Abstract

The activity of membrane enzymes could be highly determined by the order of the lipid of the membrane and the enzyme distribution. Particularly, the reordering of phospholipid substrates and the local fluctuations of the lipid phases have been included in mathematical models to explain the modulation of the activity of membrane enzymes in extracellular vesicles, liposomes, or microvesicles. The applied principles are different to those derived from the classic considerations such as 3D environment, aqueous, and homogeneous media. Instead, the lateral diffusion of enzyme and substrate and highly nonhomogeneous 2D environment determine fluctuations of enzymatic activity capable to explain metabolic effects, such as in case of peptide-induced membrane components reordering. In this chapter, we review some applications to lipid metabolizing enzymes, due to analytical results of the kinetic theory of membrane enzymes.

Keywords: phospholipid domain, substrate reordering, lipolytic enzyme, phospholipase, enzyme kinetics, lipid phase, micelle, membrane

## 1. Introduction

The so-called extracellular vesicles (EVs) are either exosomes or microvesicles, which are formed from intracellular multivesicular bodies or plasma membrane, respectively [1, 2]. The lipid content of EVs plays a key role in various pathophysiological processes [3] as well as the native proteins on their surface, many of them having functions in cellular metabolism and signal transductions, such as phospholipases [4]. Interestingly, some membrane protein-related human diseases arise from dysregulation of signal transduction pathways [2]. Moreover, some phospholipases are very important for biogenesis of EVs and there are many phospholipases in EVs [5–7]. About the lipid phase of EVs, lipid exchange between vesicles has been described [8], exosomes can vectorize some lipids acting as transport, and the lipid composition can be modified by in vitro manipulation [7]. On the other hand, microdomains of EVs could be transferred to a target membrane cells by means of membrane fusion, and as a consequence, the lipid substrate redistribution could be able to affect the activity of lipid metabolizing enzymes.

Taking into account all the abovementioned causes and effects of the EVs related to both lipid substrate reordering and their metabolizing enzymes, the understanding of the effect of lipid-substrate reordering over the enzyme activity could be

essential to the development of therapeutic purposes as well as to insight the carcinogenesis and to perform enzyme kinetics experiments. We hope that this purpose of understanding the enzyme kinetics in the lipid phase will be fulfilled at least partially in the remainder of this chapter.

Numerous processes associated with the cell membranes are mediated by the action of lipid metabolizing enzymes. Knowing how the changes of membrane properties affect the activity of these enzymes allows us to explain disease mechanisms and pharmacological activities. Specifically, the knowledge about the mechanisms of the reactions catalyzed by these lipid metabolizing enzymes can contribute to the understanding of several regulation and signaling phenomena in cells. Thus, the enzymes of the phospholipase C family (PLC) [9] are involved in lipid signaling pathways affecting levels of free calcium and protein phosphorylation [10, 11], regulating secretion, transport, metabolism, gene expression, and protein translation. Since phospholipases react in a lipid-water interface, different kinetic experimental systems have been developed using phospholipid vesicles, phospholipid and detergent mixed micelles, or phospholipid monolayers. As a first step, the watersoluble enzyme would bind to the lipid phase, then having many catalytic cycles with the lipid substrate before the enzyme returns to the aqueous solution.

To study the kinetic measurements of phospholipases, the theory known as surface dilution kinetics [12] has been applied. This theory allows to estimate the main enzyme kinetic parameters considering the effects of the substrate staying into the lipid phase ("surface dilution") on the enzyme activity. Similar to the most enzyme kinetic models, in this theory, the mass action law and the steady-state assumption for enzyme intermediaries are applied. In the calculations with regard to molecules in water phase, their concentrations are used. Instead, in the case of calculations of molecules dissolved in lipid phase, their mole fractions are used.

