical and physical systems, such as chemical waves in the Belousov-Zhabotinsky reaction, and photochemical structures for a large number of photochromic and chromogenic compounds [8-10]. However, most examples of photochemical pattern formation or stable patterns in Belousov-Zhabotinsky reactions do not arise via a Turing instability, i.e. a diffusiondriven instability, but contain convective or surface effects [1 1,12]. In biochemical systems only a few experimental studies support the ability of reaction-diffusion mechanisms to produce patterns. The recent work of Aon et al. [13] shows a phenomenon of
pattern formation in a photobiochemical system. Those authors suggested that the non-linear reduction of an electron acceptor by photosynthetic activity, when coupled to diffusion, can trigger a symmetry change that redistributes a monotonic gradient of the electron acceptor in a banding pattern. The coupling of autocatalysis with diffusion has also been postulated to account for periodic precipitation processes, namely Liesegang rings, where the threshold for crystal growth would be the non-linear mechanism participating in the phenomenon [14]. The immobilization of enzyme systems gives rise to diffusional limitations (mass-transfer constraints) on the supply of substrates or removal of products in the immediate environment of the enzymes. In consequence, local metabolite-concentration profiles may be established across the membrane width, which will, in turn, determine the kinetic behaviour of the enzymes. In addition, immobilized enzymes constitute a well-defined system from the point of view of composition, thereby allowing knowledge and control of all variables. Thus modelling of these systems with reaction-diffusion equations is straightforward [15]. Although concentration profiles can be calculated from reaction-diffusion equations, little attention has been paid to their experimental visualization [16]. In the present work, a theoretical and experimental approach to a reaction-diffusion model of two immobilized enzymes participating in acid-base metabolism is developed. The enzymes involved are glutaminase and urease, which show autocatalytic behaviour with respect to pH. When diffusion is coupled to the autocatalytic reaction by means of immobilization, different patterns of the internal pH profile of the membrane appear. Theory predicts, and experiments confirm, that the evolution of the internal pH of the membrane towards a nearly symmetric steady state, through a transient asymmetric pattern of pH, is triggered by slight perturbations of the pH in an asymmetric fashion at the boundaries.
$ Present address and address for correspondence and reprint requests: Laboratory of Biochemistry, E. C. Slater Institute, University of Amsterdam, P.O. Box 20151, IOOOHD, Amsterdam, The Netherlands. Vol. 269
116
S. Cortassa and others
MODEL DESCRIPTION Nomenclature used Parameter Description Units Enzyme of the reaction i Ei Substrate and Product of M S, and Pi the reaction i Maximal rate of enzyme i V1 at a given pH Vi M(opt) Maximal rate of enzyme i at the optimal pH Ratio of maximal rate at a Nongiven pH to maximal rate dimensional at the optimal pH of the enzyme i Ionization constants of the M Kies,i Kie 2 enzyme i pH of the unstable steady NonpH* state dimensional M-1 ,3 Normalizing constant cm e Membrane width in the direction parallel with diffusion Diffusion coefficient for H+ cm2/s DH + M O = Vi me2/DH+ Kinetic parameter for enzyme i Characteristic time constant S d= e2/DH+ of the system NonNormalized factor of the dimensional system
The model considers an artificial protein membrane in which two enzymes, E1 and E2 are homogeneously immobilized. The product of the reaction catalysed by E1 is an acid, whereas that produced by E2 is a base. For the present experimental system, El and E2are glutaminase and urease respectively. The reactions involved may be schematically represented as follows:
Si
Pi
i=
following equation [17]:
viM(pH) -Kies 10p
+
I0-pH
Kies 10-PH + Ki K
i= 1,2 (1)
where Kiesl and K'es2 are the ionization constants of enzyme i, and V1M(opt) and VtM(pH) are the maximal rates at the optimum pH or at a given pH respectively. This expression is valid only if the relation between the ionization constants of the enzyme Kiesl/Kies2 > 102 holds [17]. The optimum pH for EB activity will occur at:
