### Micro-organism and fermentation system

The mycelial culture of mushroom *Pleurotus ostreatus* was grown at 28 ± 1 °C for 5 days in a medium containing 1% glucose, 1% malt extract, 10% potato extract, and 0.15 % KH_{2}PO_{4} after inoculation with small mycelial pieces of the fungus. The enzyme production medium contained (g l^{−1}) NH_{4}H_{2}PO_{4}– 24, MgSO_{4}.7H_{2}O– 0.5, CaCl_{2}.2H_{2}O– 0.37, H_{3}PO_{4}– 0.57, FeSO_{4}.7H_{2}O– 0.25, MnCl_{2}– 0.032, NaMoO_{4}– 0.032, KH_{2}PO_{4}– 1.5 [7] in combination with a carbon source at 20 g l^{−1}. Mycelial pellets were then transferred aseptically to a 250 ml of polypropylene flask containing two 1-cm diameter glass beads and crushed to form a slurry mixture by shaking for 2 h. For submerged fermentation (smf), 0.4 gl^{−1} inoculums were added to a 50 ml enzyme production medium in a 250-ml Erlenmeyer flask, while for solid-state fermentation (ssf), the same quantity of inoculum was added to polyurethane foam (PUF) cubes of 1 g, absorbed with 50 ml of enzyme production medium in a 250-ml Erlenmeyer flask [8]. Erlenmeyer flask for solid-state fermentation containing PUF with enzyme production media was initially moistened with 75% and incubated at 85 % of the relative humidity of the incubation chamber. Fermentation was carried out for 7 or 10 days at under constant shaking at 180 rpm for smf or on solid matrix-assisted ssf under the static condition at 30 ± 1 °C.

### Carbon substrate

The white portion of the peel of pomelo (*Citrus maxima*) was initially washed by water followed by dist. water and oven-dried at 50 °C. The dried mass was grounded into a fine powder and screened through 200 meshes. The sieved mass was dried overnight at 50 °C and used as the carbon source due to its water-soluble nature. The carbon source was used at 20 gl^{−1} in all experiments of smf or ssf except the experiment of Fig. 1, where the varying concentration of carbon substrate was used. All other chemicals used were of analytical grade.

### Preparation of inert supporting matrix

Low-density polyurethane foam (17 kg m^{−3}) was used as an inert support and PUF cubes (0.5 cm^{−3}) were washed with dist. water, oven-dried at 60 °C [9]. Dried PUF cubes (1 g) were absorbed with liquid medium (50 ml) were then placed in a 250-ml Erlenmeyer flask. The PUF containing culture medium was sterilized by autoclaving under standard conditions [8].

### Enzyme extraction and biomass evaluation

The culture filtrate containing enzymes for submerged fermentation (smf) was obtained by centrifugation of the medium at 10,000 rpm at 4 °C for 10 min. The mycelia were washed with dist. water, collected by filtration using filter paper (Whatman no 1), and dried at 60 °C. For solid-state fermentation (ssf), the PUF cubes with attached mycelium were squeezed to remove the medium containing extracellular enzymes. The culture filtrate was collected by centrifugation under the above-mentioned conditions [8]. After extraction, PUF cubes were thoroughly washed with dist. water to remove any adhered particle with mycelium, immobilized with solid PUF cubes, and dried completely at 60 °C. In both the fermentation systems, biomass was evaluated as the difference of pre-weighed filter paper or pre-weighed PUF cubes [10].

*Exo*-polygalacturonase activity assay

*Exo-*polygalacturonase (*exo*-PG) activity was assayed by incubating the enzyme for 10 min at 50 °C with 0.5 % (w/v) polygalacturonic acid in 50 mM citrate buffer (pH 6.0). The released reducing groups, expressed as galacturonic acid were quantified by di-nitro salicylic acid reagent [11]. A control was simultaneously prepared taking thermally denatured enzyme. The concentration of the product (D-galacturonic acid) was compared with a D-galacturonic acid standard curve. The enzyme activity was expressed as Uml^{−1} in which, one enzyme unit is defined as the amount of enzyme required for releasing of 1 μmol of galacturonic acid per minute under the assay condition. Total carbohydrate as a substrate in the culture medium was determined by anthrone reagent following standard protocol and the yield coefficient was calculated from the experimental data based on the Monod equation.

