This section concerns simple, in vitro enzymatic systems in which the media is either of a controllable structural complexity or of an aqueous-hydrophobic nature.

A number of studies have dealt with the in vitro behaviour of immobilised enzymes [1–3]. Studies in our group, as explained in the examples given below, illustrate the emergence of a complex behaviour in cases in which simple, ‘monosteric’ (as opposed to ‘allosteric’) enzymes were inserted in various types of artificial structures.

2.1 The problem

The simplest possible behaviour for a monosteric enzyme, E, catalysing the reaction between substrate, S, and product, P, according to:

$$\mathrm{S}\begin{array}{c}\multicolumn{1}{c}{\mathrm{E}}\\ \multicolumn{1}{c}{\leftrightarrow }\end{array}\mathrm{P}$$

(1)

in a homogeneous, aqueous medium, is that described (when P=0) by Michaelis and Menten's kinetic equation [4]

$$\mathrm{V}={\mathrm{V}}_{\mathrm{m}}\mathrm{S}/({\mathrm{K}}_{\mathrm{m}}+\mathrm{S})$$

(2)

in which the italic symbol (e.g., S) characterises the concentration of the corresponding substance (e.g., S), and V_m and K_m are the maximum rate and the Michaelis constant of the reaction. The corresponding curve in the direct system of co-ordinates is a hyperbola (Fig. 1a), but it is customary to use the reciprocal system of co-ordinates {1/S,1/V} (equation (3)):

$$\frac{1}{V}=\frac{K_{\mathrm{m}}}{V_{\mathrm{m}}}\frac{1}{S}+\frac{1}{V_{\mathrm{m}}}$$

(3)

which gives a straight line, whose intercepts with the co-ordinate axes are −1/K_m and 1/V_m (Fig. 1b) [5]. In such a case, the reaction kinetics are said to be ‘monophasic’.

Now, let us consider model systems in which monosteric enzymes, E_j, are inserted at random in a gel slice, with thickness l, separating two aqueous solutions, $\mathrm{e}$ and $\mathrm{i}$ (Fig. 2) [6]. Locally, the rate, V_j, of the catalysed reaction(s) of substrate(s) S_j and product(s) P_j (4):

$${\mathrm{S}}_j\begin{array}{c}\multicolumn{1}{c}{{\mathrm{E}}_j}\\ \multicolumn{1}{c}{\leftrightarrow }\end{array}{\mathrm{P}}_j$$

(4)

is again a hyperbolic function of S_j (for P_j=0), which may be written as:

$$V_j={\mathrm{V}}_{\mathrm{m}j}{\gamma }_j{\lambda }_j$$

(5)

$${\lambda }_j={\mathrm{S}}_j/({\mathrm{K}}_{\mathrm{m}j}+{\mathrm{S}}_j)$$

(6)

In these equations V_mj, γ_j, λ_j and K_mj (assumed to be constant with respect to pH) are the maximum rate, the possible pH dependence, the substrate dependence and the Michaelis constant of enzyme E_j in solution, respectively. Within the gel slice, concentration S_j obeys the differential equation:

$$\partial {\mathrm{S}}_j/\partial \mathrm{t}={\mathrm{D}}_j({\partial }^2{S}_j/\partial {\mathrm{x}}^2)-{\mathrm{V}}_j(\mathrm{x},\mathrm{t})$$

(7)

where t is time, D_j the diffusion coefficient of S_j and x the distance along l with x=0 and x=l at the boundaries of $\mathrm{e}$ and $\mathrm{i}$. The flux, J_j of S_j at time t and at point x in the gel slice is written as:

$$J_j(\mathrm{x},\mathrm{t})=-{\mathrm{D}}_j{(\partial {\mathrm{S}}_j/\partial \mathrm{x})}_{\mathrm{x},\mathrm{t}}$$

(8)

while the rate of disappearance, $\overline{V_j}$, of S_j from one of the aqueous solutions (e.g., $\mathrm{e}$) is written

$$\overline{V_j^{\mathrm{e}}}=-{\mathrm{J}}_j(0,\mathrm{t})\mathrm{A}/{\mathrm{v}}^{\mathrm{e}}$$

(9)

where A is the surface area of the gel slice and v^e the volume of compartment $\mathrm{e}$.