Using this theory and its associated experiments, it has been proposed that many lipid metabolizing enzymes follow a mechanism composed by two binding steps of the enzyme on the lipid phase: a first binding step to the lipid phase followed by a second binding step to the substrate. More specifically, depending on the first binding step, there are two possible kinetic models: in the phospholipid-binding model, first the enzyme binds specifically to the phospholipid substrate; n the surface-binding model, first the enzyme binds to any lipid phase region. In homogeneous substrate distribution conditions, these are the kinetic equations derived for the phospholipid-binding model (Eq. 1) and the surface-binding model (Eq. 2) [12, 13]:

$$V = \frac{kC\_{E\_\Gamma}f^2}{\frac{k\_mk\_r}{C\_L} + k\_mf + f^2} \tag{1}$$

molecules such as the protein myristoylated alanine-rich C kinase substrate (MARCKS) or pentalysine (Lys5, one of the first five amino acid residues of the region of bovine MARCKS) [17]. In biological membranes, the microdomain structure and dynamics are widely diverse, considering the scaffolding of cell

lipid phase with the largest substrate molar fraction or not, respectively.

(thereby, considering an infinite number of infinitesimal domains).

2. Changes in the lipolytic enzyme activity due to substrate reordering

In the calculations of the lipase activities in membranes, it is assumed that whatever the structure of the lipid phase (micelle, liposome, or monolayer), all the lipids in the lipid-water interface expose the same area to the aqueous phase. As a consequence, the area of the lipid phase surface is proportional to the amount of lipid molecules. Then, at the beginning of the reaction, the lipid area will be

In order to find the effects of lipid substrate domain formation on enzyme activities, it is necessary for an extended mathematical formulation starting for similar principles to those of the original surface dilution kinetics theory. That is because the total activity could be integrated by each one of these substrate domains, i.e., whenever there is a phospholipid substrate (e.g., Figure 1), and therefore, the formation of domains enriched in a phospholipid substrate could either increase (inside the enriched domain) or decrease (outside the enriched domain; i.e., inside the nonenriched domain) the total enzymatic activity on the membrane. For this reason, it is useful to propose a comprehensive quantitative model that explicitly takes into account the enzyme activity in the different phospholipid phases, which here are frequently called substrate domains or simply "domains," to distinguish them from the eventual thermodynamic phases on membranes. Below, a theoretical frame for lipid binary membrane systems is shown, and then the theoretical frame is extended to a more realistic lipid phases with any number of domains, including continuous gradient of phospholipid substrate

A generalization of the surface dilution kinetics theory applied to lipolytic enzymes has been necessary for cases of nonhomogeneous substrate distribution. This is because the reordering of the phospholipid substrate could have important effects on the activity of lipolytic enzymes. In the figure, for the nonhomogeneous substrate distribution, two domain phases can be distinguished: enriched substrate domain and nonenriched substrate domain (named elsewhere as nondomain phase) depending on whether the domain phase corresponds to the

Effects of Vesicular Membranes Reordering on the Activity of Lipid Metabolizing Enzymes

DOI: http://dx.doi.org/10.5772/intechopen.85972

proteins [18].

73

Figure 1.

$$V = \frac{kC\_{E\_T}f}{\frac{k\_mk\_l}{C\_L} + k\_m + f} \tag{2}$$

where V is the rate of product formation (mol/[volume�time]), f is the mole fraction of the substrate (dimensionless), CET is the total enzyme concentration (mol/volume), CL is the total lipid concentration (mol/volume), k is the catalytic time constant (time�<sup>1</sup> ), ks is the dissociation constant (mol/volume), and km is the interfacial Michaelis constant (dimensionless).

A more complex approach must consider that phospholipids can be reordered in lateral domains [14–16] because of their interactions with either phospholipids, cytoskeleton, or charged soluble molecules, and then more adequate mathematical expressions are necessary involving phospholipid reordering. For example, phosphatidylinositol 4,5-biphosphate (PIP2) and phosphatidylserine (PS) can be reordered in lateral domains because of the direct interactions with Ca2+ or basic

Effects of Vesicular Membranes Reordering on the Activity of Lipid Metabolizing Enzymes DOI: http://dx.doi.org/10.5772/intechopen.85972

#### Figure 1.

essential to the development of therapeutic purposes as well as to insight the carcinogenesis and to perform enzyme kinetics experiments. We hope that this purpose of understanding the enzyme kinetics in the lipid phase will be fulfilled at

