Theoretical Analysis of Steady States for Ester Hydrolysis in an Enzymatic Membrane Reactor With Product Retention

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Desalination 246 (2009) 545–555

Theoretical analysis of steady states for ester hydrolysis in anenzymatic membrane reactor with product retention

Marek Staniszewskia*, Stanisław Koter baFaculty of Food Science, University of Warmia and Mazury, pl. Cieszyński 1, 10-726 Olsztyn, Poland

Tel. +48 (89) 523-3510; email: [email protected] bFaculty of Chemistry, Nicolaus Copernicus University, ul. Gagarina 7, 87-100 Toruń , Poland

Received 20 June 2007; Accepted 14 March 2008

Abstract

The activity of enzymes is considerably affected by the pH of a reaction mixture. If a product of enzymaticreaction changes pH, as in the case of ester hydrolysis, this causes the non-monotonicity of both substrate and productdependent rate expression. This dependence determines the conversion degree values as well as the multiplicity of steady states in an enzymatic membrane reactor (EMR), i.e. a continuous stirred-tank bioreactor with an enzymerecycle. The transport properties of a membrane used also can modify the static behaviour of the system. The resultsof multiplicity analysis of enzymatic reaction which produces a weak acid in EMR are presented. To estimate theretention of a product by a membrane, the theory of irreversible thermodynamics by Kedem and Spiegler has beenused. On the bifurcation diagrams a trivial steady state (at the conversion degree going to zero) and the non-trivialsteady states are shown. Limits of regions of different multiplicity are localised. The effects of product acidity,enzyme properties and transport properties of a membrane on the structure of steady states are discussed. A speciallywritten software in Delphi™ has been used for the calculations.

Keywords: Enzymatic membrane reactor; Enzyme activity; Reactor performance; Steady-state multiplicity;Bifurcation

1. Introduction

The activity of enzymes is considerablyaffected by the pH of reaction mixture. Themaximum of activity is observed at a strictly

*Corresponding author.

defined value of pH dependent on the nature of an enzyme [1–5].

The maximum activity of an extracellular alkaline lipase from thermophilic Bacillus is pre-sent at pH 8.0–9.0 [6]. This enzyme of molecular weight ~37 kDa exhibits catalytic properties in

both the hydrolysis and the synthesis of esters in

doi:10.1016/j.desal.200 .0 .08 3 7 0

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M. Staniszewski, S. Koter / Desalination 246 (2009) 545–555

pH range 4.5–12. The catalytic action is pro-moted in the presence of Ca2+, Na+, Mg2+ and Ba2+

ions.The aminopeptidase from a cell-free extract of

Lactobacillus helveticus has a trimeric structure

and a molecular mass of ~129 kDa [7]. Theenzyme is optimally active in pH ranging 4.5–8.5(with maximum at pH 7.0) and strongly activated

by Co2+ ions. The aminopeptidase shows activitytoward dipeptides and tripeptides that havehydrophobic amino acids (Leu, Ala and Phe) or diaminocarboxylic acids (Lys and Arg) at the

N-termini.Monocyclic enzyme cascades may have either

one or many steady states depending on its struc-

ture [8]. This fact plays an important role as aregulator of biochemical reactions in living cells.Fluctuations in metabolites concentration aredetected through interactions between enzymes.The response of a cascade causes the regulationof the concentrations.

The multiplicity of steady states of differentcontinuous chemical reactors was extensivelystudied especially for exothermic reactions. Oneof the published examples is the analysis of dynamics of continuous solution polymerization

reactor with copolymerization of vinyl acetateand methyl methacrylate [9].

The phenomenon of steady-state multiplicitywas observed for enzyme catalysed reactions inflow reactors. Multiple steady states were experi-mentally studied in a continuous stirred tank reactor (CSTR) during hydrolysis of sucrose byinvertase [10]. Two different steady states werefound at the sucrose concentration equal to0.135 M and 0.7 M depending on the initial con-

centration of substrate in the reactor ranging from0.1M to 1.0M. This effect is a consequence of anticompetitive inhibition of the enzyme bysucrose.