2.2 Kinetic behaviour of a monosteric enzyme inserted in a gel slice separating aqueous compartments

In the simplest possible case, i.e. when the system is monoenzymatic (index j thus may be deleted), the concentration of S is the same in $\mathrm{e}$ and $\mathrm{i}$ (S^e=Sⁱ), the system is without any pH effect (γ=constant) and the stationary state has been established everywhere in the gel slice (∂S/∂t=0), one may consider the dimensionless, diffusion–reaction parameter:

$$\alpha \mathrm{l}=\sqrt{V_{\mathrm{m}}\gamma {\mathrm{l}}^2/{\mathrm{K}}_{\mathrm{m}}D}=\sqrt{{\tau }_{\mathrm{diff}}/{\tau }_{\mathrm{react}}}$$

(10)

In equation (10), τ_diff and τ_react are the characteristic times for the diffusion of S and for the reaction, respectively, with:

$${\tau }_{\mathrm{diff}}={\mathrm{l}}^2/\mathrm{D}\mathrm{and}{\tau }_{\mathrm{react}}={\mathrm{K}}_{\mathrm{m}}/{\mathrm{V}}_{\mathrm{m}}\gamma$$

(11)

The larger the contribution of the reaction and the smaller that of diffusion, the larger is the value of αl and vice versa. Computing the dependence of $-1/\overline{V^{\mathrm{e}}}$ as a function of K_m/S^e (reciprocal system of co-ordinates), the case when αl⪡1 gives a linear plot (i.e. there are Michaelis–Menten kinetics in the gel slice at all values of substrate concentration and the system behaviour thus remains monophasic) with the apparent K_m being equal to the actual K_m of the enzyme in solution and $\overline{V_{\mathrm{m}}}$ being proportional to the V_m of the enzyme in solution. However, as αl increases, the plot of $\{{\mathrm{K}}_{\mathrm{m}}/{\mathrm{S}}^{\mathrm{e}},-1/\overline{V^{\mathrm{e}}}\}$ ceases to be linear. The curve has a monotonous negative curvature and its extremities (low and high S values) can be approximated by two different straight lines (dual-phasic behaviour). In this case, although there is a single monosteric enzyme present in the system, one could be tempted to wrongly interpret the kinetic data as corresponding to the presence of two different monosteric enzymes (one detectable at low and the other at high substrate concentrations). With pH-dependent systems (i.e. when the reaction produces or consumes protons or hydroxyl groups), situations are encountered in which the plot in the reciprocal system of co-ordinates $\{{\mathrm{K}}_{\mathrm{m}}/{\mathrm{S}}^{\mathrm{e}},-1/\overline{V^{\mathrm{e}}}\}$ ceases to be linear and possibly becomes sigmoidal. Therefore, even with an enzyme of the Michaelis–Menten type, it is possible to obtain in the reciprocal system of co-ordinates (Fig. 3) a complex, non-linear (e.g., dual-phasic or sigmoidal) kinetic behaviour by imposing appropriate (and possibly rather simple) structural conditions.

2.3 Two-enzyme artificial first-order transport systems

Let us consider again [6] a model system made of a gel slice separating two aqueous compartments, $\mathrm{e}$ and $\mathrm{i}$, but now let us insert two monosteric enzymes, E₁ and E₂, differing in their optimum pH value and catalysing the associated reactions

$$\mathrm{S}+\mathrm{XY}\begin{array}{c}\multicolumn{1}{c}{{\mathrm{E}}_1}\\ \multicolumn{1}{c}{\leftrightarrow }\end{array}\mathrm{PX}+\mathrm{Y}$$

(12)

$$\mathrm{PX}\begin{array}{c}\multicolumn{1}{c}{{\mathrm{E}}_2}\\ \multicolumn{1}{c}{\leftrightarrow }\end{array}\mathrm{S}+\mathrm{X}$$