Numerous processes associated with the cell membranes are mediated by the action of lipid metabolizing enzymes. Knowing how the changes of membrane properties affect the activity of these enzymes allows us to explain disease mechanisms and pharmacological activities. Specifically, the knowledge about the mechanisms of the reactions catalyzed by these lipid metabolizing enzymes can contribute to the understanding of several regulation and signaling phenomena in cells. Thus, the enzymes of the phospholipase C family (PLC) [9] are involved in lipid signaling pathways affecting levels of free calcium and protein phosphorylation [10, 11], regulating secretion, transport, metabolism, gene expression, and protein translation. Since phospholipases react in a lipid-water interface, different kinetic experimental systems have been developed using phospholipid vesicles, phospholipid and detergent mixed micelles, or phospholipid monolayers. As a first step, the watersoluble enzyme would bind to the lipid phase, then having many catalytic cycles with the lipid substrate before the enzyme returns to the aqueous solution. To study the kinetic measurements of phospholipases, the theory known as surface dilution kinetics [12] has been applied. This theory allows to estimate the main enzyme kinetic parameters considering the effects of the substrate staying into the lipid phase ("surface dilution") on the enzyme activity. Similar to the most enzyme kinetic models, in this theory, the mass action law and the steady-state assumption for enzyme intermediaries are applied. In the calculations with regard to molecules in water phase, their concentrations are used. Instead, in the case of calculations of molecules dissolved in lipid phase, their mole fractions are used. Using this theory and its associated experiments, it has been proposed that many lipid metabolizing enzymes follow a mechanism composed by two binding steps of the enzyme on the lipid phase: a first binding step to the lipid phase followed by a second binding step to the substrate. More specifically, depending on the first binding step, there are two possible kinetic models: in the phospholipid-binding model, first the enzyme binds specifically to the phospholipid substrate; n the surface-binding model, first the enzyme binds to any lipid phase region. In homogeneous substrate distribution conditions, these are the kinetic equations derived for the phospholipid-binding model (Eq. 1) and the surface-binding model (Eq. 2) [12, 13]:

<sup>V</sup> <sup>¼</sup> kCET <sup>f</sup>

<sup>V</sup> <sup>¼</sup> kCET <sup>f</sup> kmks

where V is the rate of product formation (mol/[volume�time]), f is the mole fraction of the substrate (dimensionless), CET is the total enzyme concentration (mol/volume), CL is the total lipid concentration (mol/volume), k is the catalytic

A more complex approach must consider that phospholipids can be reordered in lateral domains [14–16] because of their interactions with either phospholipids, cytoskeleton, or charged soluble molecules, and then more adequate mathematical

expressions are necessary involving phospholipid reordering. For example, phosphatidylinositol 4,5-biphosphate (PIP2) and phosphatidylserine (PS) can be reordered in lateral domains because of the direct interactions with Ca2+ or basic

kmks

time constant (time�<sup>1</sup>

72

interfacial Michaelis constant (dimensionless).

2

), ks is the dissociation constant (mol/volume), and km is the

<sup>2</sup> (1)

CL <sup>þ</sup> km <sup>þ</sup> <sup>f</sup> (2)

CL þ km f þ f

least partially in the remainder of this chapter.

Extracellular Vesicles and Their Importance in Human Health

A generalization of the surface dilution kinetics theory applied to lipolytic enzymes has been necessary for cases of nonhomogeneous substrate distribution. This is because the reordering of the phospholipid substrate could have important effects on the activity of lipolytic enzymes. In the figure, for the nonhomogeneous substrate distribution, two domain phases can be distinguished: enriched substrate domain and nonenriched substrate domain (named elsewhere as nondomain phase) depending on whether the domain phase corresponds to the lipid phase with the largest substrate molar fraction or not, respectively.

molecules such as the protein myristoylated alanine-rich C kinase substrate (MARCKS) or pentalysine (Lys5, one of the first five amino acid residues of the region of bovine MARCKS) [17]. In biological membranes, the microdomain structure and dynamics are widely diverse, considering the scaffolding of cell proteins [18].