Steady states of a two-substrate enzymaticreaction taking place in a CSTR were theoreti-cally investigated [11]. The authors determinednumerically the range of kinetic constants of an

enzyme where the hysteresis and bistabilityoccur.

The operation of an isothermal, non-ideal,tank enzyme reactor was theoretically studiedusing a non-Michaelis kinetics in [12]. The sys-

tem exhibits steady states of multiplicity three.A higher conversion was obtained at a higher feed concentration.

Similar phenomena were found in fermentors.The theoretical dynamics analysis of cultivationof aerobic microorganisms in a well-stirred reac-tor was previously described [13]. The authorsdeveloped a model based on the mass balanceequations for substrate, biomass and oxygenconcentrations with the use of a Monod–Monod

type expression and presented the phase portraitsof the system.The dynamics of anaerobic fermentation

process oriented for ethanolic fermentation wereinvestigated [14]. The four-dimensional modelwas used to simulate the static and dynamic be-haviour of the continuous stirred tank fermentor.

A theoretical analysis of dynamics of a membrane fermentor has been presented [15]. Themodelled system consisted of ethanolic fermen-tation coupled with diffusional removal of

ethanol. Selected operating parameters wereinvestigated as the bifurcation parameters. Gene-rally it was stated that the presence of a mem-

brane stabilises the process. More detailed modelanalysis of effect of the in-situ membrane ethanolremoval on the ethanol production in a membranefermentor was published [16,17].

If a product of enzymatic reaction influencesthe pH of reaction environment, as in the case of an ester hydrolysis, this causes the non-mono-

tonicity of the substrate and product dependentrate expression. The systems of this type aredifferent from those dependent on a substrateonly [18]. This dependency influences a multi-

plicity of steady states and the conversion degreein an enzymatic membrane reactor (EMR), i.e. acontinuous stirred-tank bioreactor with fullenzyme recycle. The selective properties of a

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membrane used, i.e. the retention of reagents, canmodify the static behaviour of this system. Thecontrol of operating parameters of EMR is of great importance in respect to the effectiveness of

process and the quality of product.

The aim of this study is to formulate a modelof enzymatic membrane reactor producing a weak acid and to explore the static behaviour of systemincluding the transport properties of a membrane.

2. Model equations

2.1. Dependence of enzyme activity on pH

pH is an important parameter influencing theactivity of enzymes, which are commonly the

protein compounds. To describe the dependenceof maximal reaction rate r max on pH of thereaction mixture, the following four-parameter expression is used:

(1) C

opt0max max

pH pH pH exp

r r A B

In Eq. (1) the values of B determine the range of pH in which enzyme shows activity, whereas the

A and C parameters influence the shape of curve.It is assumed that the enzyme activity does notchange with time.

The value of pH in a reactor is calculated from

(2) 2 4

pH log2

K K KPP

where K is the dissociation constant and P is aconcentration of weak acid P in the reactor. In

Eq. (2) the assumption that only dissociating pro-duct P determines the acidity of reaction mixturehas been made.

2.2. Rate of enzymatic hydrolysis

It is assumed that the inactivation and thelosses of an enzyme in EMR are insignificant.

The rate of hydrolysis of a substrate catalysed byan enzyme is expressed according to theMichaelis–Menten scheme [19]:

(3) max

M

( ),

r P S r S P

K S

The reaction rate r in the reactor depends on thesubstrate and product concentrations because of the relation between r max and pH influenced bythe product [Eqs. (1) and (2)]. The type of thisdependence is different in comparison to a com-

petitive or non-competitive inhibition of product[20]. This fact has important meaning for thestructure of steady states of the reactor.

2.3. Product retention

To estimate the retention of a product theapproach based on irreversible thermodynamicshas been used. The model of a solute transportthrough a reverse osmosis membrane treated as a

black box was proposed by Kedem and Katchal-sky [21]. This model was improved by Kedemand Spiegler [22] by applying the linear transportequations on a local level, not to the wholemembrane. This assumption gives the expressionin which the solute retention, R, is directly relatedto the volume flux J v [23]:

11

11 exp

v

R

J (4)

In Eq. (4) the reflection coefficient, σ, denotes themaximal retention reached at a very high flow J v.