(13)

at random in the gel slice. The overall reaction

$$\mathrm{XY}\leftrightarrow \mathrm{X}+\mathrm{Y}$$

(14)

is assumed to be exergonic in the direction of the splitting of XY into X and Y. Different pH values, pH^e and pHⁱ, are imposed on compartments e and i to create a pH gradient in the gel slice, and the pH^e and pHⁱ values are chosen so as to make E₁ active only in a layer, l₁, of the gel slice close to compartment $\mathrm{e}$ and E₂ active only in a layer, l₂, close to compartment $\mathrm{i}$ (Fig. 4). Under such conditions, substrate S diffusing from compartment $\mathrm{e}$ is transformed into PX by enzyme E₁ in layer l₁; the part of PX which diffuses towards l₂ is transformed back into S by enzyme E₂; this newly-formed S diffuses partially towards compartment $\mathrm{i}$ and, depending on the concentration profiles in the gel slice, the overall process can produce an uphill transport of S at the expense of the energy provided by the exergonic splitting of XY. The system under consideration thus is a model of a first-order [7] process of transport.

2.4 Monoenzymatic artificial second-order transport systems

Yeast alcohol deshydrogenase (ADH) catalyses the reversible reaction:

(16)

A gel slice, G, containing a homogeneous distribution of ADH was inserted between two barriers, B, that prevented enzyme leakage and were practically impermeable to NAD⁺ while being permeable to NADH and OH⁻. Gel slice and barriers were then used to separate two aqueous compartments, $\mathrm{e}$ and $\mathrm{i}$, containing identical mixtures of alcohol, acetaldehyde, NADH and phosphate buffer, but no NAD⁺ (Fig. 5). Constant pH values, more alkaline in $\mathrm{i}$ than in $\mathrm{e}$, were imposed in the aqueous compartments, thus creating a pH gradient in the gel slice. After an initial decrease of the NADH concentration in both compartments, corresponding to NADH equilibration of the gel slice with the bathing solutions, NADH concentration began to increase in $\mathrm{i}$, while it continued to decrease in $\mathrm{e}$. After a few days, the NADH concentration in $\mathrm{i}$ was higher than the initial NADH concentrations in the aqueous compartments and almost twice as high as the final concentration in $\mathrm{e}$ (Fig. 6). The qualitative interpretation is as follows. At the outset of the experiment, NADH diffuses from $\mathrm{e}$ and $\mathrm{i}$ into the gel slice. In the part of the gel close to $\mathrm{e}$, where the concentration of H⁺ is the highest, NADH is transformed into NAD⁺ as a consequence of the pH-dependence of the enzyme, then NAD⁺ diffuses into the gel but cannot diffuse into $\mathrm{e}$ and $\mathrm{i}$ because of the barriers B. In the part of the gel close to $\mathrm{i}$, where the concentration of H⁺ is lowest, NAD⁺ tends to be transformed back into NADH, thus increasing the NADH concentration in this part of the gel. As a consequence, NADH tends to diffuse toward compartment $\mathrm{i}$. This corresponds to an uphill transport of NADH driven by the pH gradient (a transport of OH⁻ ions from $\mathrm{i}$ to $\mathrm{e}$ rather than a transport of H⁺ ions from $\mathrm{e}$ to $\mathrm{i}$), while electric neutrality in compartments $\mathrm{e}$ and $\mathrm{i}$ is ensured via exchanges of saline ions provided by the buffer. Such a system is therefore a model of second-order active transport [7].

2.5 Enzyme-catalysed reactions in an aqueous/hydrophobic reaction medium

Decreasing the polarity of the reaction medium by decreasing its water content led to an exponential increase in the apparent K_m for the hydrolysis of the dipeptide (see above) and in the solubility of this dipeptide (Fig. 7). Since the interactions between substrate and the active site of α-chymotrypsine are mainly hydrophobic, this means that decreasing the water content of the reaction medium tends to favour the interactions of the dipeptide with the solvent mixture and thus to weaken the interactions of the dipeptide with the active site of the enzyme. Moreover, it has been established that the ratio of the actual rate constants for the formation/dissociation of the enzyme–substrate complex in the solvent mixture is given by the corresponding ratio in aqueous medium divided by the equilibrium constant for the transfer of the dipeptide from water to the solvent mixture (which is equal to the ratio of the dipeptide solubility in the solvent mixture to that in water).