In order to find the effects of lipid substrate domain formation on enzyme activities, it is necessary for an extended mathematical formulation starting for similar principles to those of the original surface dilution kinetics theory. That is because the total activity could be integrated by each one of these substrate domains, i.e., whenever there is a phospholipid substrate (e.g., Figure 1), and therefore, the formation of domains enriched in a phospholipid substrate could either increase (inside the enriched domain) or decrease (outside the enriched domain; i.e., inside the nonenriched domain) the total enzymatic activity on the membrane. For this reason, it is useful to propose a comprehensive quantitative model that explicitly takes into account the enzyme activity in the different phospholipid phases, which here are frequently called substrate domains or simply "domains," to distinguish them from the eventual thermodynamic phases on membranes. Below, a theoretical frame for lipid binary membrane systems is shown, and then the theoretical frame is extended to a more realistic lipid phases with any number of domains, including continuous gradient of phospholipid substrate (thereby, considering an infinite number of infinitesimal domains).

### 2. Changes in the lipolytic enzyme activity due to substrate reordering

In the calculations of the lipase activities in membranes, it is assumed that whatever the structure of the lipid phase (micelle, liposome, or monolayer), all the lipids in the lipid-water interface expose the same area to the aqueous phase. As a consequence, the area of the lipid phase surface is proportional to the amount of lipid molecules. Then, at the beginning of the reaction, the lipid area will be

constant regardless of any substrate reordering. However, depending on the enzymatic model, the substrate reordering effectively could change the enzyme activity, having many differences between homogenous or nonhomogeneous substrate distributions.

To understand how the substrate lateral reordering might affect the enzyme kinetics, a mathematical approach has been developed for the models having two steps binding between the enzyme and the lipid phase. First, the simple nonhomogeneous case of a binary condition is considered where the substrate can be distributed in two coexisting lateral phases: an enriched domain and a nonenriched domain (usually named nondomain). Finally, a more general expression corresponding to any gradients of substrate molar fraction will be shown.

As in the case of homogenous distribution of substrate on lipid phase [12], in the simple nonhomogeneous distribution given by a binary substrate distribution (i.e., two mixed lipid molecules, one of them being the substrate), the kinetics surface dilution theory is applied to the surface-binding model and to the phospholipid-binding model [19]:

1. For the phospholipid-binding model (Figure 2A), the enzyme activity (V) depends on the substrate reordering in according to

$$V = \frac{kC\_{Er}\left(a\_1f\_{S^4}^2 + a\_2f\_{S^2}^2\right)}{\frac{k\_mk\_r}{C\_L} + k\_mf + a\_1f\_{S^4}^2 + a\_2f\_{S^2}^2} \tag{3}$$

where f <sup>S</sup><sup>i</sup>and ai are the substrate molar fraction and the fraction of the total lipid

Effects of Vesicular Membranes Reordering on the Activity of Lipid Metabolizing Enzymes

According to Eq. (4), f is the average of the substrate molar fraction weighed by

2 <sup>S</sup><sup>1</sup> þ a<sup>2</sup> f

the average of the square of the substrate mole fraction weighted by the phase

<sup>L</sup> þ km f þ f

respectively. Therefore, there are for V minimal and maximal values (Vmin and

<sup>V</sup>min <sup>¼</sup> ETkf <sup>2</sup> kmks

<sup>V</sup>max <sup>¼</sup> ETkf kmks

We can see that Vmin corresponds to V for a lipid homogeneous phase (Eq. 1)

Curiously, in spite of Eq. 8 deduced for the phospholipid-binding model with any substrate distribution, this equation is equal to Eq. 2, which corresponds to the case of surface-binding model with homogeneous substrate distribution on the

According to Eq. 3, if the homogeneous distribution of the substrate on the

2. On the other hand, for the surface-binding model (Figure 2B), the enzymatic

The differences between the behaviors of both enzymatic models are much more than the differences between the corresponding equations for the enzymatic activities (Eqs. 3 and 9). Following this theory, important differences exist in both the ratio of the substrate regarding the two substrate domains and the total enzyme

enzyme activity will increase. In particular, the total enzyme activity increases when the recruitment of substrate to the enriched-substrate domain (e.g., phase 1) increases, due to an increase of either the domain mol fraction ( f <sup>S</sup><sup>1</sup> ) or the exten-

activity in a two-phase membrane equals to the enzyme activity in a

<sup>V</sup> <sup>¼</sup> CET kf kmks

membrane is broken (i.e., substrate reordering such that 〈 f

homogeneous lipid phase, following the equation:

and applying the same restrictions given by Eqs. 4 and 5.