The values of retention coefficient of low mole-cular weight compounds depend mainly ontransport properties of thin charged layer of amembrane, which exhibits both the sieve effectand the charge effect related to valence of a co-ion. The retention of a product greater than zeroincreases its concentration in the reactor and thusit modifies the static behaviour of the EMR.

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2.4. Calculating of steady states of a reactor

The dynamics of isothermal, constant density,continuous stirred tank reactor, i.e. the time deri-vatives of substrate and product concentrations in

the reactor (Fig. 1), where the enzymatic reaction(5) characterised by the rate r takes place, isdescribed by Eqs. (6) and (7) resulting from themass balance.

S P + Q (5)

In reaction (5) S is an ester, P is a weak acid andQ is an alcohol.

(6) in outdd

S V S S F rV t

(7)out

d

d

PV P F rV

t

In Eq. (6) S in is the substrate concentration atthe inlet. It is assumed that the product concen-trations at the inlet Pin = 0 and Qin = 0. r is thereaction rate calculated according to Eq. (3). For the steady-state operation Eqs. (6) and (7) take

the form:

(8) in out0F

S S r V

(9)out0F

P r V

The unknown Pout is related to P by the retentioncoefficient R:

(10)out1 P RP

Substituting Pout from Eq. (10) into (9) oneobtains the equation for the steady state of EMR:

(11) 0 1F

P R r V

Fig. 1. Schematic drawing of the EMR.

By summing Eqs. (8) and (9) the expression for S out is derived:

(12)Eq.(10)

out in out in (1 )S S P S P R

Further it will be assumed that the retention of thesubstrate S and product Q as non-ionic compounds by the membrane is negligible, i.e. S out = S

and Qout = Q. The concentrations of reagents for the steady-state operation can be found by thenumerical solution of Eqs. (11) and (8) for variables P and S .

The calculations of conversion degree X in areactor, defined as

(13)

Eq. (12)

in out

in in

(1 )S S P R X S S

have been done using the bisection method [24].A specially written software in Delphi™ has beenused for the model analysis and for drawing the

bifurcation diagrams and sections of catastrophicsets.

3. Results and discussion

3.1. Bifurcation diagrams

3.1.1. Influence of volume flow

Bifurcation diagrams give much basic infor-mation about steady states of the system. Theyshow the situation of the steady states as well asthe values of the state variables. There are twotypes of steady states calculated from the model

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Fig. 2. Steady-state conversion degree X as a function of volume flow of feed F ; the trivial steady state at pH ~7,

in which X is close to zero, is present in the whole rangeof F .

equation [Eq. (11)] and shown in Figs. (2)–(4).The trivial steady state, at the product concen-tration near zero, is present in the whole range of

bifurcation parameters F , S in, r max. This steadystate appears if the pH of reaction mixture is far from the optimum pH of an enzyme (for pH >

pHopt, close to the neutral pH) resulting in residual

enzyme activity. The non-trivial steady states of non-zero conversion degree are present only in acertain range of volume flow F . In the case of multiplicity higher than one, the steady-state con-centrations depend on the initial pH in a reactor.

Fig. 2 shows the bifurcation curves X = f (F )calculated from Eq. (11) with the volume flow F

as a bifurcation parameter. The values of other parameters are as follows: S in = 20 mol m!3, pK =6.0, r max = 0.2 mol s!1, K M = 2 mol m!3, pHopt =

4.3, A = 3.0, B = 0.5, C = 3.0. Two non-trivialsteady states are present only at values F notexceeding 0.15 m!3 s!1. The curves representingnon-trivial steady-state conversions join at F =0.15 m3 s!1 (conversion degree about 0.05). At F

going to zero the higher conversion degree (upper curve) goes to unity whereas the lower one goesto zero.

Fig. 3. Steady-state conversion degree X as a function of substrate concentration in a feed S in.

3.1.2. Influence of substrate concentration in

a feed

The bifurcation curves X = f (S in) are shownin Fig. 3. The calculations have been performedfor the following values of parameters: F =0.06 m3 s!1, pK = 6.0, r max = 0.2 mol s!1, K M =2 mol m!3, pHopt = 4.3, A = 3.0, B = 0.5, C = 3.0.