pHi
PKiesl + pKes2
(opt)
[PJ]/[P2] < 1,
and may be verified experimentally close to this pH. Although the expression may look rather ad hoc, it may be obtained by expanding the function pH = pH* + log [(/3 [P2] + 1)/(3 [P1]+ 1)] as a Taylor series and neglecting the quadratic and higher-order terms. This function was derived from the following reasoning: since P1 is an acid, an increase in its concentration will lead to a decrease in the pH with respect to pH*; the converse is true for an increase in the concentration of the alkaline product, P2. Unity is added to the [Pi] values in order to account for their positive contributions to the change of pH (since for an argument of the logarithm lower than unity the logarithmic function will be negative); then only the non-dimensional ratio (/3 [P2] + 1)/(, [P1] + 1) is allowed to reach values over or under unity. From the experiments with reaction products of glutaminase and urease the order of magnitude of ,3 was estimated as 1 M-1. Under these conditions, the temporal change of pH at any point in the membrane width (the spatial co-ordinate actually taken into account) (OpH/Ot) is accounted for by a diffusional term (J2pH/ax2) and a reaction term [o-F(pH)]. Thus the kinetic equation describing the pH dynamics is a reactiondiffusion equation:
apH a2pH
OF(pH) = ° Boundary conditions: pH(O,t) = pHo and pH(l,t) =pH
1,2
The model assumes that both enzymes obey Michaelis-Menten kinetics and that the substrates are present in excess to assure zero-order kinetics for both reactions. Moreover, the conditions of the reaction were assumed to allow the substrate concentrations to remain constant inside the membrane; that is, substrate consumption was assumed to be negligible compared with initial concentrations. The pH-dependence of the enzymic activity may be treated using the classical assumption that the enzymes behave as dibasic acids, exhibiting two ionization equilibria. Of the three forms in which each enzyme may exist, only the monoprotonated species shows activity. The pH-dependence may be expressed by the
V iM(opt)
(pK'es2 + pK2esi) This expression is strictly valid only around pH*, where
a,
Ei
-
The pH optima for the activities of E, and E2 should be different, with that of E, lower than that of E2 (Fig. 1) in order to obtain an asymmetric pattern (see below). Another condition required by the model is that the immobilized enzymes are homogeneously distributed and that only substrates and products can diffuse in the membrane. The instantaneous pH is assumed to be described by the following phenomenological expression: pH = pH* +,/ ([P2/[P1]) (3) with pH*, the pH of the unstable steady state, given by
2
i = 1, 2
(2)
-
-
(4)
Initial conditions: pH(x,O) = g(x) where F(pH) =a2-al VIM (opt) e2 DH+
T2
e VM(opt) DH+
C2; 0. = /13 = AXT2 where pH(O,t) and pH(l,t) indicate the pH values at any time, t, at the boundaries of the membrane width (spatial co-ordinates 0 and 1). The initial condition is accounted for by the function g(x) describing the spatial distribution of pH at time 0. In this case g(x) is the equation of a straight line with slope (pHo-pHj)/e and y-intercept pH1. In the model we have chosen o- = q2 in order to reduce the number of parameters to be investigated. The experimental conditions were selected so that V'M(Opt)= V2M(Opt) to fulfil alT
OI1
=
=
(r
Eqn. (4) has been made non-dimensional by introduction of the factors e2/DH+ and e, the membrane width, as characteristic time and space units respectively. This characteristic time is proportional to the time constant of the system, equal to 0 = e2/DH+. Thus the parameter o- contains kinetic and spatial information about the system. Mathematically, the existence and uniqueness of a solution to the system of eqns. (4) can be proved by the Leray-Schauder-Tychonov theorem [18]. The numerical
1990
Pattern formation in an immobilized bienzyme system
117
simulation is done with an explicit-finite-difference method. pH n is an approximation of pH(iAx,nAt), where Ax and At are the space and the time steps respectively, and i and n the number of the spatial and temporal steps being considered. The approximation is defined by the following scheme:
pHin+ -pHi5 pHi+l + pHiAt
2pHi
oF(pH
n)
=
Glutaminase
0;
Ax2
0 1I