Frogs maintain the saline concentration of their internal medium, even when immersed in freshwater ponds, due to the activity of a Mg-dependent Na-K-ATPase that creates a steady transepithelial potential difference in the range of a few tens to more than a hundred millivolts (positive inside). This active ion-pumping system has been studied in great detail, especially by Ussing's group [11,12], using the ventral skin of frogs mounted between two aqueous compartments filled with appropriate saline solutions. Maintaining the electrical potential of frog skin in these conditions requires sodium in the external medium. Lithium is the only ion that can be used instead [13]. However, when all or part of the external sodium is replaced by lithium, the electric potential frequently oscillates [14–16], which it never does in the absence of lithium (Fig. 8).

A body of evidence obtained by others, cited in Lassalles et al. [17], demonstrates that all the ionic processes are controlled at the level of the skin epithelium, which comprises a few layers of cells separating the external medium $\mathrm{e}$ from the internal medium $\mathrm{i}$. The epithelium may be modelled as follows (Fig. 9): it is made of two main compartments, C₁ and C₂; the membrane, $\mathrm{a}$, at the external face of the epithelium is relatively permeable to Na⁺ and Li⁺ whilst it is almost impermeable to K⁺; the membrane, $\mathrm{b}$, between compartments C₁ and C₂ is permeable to K⁺ but not to Na⁺ and contains the Na-K-ATPase pumping sodium and lithium from C₁ to C₂ and potassium from C₂ to C₁; the face, c, between epithelium and internal medium is not an actual membrane, but may be considered rather as a non-selective diffusive barrier.

In studies of the oscillatory process, our group [17–20] has shown experimentally that (i) when no transepithelial potential is imposed, sustained oscillations with a period of about 10 min are maintained for several hours, (ii) an oscillation of the Na⁺ influx accompanies the electric oscillation (the two oscillations have approximately the same period but are not in phase), (iii) under conditions of imposed potential the transepithelial electric current has damped oscillations, (iv) the shape of the oscillations in potential is quite variable (from almost sinusoidal to very complex), (v) theophyllin (which induces an accumulation of cyclic AMP within the cells) promotes a significant decrease in the mean electric potential of the skin but does not affect the characteristics of the oscillation very much, (vi) important factors influencing the oscillations include temperature, the permeability of the external membrane to lithium and the potassium concentration in the internal medium, (vii) no evident correlation exists between skin area and the characteristics of the oscillations, which may therefore have a local origin (possibly in local oscillators at the level of the cell) and the coupling of these oscillators would generate the macroscopic oscillations and (viii) synchronisation of local oscillators can be controlled by varying the coupling resistance in the absence of diffusion, which is consistent with electrical coupling rather than diffusion being responsible for the synchronisation.

After extensive numerical simulation of the oscillatory process, it proved impossible to obtain all the above characteristics unless the model had a few well-defined properties, the following ones in particular.

• • If £₁ and £₂ represent the volumes of compartments C₁ and C₂ per unit surface of epithelium, they obey the relation £₁/£₂⪡1, i.e. compartment C₂ has to be much larger than compartment C₁. This makes it possible to choose one of the various possible modes of transport of Na⁺ through the epithelium proposed in the literature and cited in Lassalles et al. [17]. Indeed, £₂⪢£₁ does not support the idea that compartment C₁ merely corresponds to most of the cytoplasmic volume and C₂ is restricted to a few endoplasmic cisternae, whereas it strongly supports the idea that compartment C₁ corresponds to a few cytoplasmic vacuoles transporting Na⁺ (and Li⁺) and compartment C₂ is part or all of the intercellular spaces and possibly some endoplasmic cisternae (in this case, most of the cell cytoplasm would remain inactive in Na⁺ (or Li⁺) transport).
• • The main variables are the concentration of Li⁺ in compartment C₁ and that of K⁺ in compartment C₂. When the characteristic times τ₁ and τ₂ are defined as corresponding to Li⁺ modifications in C₁ and K⁺ modifications in C₂, respectively, the ratio of the characteristic times obeys the relation 0<τ₁/τ₂<1 with 1 min⩽τ₁⩽5 min.
• • Two parameters, ρ and u, represent the relative importance of the active to the passive fluxes of Li⁺ and K⁺, respectively, such that ρ⪢1 and u⪢1, with ρ⩽u⩽2ρ (i.e., ρ and u are of the same order of magnitude).
• • P represents the permeability coefficients of interfaces $\mathrm{a}$, $\mathrm{b}$ and $\mathrm{c}$ for ions K⁺ and Li⁺, such that P_K^b/P_K^c⪡1 and P_Li^a/P_K^c⪡1.