Unlike in the case of diagram X = f (F ) (Fig. 2),

the maximum of conversion degree equal to 0.47is present at the inlet substrate concentrationabout 2.8 mol m!3. The steady state has multi-

plicity three for the inlet substrate concentrationover 2.0 mol m!3. For lower inlet concentrationonly a trivial steady state of multiplicity oneexists. At increasing inlet substrate concentrationthe conversion degree of both non-trivial steadystates decreases. This fact can be explained as asaturation effect of Michaelis–Menten typekinetics applied in the model.

3.1.3. Influence of maximal reaction rate

The maximal reaction rate r max is related to theamount of enzyme in the reactor. In the calcu-lations of bifurcation curves with respect to r max

(Fig. 4), the following values have been assumed:F = 0.06 m3 s!1, S in = 20 mol m!3, pK = 6.0, K M =

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Fig. 4. Steady-state conversion degree X as a function of the maximal reaction rate r max.

2 mol m!3, pHopt = 4.3, A = 3.0, B = 0.5, C = 3.0.Similar to the case of diagram X = f (S in) (Fig. 3),the non-trivial steady states are present startingfrom a certain value of maximal reaction rate r max

equal to 0.9 mol s!1. The upper curve approachesasymptotically to unity with increasing value of r max, whereas the lower one drops below the valueof 0.01.

3.2. Structure of the steady state

3.2.1. Effect of pK of a product

Fig. 5 presents the sections of catastrophic setwith plane (F , pK ) for values: S in = 20 mol m!3,r max = 0.2 mol s!1, K M = 2 mol m!3, pHopt = 4.3,

A = 3.0, B = 0.5, C = 3.0, ω = 8×10!6 m s!1. Thelines separate the planes into regions of multi-

plicity one (1 SS) and three (3 SS). For lower

values of pK up to pK = 5.8 the region of multiplicity three covers a higher range of volume flowF , thus increasing the range of operating para-meters of high conversion and productivity. Theincrease of the reflection coefficient of productincreases the area of the region of multiplicitythree for higher values of volume flow F

(Figs. 5B, 5C).

A

B

C

Fig. 5. Section of catastrophic set of EMR with plane(F , pK ) for reflection coefficients σ = 0.0 (A), σ = 0.3(B), σ = 0.6 (C).

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3.2.2. Effect of optimal pH for the enzyme

In Fig. 6 the sections of catastrophic set with plane (F , pHopt) are presented for the followingvalues: S in = 20 mol m!3, r max = 0.2 mol s!1, K M =2 mol m!3, A = 3.0, B = 0.5, C = 3.0, ω = 8×10!6

m s!1. A region of multiplicity three is localisedin the range of pHopt 4.4–4.8 for a whole range of volume flow F . The region is surrounded by theregions of multiplicity one. The range of pHopt

corresponding to the steady states of multiplicitythree decreases with increasing value of F . Theincrease of the reflection coefficient of productshifts slightly the limits between the regions inthe direction of lower pHopt (Figs. 6B, 6C).

3.2.3. Effect of A parameter

Fig. 7 shows the sections of catastrophic setwith plane (F , A) for values: S in = 20 mol m!3,r max = 0.2 mol s!1, K M = 2 mol m!3, pHopt = 4.3,

B = 0.5, C = 3.0, ω = 8×10!6 m s!1. The region of multiplicity three is situated under the line sepa-rating the plane. The decrease in the value of A

parameter from 7 to 1 causes an increase of therange of volumetric flow F in which high con-versions appear. A higher value of A means a

lower range of pH where the enzyme exhibitscatalytic activity. The increase of the reflectioncoefficient of product increases significantly thearea of the region of multiplicity three in direc-tion of high values of volumetric flow F

(Figs. 7B, 7C).