2

a<sup>1</sup> f <sup>S</sup><sup>1</sup> þ a<sup>2</sup> f <sup>S</sup><sup>2</sup> ¼ f (4)

a<sup>1</sup> þ a<sup>2</sup> ¼ 1 (5)

<sup>S</sup><sup>2</sup> (6)

<sup>2</sup> (7)

<sup>L</sup> <sup>þ</sup> km <sup>f</sup> <sup>þ</sup> <sup>f</sup> (8)

2 <sup>S</sup><sup>i</sup>〉 . f 2

CL <sup>þ</sup> km <sup>þ</sup> <sup>f</sup> (9)

<sup>2</sup> and f,

), then the

area in the ith phase (i = 1 or 2), respectively, and such that

DOI: http://dx.doi.org/10.5772/intechopen.85972

In Eq. 3, we can see that V depends hyperbolically on

f 2 Si � <sup>a</sup><sup>1</sup> <sup>f</sup>

area. These mean values have minimal and maximal values equal to f

and

the domain areas.

Vmax, respectively):

[12, 13]

membrane.

sion of the domain (a1).

binding to membrane.

75

#### Figure 2.

A and B represent two different Lipolytic enzyme kinetic models for a lipid phase with two substrate domains: enriched substrate domain and nonenriched substrate domain. L, lipid molecule; S, phospholipid substrate; E, lipolytic enzyme; ES and ESS, enzyme-substrate complexes; and P, product. Individual kinetic constants are shown (k, k<sup>0</sup> , k1, k<sup>0</sup> <sup>1</sup>, k�<sup>1</sup>, k<sup>0</sup> �<sup>1</sup>, k2, k<sup>0</sup> <sup>2</sup>, k�<sup>2</sup>, k<sup>0</sup> �<sup>2</sup>). In the text: km � <sup>k</sup>�<sup>2</sup> <sup>þ</sup> <sup>k</sup> <sup>k</sup><sup>2</sup> and ks � <sup>k</sup>�<sup>1</sup> <sup>k</sup><sup>1</sup> . The association of symbols and parameters to a particular domain is indicated by mean of primed or nonprimated signs in each case.

Effects of Vesicular Membranes Reordering on the Activity of Lipid Metabolizing Enzymes DOI: http://dx.doi.org/10.5772/intechopen.85972

where f <sup>S</sup><sup>i</sup>and ai are the substrate molar fraction and the fraction of the total lipid area in the ith phase (i = 1 or 2), respectively, and such that

$$a\_1 f\_{\mathbb{S}^1} + a\_2 f\_{\mathbb{S}^2} = f \tag{4}$$

and

constant regardless of any substrate reordering. However, depending on the enzymatic model, the substrate reordering effectively could change the enzyme activity, having many differences between homogenous or nonhomogeneous substrate

To understand how the substrate lateral reordering might affect the enzyme kinetics, a mathematical approach has been developed for the models having two

nonhomogeneous case of a binary condition is considered where the substrate can

nonenriched domain (usually named nondomain). Finally, a more general expression corresponding to any gradients of substrate molar fraction will be shown. As in the case of homogenous distribution of substrate on lipid phase [12], in the simple nonhomogeneous distribution given by a binary substrate distribution (i.e., two mixed lipid molecules, one of them being the substrate), the kinetics surface dilution theory is applied to the surface-binding model and to the

1. For the phospholipid-binding model (Figure 2A), the enzyme activity (V)

CL þ km f þ a<sup>1</sup> f

A and B represent two different Lipolytic enzyme kinetic models for a lipid phase with two substrate domains: enriched substrate domain and nonenriched substrate domain. L, lipid molecule; S, phospholipid substrate; E, lipolytic enzyme; ES and ESS, enzyme-substrate complexes; and P, product. Individual kinetic constants are

�<sup>2</sup>). In the text: km � <sup>k</sup>�<sup>2</sup> <sup>þ</sup> <sup>k</sup>

and parameters to a particular domain is indicated by mean of primed or nonprimated signs in each case.

