Pergamon
Chemical Enoineering Science, Vol. 52, No. 15, pp. 2629 2639, 1997 PII:
S00092509(97)00078X
(~, 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009 2509/97 $1%00 + 0.00
Mass transfer and growth kinetics in filamentous fungi F. LfpezIsunza,** C. P. LarraldeCorona* and G. ViniegraGonz~lez* *Departamento de Ingenieria de Procesos e Hidrfiulica; tDepartamento de Biotecnologia, Universidad Autfnoma Metropolitana Iztapalapa, Mfxico 09340, D.F., Mexico (Accepted 1 August 1996)
AbstractA mechanistic model has been developed to describe mass transfer and growth kinetics in filamentous fungi on solid substrate before branching takes place. The model accounts for substrate diffusion in the solid medium, uptake and release of substrate at the fungal wall membrane by a carrier, and formation of wall precursors coupled to mass transport by diffusion and convection inside the hypha. Model simulations for a range of substrate concentration showed good qualitative agreement with experimental data, when measured values of hyphal diameter and maximum length of a germ tube were employed. The model was used to evaluate the relative contribution of convective flow to the overall mass transport inside the hypha, to explore the magnitude of substrate concentration microgradients in the solid medium, and to assess the role of transport and kinetic parameters in the behaviour of hyphal growth. ~/ 1997 Elsevier Science Ltd
Keywords: Mechanistic model; mass transfer; diffusion and convection; growth kinetics; solid substrate; filamentous fungi; moving boundary.
INTRODUCTION On the ocassion of the celebration of John Villadsen's 60th anniversary, we present an application of the biochemical reaction engineering principles to the analysis of the phenomena of mass transfer and growth kinetics in filamentous fungi, and the numerical solution of the mechanistic model by the method of orthogonal collocation; two subjects in which J. Villadsen has made important contributions (Villadsen and Stewart, 1967; Villadsen and Michelsen, 1978; Nielsen and Villadsen, 1992, 1994). The development of a model to describe the hyphal growth of filamentous fungi on solid substrates should consider the description of mass transport of nutrients in the medium, through the fungal membrane at the wall, and within the hyphae where, simultaneously, several biochemical reactions occur. Fungi and actynomicetes are important organisms for the production of food materials, pharmaceuticals, intermediate chemicals, enzymes, feedstuffs and agricultural materials. They are also important biomaterials to be used in bioprocess engineering. Filamentous fungi are, in particular, the most important microorganisms for the processing of solid sub
* Corresponding author.
strates because of their physiological capabilities and hyphal mode of growth (Mitchell, 1992). Filamentous fungi have a vegetative phase made of structures called mycelia which grow by their tips as binary branching trees (ViniegraGonzalez et al., 1993). Each mycelium is like a forest where an individual tree is called hypha. Each hypha usually starts from a spore which gives rise to one or several cylindrical outgrowths called germinal tubules, where material is transported towards the tip to support apical growth (Jennings, 1976, 1978). Hyphae are encased in a rigid matrix of an heteropolymer called chitin whose building blocks are made from glucosamine. Growth at the tip is possible if a destructiveconstructive cyclic process of chitin allows for cell wall enlargement (BartnickiGarcia, 1973). This is in agreement with the observations made by Gooday (1971), who found that deposition of radiolabelled glucosamine was proportional to hyphal enlargement, and that pulses of hot glucosamine were located at the tip of the hyphal branches with practically no labelled species in the subapical regions. It has been recognised that cytoplasmic vesicles play an important role in apical growth of mycelial organisms (BartnickiGarcia, 1990). Two kinds of vesicles of different sizes, performing different functions, could be distinguished: macrovesicles and microvesicles. We now quote from BartnickiGarcia (1990): "the polymers and enzymes that compose the