3.2.4. Effect of C parameter

In Fig. 8 the sections of a catastrophic set with plane (F , C ) is shown for the following values:

S in = 20 mol m!3

, r max = 0.2 mol s!1

, K M =2 mol m!3, pHopt = 4.3, A = 3.0, B = 0.5, ω =8×10!6 m s!1. The region of multiplicity three issituated generally at high value of C parameter.For C = 3.0 the steady state of multiplicity threeis achieved up to the volume flow F = 0.15 m3 s!1.A higher value of C parameter means a more flatshape of a pH dependency on an enzyme activity

A

B

C

Fig. 6. Section of catastrophic set of EMR with plane(F , pHopt) for reflection coefficients σ = 0.0 (A), σ = 0.3(B), σ = 0.6 (C).

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A

B

C

Fig. 7. Section of catastrophic set of EMR with plane(F , A) for reflection coefficients σ = 0.0 (A), σ = 0.3 (B),σ = 0.6 (C).

A

B

C

Fig. 8. Section of catastrophic set of EMR with plane(F , C ) for reflection coefficients σ = 0.0 (A), σ = 0.3 (B),σ = 0.6 (C).

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—this parameter does not affect the range of pHwhere enzyme is active. Comparable to the caseof A parameter the increase in retention of a

product increases the area of the region of multiplicity three (Figs. 8B, 8C).

4. Conclusions

A model for an enzymatic membrane reactor with reaction-producing weak acid has beenformulated and simulated. The results of analysisof static behaviour of the EMR are presented in

bifurcation diagrams and sections of catastrophicset on selected planes of operating parameters for different reflection coefficients of the product.

The effects of product acidity, enzyme propertiesand transport properties of a membrane on thestructure of steady state have been discussed.

The results of the analysis allow the formula-tion of the following:C The bifurcation diagrams: the steady-state

conversion degree X versus a bifurcation para-meter have different characters, depending onthe bifurcation parameter (F , S in and r max).Contrary to the other diagrams, X = f (S in)shows the maximum equal to 0.47 at S in = 2.8mol m!3. The position of the maximum de-

pends on the volume flow F . In the region of non-trivial steady states for fixed operating

parameters, the higher conversion degree of substrate is obtained at lower initial pH in thereactor.

C The higher the difference between the pK of a product and the pHopt of an enzyme, the higher the range of operating parameters in whichhigh effectiveness of conversion is reached.

C

For a given value of pK of a product thereexists an optimal value of pHopt of the enzymeassuring high conversion and production rate.

C The increase in the range of pH in which theenzyme exhibits a high catalytic activityresults in increase of the range of operating

parameters giving the high conversion of a substrate.

C The shape of pH dependence of enzyme acti-vity influences the structure of steady states of EMR.

C The retention of the product affects the cata-lytic activity of enzyme. In all investigated

cases the increase in the product retentionincreases the region of operating parameters atwhich the non-trivial steady states of practicalimportance appear. However, at a high enoughconcentration of product P, the rate of thereaction can decrease.

5. Symbols

A — Constant in expression for enzymeactivity Am — Working membrane area, m2

B — Constant in expression for enzymeactivity

C — Constant in expression for enzymeactivity

F — Volume flow, m3 s!1

J v — Volume flux across a membrane(=F / Am), m3 m!2 s!1

K — Dissociation constant of a product

K M — Michaelis–Menten constant of anenzyme, mol m!3

P — Concentration of product P in areactor, mol m!3

Pin — Concentration of product P at inlet,mol m!3

Pout — Concentration of product P atoutlet, mol m!3

pH — pH in a reactor pHopt — Optimum value of pH for an

activity of enzymeQ — Concentration of product Q in areactor, mol m!3

Qin — Concentration of product Q atinlet, mol m!3

Qout — Concentration of product Q atoutlet, mol m!3

R — Retention coefficient of product

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r — Reaction rate, mol m!3 s!1

r max — Maximal reaction rate, mol m!3 s!1

r 0max — Maximal reaction rate at pH = pHopt, mol m!3 s!1

S — Substrate concentration in a

reactor, mol m!3

S in — Substrate concentration at inlet,mol m!3

S out — Substrate concentration at outlet,mol m!3

X — Conversion degree of a substrateV — Working volume of a reactor, m3

Greek

σ — Reflection coefficientω — Salt permeability, m s!1

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Documents

Theoretical Analysis of Steady States for Ester Hydrolysis in an Enzymatic Membrane Reactor With Product Retention