#### Theory of bioprocess modeling

In a fermentation system, the microbial growth curve assumes a sigmoid shape and the growth curve can be predicted using Velhurst-Pearl logistic equation [12] as

$$ dX/ dt={\mu}_{\mathrm{max}}X\left(1-X/{X}_{\mathrm{max}}\right) $$

(1)

Where *μ*_{max} is the maximum specific growth rate (h^{−1}) and *X*_{max} is the maximum attainable biomass concentration (gl−^{1})

The integrated form of Eq. (1) with the initial condition *X* = 0 at *t* = 0 which gives as

$$ X={X}_0^{e^{\mu \max\;t}}/1-\left({X}_0/{X}_{\mathrm{max}}\right)\left(1-{e}^{\mu\;\max }t\right) $$

(2)

The kinetic parameter *μ*_{max} can be determined after rearranging Eq (2) as

$$ {\ln}^{X_{\mathrm{max}}}/{X}_0={X}_{\mu }t-\ln \left(\zeta /1-\zeta \right) $$

(3)

where *ζ* = *X*/*X*_{max,} i.e., the dimensionless variable of relative growth

If the experimental data is well-fitted in the equation, then a plot of ln(*ζ*/1 − *ζ*) versus time (*t*) give a straight line of the slope, *μ*_{max} and the intercept is − ln(*X*_{max}/*X*_{0}) [13]

Among the various model reported in the literature to express quantitatively the production rate of a metabolic compound, the classic Luedeking and Piret kinetic model [14] has a wide application in the microbial fermentation system. Kinetics of product formation, i.e., enzyme (E) can be mathematically expressed as

$$ dE/ dt={\alpha}^{dX}/ dt+\beta X $$

(4)

where *α* and *β* are empirical coefficient that may vary with fermentation conditions. It states that the product formation rate varies linearly with both the instantaneous cell mass concentration (*X*) and growth rate (*dX*/*dt*).

If we consider the product yield coefficient of the enzyme (E) in terms of biomass (*Y*_{E/X}), then the Eq. (4) can be expressed as

$$ dE/ dt={Y}_{E/X} dX/ dt+\beta X $$

(5)

The coefficient *β* may be negative, positive or zero value, because Luedeking-Piret model is related to growth-associated or non-growth-associated product formation.

The Eq. (5) may be arranged as a function of biomass (where *E* = *E*_{0} at *t* = 0) which gives as

$$ {E}_{(t)}={E}_0+{Y}_{E/X}\left(X-{X}_0\right)+{\beta}^{X_{\mathrm{max}}}/{\mu}_{\mathrm{max}}\kern0.5em \ln \left[\frac{X_{\mathrm{max}}-{X}_0}{X_{\mathrm{max}}-X}\right] $$

(6)

at a respective time (*t*) of fermentation [15]

Defining *λ* = (*E* − *E*_{0}) and *ζ* = *X*/*X*_{max,} the Eq. (6) can be re-arranged in the equitation as

$$ \lambda ={Y}_{E/X}{X}_{\mathrm{max}}\left\{\left(\zeta -{\zeta}_0\right)+\sigma \ln \left[\frac{1-{\zeta}_0}{1-\zeta}\right]\right\} $$

(7)

where *σ* = *β*/*Y*_{E/X}*μ*_{max,} i.e., the ratio between the rate of secondary formation or breakdown of the product as related to the maximal rate of product formation (*Y*_{E/X}*μ*_{max}).

The variable *λ*, which represents the increase of product formation is very important since the plot derived from *λ* versus *ζ* as Eq. (7), the shape of the production curve and the presence of product breakdown can easily be determined and compared between two types of fermentation system [15].

#### pH and temperature stability of crude *exo*-polygalacturonase

In the determination of pH stability, crude *exo*-polygalacturonase was diluted with 0.1 M of different buffer system (pH 3.0–pH 9.0) and incubated at 37 °C for 2 h. The residual enzyme activity was determined following the standard method. The temperature stability of the enzyme was also determined by measuring the residual activity at different intervals after incubating the enzyme at different temperatures (30–70 °C) in 50 mM citrate buffer (pH 6.0).