2629
2630
F. LdpezIsunza et al.
amorphous phase of the wall are secreted in macrovesicles, whereas the enzyme that makes the microfibrillar chitin skeleton of the wall of most fungichitin synthetase is delivered separately in microvesicles". Macrovesicle components are synthesized internally during the process of endomembrane differentiation, whereas microvesicle components are assembled at or near the Wallmembrane interface. Several models have been proposed in the literature to describe hyphal growth of a fungal colony by apical enlargement and branching. In general, they can be divided into cell models based on the observed cellular mechanisms, and population models which consider a large number of intracellular components and reactions. Both types had been summarized by Nielsen (1992). Here only some cell models are discussed in connection with this work. Prosser and Trinci (1979) were apparently the first to present a model where hyphal growth and branching are based on the observed cellular mechanisms. The model relates hyphal growth with the productio~a and transport of vesicles containing wall precursors and/or the enzymes required for wall synthesis, and its absorption at the tip to produce hyphal elongation. Their finite difference model predicted changes in hyphal length and in the number and positions of branches and septa, that were in good agreement with experimental data. BartnickiGarcia et al. (1989) proposed an empirical model to describe the morphological development of mycelia, where vesicles coming from the subapical region are assembled in a distribution centre, the 'spitzenkorper', where they are later randomly distributed to the surrounding cell wall. This model describes an ideal conical shape of the hypha in terms of a contangent function with focus in the 'spitzenkorper'. Two main parameters are used: the rate of increase in area which is equal to the number of vesicles released by the "vesicle supplied centre' (VSC) per unit time, and the rate of linear displacement of the VSC. The model describes qualitatively the apical growth and hyphal morphogenesis. Georgiou and Shuler (1986) have shown that an important feature of fungal cultures grown on solid substrates, such as agar plates, is the creation of concentration gradients of substrate in the medium around the hypha as growth takes place. Their model takes into account the diffusion of the substrate in the agar gel and considers the biomass in the colony in terms of four different species (vegetative, competent, conidiophore and conidial). Their simulation studies suggest that mass transfer limitations may be responsible for the proliferation of differentiated structures, and the predicted gradients of glucose concentration are qualitatively in agreement with those measured in fungal colonies on agar plates by Robson et al. (1987). Aynsley et al. (1990) proposed a model for the growth of mycelial fungi in a submerged culture, and this considers that the substrate is absorbed from the surrounding medium uniformly along the hypha,
where it is transformed into a wall precursor following a Monod kinetics. It is assumed that no mass transfer limitations exists, and that the precursor is transported (convectively) at a constant rate to the hyphal tip. Here the precursor is transformed, via MichelisMenten kinetics, into a cell wall material. Hyphal tip grows at a rate proportional to the production of wall material. Mitchell et al. (1991) presented a model for the growth of Rhizopus oligosporus on a solid substrate, in which glucoamylase is excreted to the solid medium where it diffuses to perform the hydrolysis of starch, releasing glucose which diffuses from the medium towards the mycelium. The glucose absorbed by the mycelium is converted immediately into biomass according to a classical Monod kinetics. They obtained good agreement between model simulations and experimental data. Yang et al. (1992) developed a model for apical growth, septation and branching of mycelial microorganisms. The model contains a deterministic part based on the diffusion and production of wall precursors to describe tip growth and septation, and a stochastic part to account for the branching processes (tip growth directions and outgrowth orientation of branches). The model describes the morphological development of mycelia up to the formation of pellets very closely to experimental observations. The common features in the aforementioned models are: (a) the production of wall precursors; (b) the mass transport of wall precursors either by diffusion or by convection; (c) the hyphal enlargement at the tip, but not as a moving boundary problem; (d) the growth of the fungal colony by branching. In this work we present a model that considers the hyphal growth as the combination of processes occurring simultaneously in three regions: (1) the solid medium, where substrate diffuses towards the hyphal wall membrane obeying Fick's law; (2) the hyphal membrane, where uptake and release of substrate by a carrier is perfomed at both ends, and where the counterdiffusion of carrier and carriersubstrate complex inside the membrane takes place; (3) the internal part of the hypha, where mass transport by convection and diffusion of substrate and precursors is coupled to wall formation at the tip. THE EMPIRICAL BASIS OF THE M O D E L
The development of the model has been based on the works by Prosser and Trinei (1979), Jennings (1976, 1978) and BartnickiGarcia (1973, 1990), The
Mass transfer and growth kinetics in filamentous fungi model considers that a germinated fungal spore grows on a solid substrate, describing hyphal elongation before branching takes place. Basic assumptions are: (1) Diffusion of substrate (Sm) in the solid medium towards the hyphal wall is in one dimension and obeys Fick's law. The diffusion path in the solid medium has a total length Lm (the typical radius of a Petri dish), which is, at least, two orders of magnitude larger than the observed maximum length attained by the main hypha (germ tube), as shown schematically in Fig. 1. (2) The intake of S,, through the cell membrane at the wall is performed by a passive but specific transport process, in which there is a high affinity for the substrate at the outer part of the fungal membrane and a low affinity at the inner part (Jennings, 1976). Within the membrane all components move by diffusion according to Fick's law. The uptake at the external part of the membrane involves the adsorption of substrate Sm by carrier A to form the carriersubstrate complex 0s, according to:
BartnickiGarcia, 1990), according to Ep
S
,P.