<sup>k</sup><sup>2</sup> and ks � <sup>k</sup>�<sup>1</sup>

<sup>k</sup><sup>1</sup> . The association of symbols

2 <sup>S</sup><sup>1</sup> þ a<sup>2</sup> f

2 <sup>S</sup><sup>1</sup> þ a<sup>2</sup> f

2 S2

> 2 S2

(3)

depends on the substrate reordering in according to

<sup>V</sup> <sup>¼</sup> kCET <sup>a</sup><sup>1</sup> <sup>f</sup>

kmks

steps binding between the enzyme and the lipid phase. First, the simple

Extracellular Vesicles and Their Importance in Human Health

be distributed in two coexisting lateral phases: an enriched domain and a

distributions.

Figure 2.

shown (k, k<sup>0</sup>

74

, k1, k<sup>0</sup>

<sup>1</sup>, k�<sup>1</sup>, k<sup>0</sup>

�<sup>1</sup>, k2, k<sup>0</sup>

<sup>2</sup>, k�<sup>2</sup>, k<sup>0</sup>

phospholipid-binding model [19]:

$$a\_1 + a\_2 = 1\tag{5}$$

According to Eq. (4), f is the average of the substrate molar fraction weighed by the domain areas.

In Eq. 3, we can see that V depends hyperbolically on

$$
\langle \langle f\_{\mathcal{S}^i}^2 \rangle \equiv a\_1 f\_{\mathcal{S}^1}^2 + a\_2 f\_{\mathcal{S}^2}^2 \tag{6}
$$

the average of the square of the substrate mole fraction weighted by the phase area. These mean values have minimal and maximal values equal to f <sup>2</sup> and f, respectively. Therefore, there are for V minimal and maximal values (Vmin and Vmax, respectively):

$$V\_{\min} = \frac{E\_T kf^2}{\frac{k\_m k\_s}{L} + k\_m f + f^2} \tag{7}$$

We can see that Vmin corresponds to V for a lipid homogeneous phase (Eq. 1) [12, 13]

$$V\_{\text{max}} = \frac{E\_T kf}{\frac{k\_m k\_r}{L} + k\_m f + f} \tag{8}$$

Curiously, in spite of Eq. 8 deduced for the phospholipid-binding model with any substrate distribution, this equation is equal to Eq. 2, which corresponds to the case of surface-binding model with homogeneous substrate distribution on the membrane.

According to Eq. 3, if the homogeneous distribution of the substrate on the membrane is broken (i.e., substrate reordering such that 〈 f 2 <sup>S</sup><sup>i</sup>〉 . f 2 ), then the enzyme activity will increase. In particular, the total enzyme activity increases when the recruitment of substrate to the enriched-substrate domain (e.g., phase 1) increases, due to an increase of either the domain mol fraction ( f <sup>S</sup><sup>1</sup> ) or the extension of the domain (a1).

2. On the other hand, for the surface-binding model (Figure 2B), the enzymatic activity in a two-phase membrane equals to the enzyme activity in a homogeneous lipid phase, following the equation:

$$V = \frac{C\_{E\_T}kf}{\frac{k\_mk\_l}{C\_L} + k\_m + f} \tag{9}$$

and applying the same restrictions given by Eqs. 4 and 5.

The differences between the behaviors of both enzymatic models are much more than the differences between the corresponding equations for the enzymatic activities (Eqs. 3 and 9). Following this theory, important differences exist in both the ratio of the substrate regarding the two substrate domains and the total enzyme binding to membrane.

The ratio between the enzymes binding into the two domains (indicated as primed and nonprimed) is described as follows:

for the phospholipid-binding model:

$$\frac{E\_{\text{S}} + E\_{\text{SS}}}{E\_{\text{S}}^{'} + E\_{\text{SS}}^{'}} = \frac{f\_{\text{d}} \left(k\_{m} + f\_{\text{d}}\right)}{f\_{n} \left(k\_{m} + f\_{n}\right)}\tag{10}$$

distribution could change the enzyme activity. However, another effect must be considered when there are domains that have been induced by soluble peptides (e.g., basic peptides such as pentalysine), which interact directly with the phospholipid substrate of the membrane (i.e., acidic phospholipid such as PIP2). In this case, such interaction could be enough to consider a competitive effect over the enzyme activity; i.e., in the lipid-water interface, the domain-inducing peptide would compete with the enzyme for the substrate, because there would be less free substrate to bind to the enzyme. Then, we have proposed that the superposition of both redistribution and competitive effects may explain some results in the litera-