Embedding simple Michaelis–Menten enzymes in a gel slice allows them to catalyse not only scalar processes but also vectorial ones, including the uphill transport of a substrate between two compartments. Moreover, a system containing a single species of a simple enzyme can be made to behave as if it contained two different enzymes (or transporters) or an allosterically controlled enzyme (or transporter). These effects are more pronounced the larger the term characteristic of the reaction (V_mγ/K_m) compared with that of substrate diffusion (D/l²). The existence of a pH feedback also tends to increase these effects. In the presence of a partially hydrophobic environment (which is likely to be the case with membrane-bound enzymes or transport systems in vivo), the measured kinetic parameters are usually different, sometimes very different from those characteristic of the protein behaviour in homogeneous, aqueous solutions. Another group [21,22] has also shown that it is possible to induce oscillatory phenomena in immobilised enzyme systems or spatio-temporal pattern formation in immobilised bienzymatic systems.

These results are of general relevance to the interpretation of kinetic data on membrane-bound enzymes or transport systems. In reciprocal co-ordinates, such experimental data are often better fitted by two straight lines (one for the high and the other for the low concentrations) than by a single line. In line with the original proposal of Epstein and co-workers [23–25], most authors interpret such data as revealing the presence of two different membrane mechanisms of reaction or transport, with the apparent K_m and V_m of the membrane processes being equated to the actual K_m and V_m of the catalytic proteins involved. Similarly, sigmoidal experimental curves have been taken as indicative of an allosteric character of the membrane-bound active proteins [26]. Our results clearly show that such interpretations can be wrong, since the complexity of the kinetic behaviour may as well correspond to the partially hydrophobic nature and other features of the cellular structures in which the enzyme or transport systems are inserted (for a more detailed discussion, see Thellier et al. [27]). Indeed, an alternative interpretation to allosteric interaction has been given for sigmoidal kinetic behaviour in an actual biological system in vivo [28].

A real cell membrane is obviously not the same as a thick homogenous gel slice or a water/hydrophobic mixture. Nevertheless, the values of the parameters used in the experiments reviewed here cover the range of biological possibilities [6,8,9]. Moreover, since real membranes are far more complex than our gel slices or aqueous/hydrophobic mixtures, it is likely that the effects of membrane structure and/or hydrophobicity in vivo are even more numerous and diverse than they are in vitro.

In the artificial transport systems discussed here (sections 2.3 and 2.4), a functional asymmetry such as a proton gradient sufficed to induce an active uphill transport even under symmetrical structural conditions. Therefore, although all the active-transport systems isolated until now correspond to proteins asymmetrically anchored in a membrane, one cannot exclude the possibility that structurally symmetrical systems of transport do exist in as yet unstudied organisms or did once exist earlier in evolution.

Situations exist in biology (see Vincent et al. [9]) where the same enzymatic activity is present on each side of a membrane and where the ionic conditions are different in the media on each side. Such a situation, which closely resembles the artificial ones studied here, is important in the homeostasis of inorganic ions in cellular compartments or in the operation of relay and amplification mechanisms (in which the direction of transfer depends on differences in ion concentrations in the compartments). Moreover, the frog epithelium in vivo resembles some of the in vitro set-ups described here (sections 2.3 and 2.4), since it can be modelled as a thick two-compartment slice with both passive ion diffusion and active ion transport by an enzyme.

It is remarkable that increasing the complexity of the behaviour of the living frog-skin system by inducing electric oscillations via lithium addition to the external solution provided information on this system (such as the relative size of compartments and the values of other parameters) that could not have been provided by a conventional, reductionist approach. This may be the basis for the introduction of ‘complexification’, as opposed to reductionism, as a method for studying those properties of a complex system that are destroyed when adopting the conventional, reductive, biochemical approaches.

Further study of the model systems discussed above would benefit from knowledge of the distribution of ions. This may be provided by secondary ion mass spectrometry (SIMS), a method largely used by physicists for materials studies (see, e.g., [29,30]), now available for application to biological specimens [31–33] after appropriate preparation of dehydrated tissue sections [34,35] or using frozen hydrated samples [36].