Effects of Vesicular Membranes Reordering on the Activity of Lipid Metabolizing Enzymes

3.1 The kinetic effects of peptide induction of phospholipid domains

In order to calculate the effects on the PLC-β activity (a lipolytic enzyme) on PIP2 (lipid substrate) due to pentalysine-induced domain formation, it has been assumed that the stoichiometry of binding is one lipid substrate per one domaininducing peptide [19]. In case of larger stoichiometry for the phospholipid binding to the peptide (as Kim et al. describes [20]), this would imply that the competitive effect from peptides tends to decrease the enzyme activity more dramatically at low substrate molar fractions in the 1:1 stoichiometric case. Then, to estimate the amount of substrate bound to all the domain-inducing peptides in any lipid phase, it was assumed that the domain-inducing peptide near the surface of the lipid phase is in equilibrium with the phospholipid substrate, obeying a Langmuir isotherm, and this peptide concentration was determined by the electrochemical equilibrium in according with a Boltzmann-like relationship, which included the membrane potential in the lipid phase and the peptide concentration in the bulk solution [21]. Moreover, knowing the substrate binding to the domain-inducing peptide, free substrate can be calculated, and then the molar fraction of free substrate can be taken into account into the deduced previous kinetic models (Eqs. 3 and 9).

As a result, if there is competition effect due to peptide binding substrate, in case of the surface-binding model, the enzymatic activity will always diminish because the substrate reordering has no effect in the enzyme activity. Instead, in case of the phospholipid-binding model with peptide-induced breakage of substrate homogeneity the enzyme activity may either increase or decrease depending on the difference between the competitive effect (diminishing the enzyme activity) and the substrate distribution effect (increasing the enzyme activity) [19]. A theoretical estimation of PLCβ, acting on PIP2 as substrate, and having enriched substrate domain induced by pentalysine, has been shown in Figure 4 of Salinas et al. 2005 [19]. A maximum

for an enriched domain, with acute declination for others, are shown.

domains

77

normalized area ai and molar fraction fi of S. The total conservation equation is given as:

4. Lypolitic enzyme activity in lipid phases with multiple substrate

The above kinetics expressions can be generalized to any amount of substrate domains, even to infinite number of domains, and this latter is very useful for modeling any kind of substrate distribution in the total lipid phase [22]. In this extended theoretical frame, we have the following:

In homogenous condition, f is the molar fraction of phospholipid substrate. S reorders into n homogeneous domains, with the ith domain (i = 1, 2,…, n) with

ture that appear as contradictory. [19]

DOI: http://dx.doi.org/10.5772/intechopen.85972

for the surface-binding model:

$$\frac{E\_S + E\_{\rm SS}}{E\_S' + E\_{\rm SS}'} = \frac{k\_m + f\_d}{k\_m + f\_n} \tag{11}$$

Furthermore, there are different expressions for the molar concentration of total enzyme binding to the lipid phase, ½ � EB :

Naming the molar concentration of total enzyme binding to the lipid phase ½ � EB , we have:

$$\begin{aligned} [E\_B] & \equiv La(E\_S + E\_{\rm SS}) + L(1 - a) \left( E\_S' + E\_{\rm SS}' \right) \\ &= E\_T \left( \frac{k\_m f + \langle f\_i^2 \rangle}{\frac{k\_m k\_s}{L} + k\_m f + \langle f\_i^2 \rangle} \right), \end{aligned} \tag{12}$$

for the phospholipid-binding model, and

$$\begin{aligned} [E\_B] & \equiv La(E\_S + E\_{\rm SS}) + L(1 - a) \left( E\_S^{'} + E\_{\rm SS}^{'} \right) \\ &= E\_T \left( \frac{k\_m + f}{\frac{k\_m k\_s}{L} + k\_m + f} \right), \end{aligned} \tag{13}$$

for the surface-binding model.

Then, due to the difference between Eqs. 12 and 13, the two enzymatic models could be easily distinguishable by means of the observed change in the total lipid metabolizing enzyme binding to the lipid phase under substrate reordering: phospholipid-binding model predicts changes in total membrane enzyme upon domain formation, unlike the surface-binding model, in which there are no changes in total membrane enzyme, whatever the distribution of the substrate is.