The rate of formation of wall precursor is given by MichaelisMenten kinetics: ~p

k'pEpS K st + S
(3) "
Furthermore, species S and P move by diffusion and convection inside the hypha (Jennings, 1976; BartnickiGarcia, 1990) towards the tip. (4) At the tip S is transformed into component Q (an adsorbed quantity in g/cm 2) which, together with P, react to form a new wall (Prosser and Trinci, 1979; BartnickiGarcia, 1990).
Sm .}A ~ Os. In a similar way, the release at the inner part of the membrane of substrate S and the regeneration of carrier A, is given by
G S
~Q Ew
Q + P
,Cell Wall.
Both bioreactions follow a MichaelisMenten kinetics according to 9~q

k'qEqS K'q+ S
9~w 
The rate of uptake at the outer part of the membrane follows a MichelisMenten kinetics, whereas the rate of release is first order. These rates are given by
kaSmA Ka+Sm
 
(1)
Nd = k~ 0s.
(2)
(3) Inside the hypha component S is consumed to form wall precursor P (Prosser and Trinci, 1979;
Solid Substrate Medium Sm


G+P
z=O
z
S, P
(5)
Rate constants in the above kinetic expressions are related to substrate intake and release at the wall membrane, and the formation of wall precursors and wall synthesis; their numerical values are given in the notation.
Empirism, to what extent? (I) It has been observed experimentally for Asper9illus niger when grown under decreasing levels of substrate (glucose) concentration produces germ
A, Os
wall membrane ;
r' 0
(4)
k" EwQP
0~~ S + A .
~a
2631
Hypha
z=Lh(t)
Fig. 1. Schematic description of the three regions where mass transfer and growth kinetics takes place (the diagram is not to scale). Simbols are: Sm, substrate in the solid medium; (A, G) carrier and carriersubstrate complex at the wall membrane; (S, P) substrate and wall precursor inside the hypha; (Q) adsorbed specie formed at the tip to produce new wall.
F. L6pezIsunza et al.
2632
Table 1. Experimental values of germ tube and diameter of Aspergillus niger for various levels of glucose concentration (LarraldeCorona, 1996) Glucose (g/l)
Dh (/~m)
L~ (#m)
10 40 70 120 300
6.32 5.44 4.95 4.85 3.37
461.4 311.5 276.9 157.0 51.9
vesicles into a single component, which is formed and adsorbed at the tip. The corresponding dimensionless quantity q is a variable defined as the mass of adsorbed Q referred to the total hyphal volume with respect to the biomass density, given by
Q
q  phLh(t)
(9)
were Ph and Lh(t ) a r e the biomass density and length of the hypha at time t. THE MASS CONSERVATION EQUATIONS
tubules with larger lengths (Lc) and diameters (Dh) , a s shown in Table 1. The explanation for this is not clear yet. However, it has been suggested that this behaviour is related to osmotic effects (Jennings, 1976). The ability of some fungi to achieve different magnitudes of Dh and Lc for different substrate concentrations in solid media is a phenomena that, in the opinion of the authors, cannot be predicted a priori if other phenomena are not considered in the model during the early stages of germination. Therefore, values of parameters like Lc and Dh should be considered as input values for model simulations. It is important to note that because of the moving boundary nature of this model, Lc is also an important variable in the definition of the kinetic parameters. (II) It has been observed in the development of the model that the formation of wall precursors and new wall may be influenced by the amount of biomass during hyphal growth. For this reason, two versions of the same model have been developed. Model I considers that the concentration of enzyme Ep is proportional to the amount of biomass in the hypha, according to k'pEp = kvph(~/4)DZ Lh(t).
(6)
On the other hand, model II assumes that Ev is constant. (III) Finally, it is important to mention that model derivation has considered some important relationships concerning the following: (1) Kinetic parameters k" and k~ (in cm/s) for substrate intake and release, used in eqs. (1) and (2), involve surface reactions, and are related to volumetric parameters k. and ka (in l/s) in terms of the ratio of volume to external surface area of the hypha, in the following way: kaDh 4
(7)
kaDh k~, =  
(8)
k~ 
4
Therefore, the above expressions were used in eqs (1) and (2), by substituting parameters k; and k} in the model. (2) An important species in the kinetic model is Q (g/cm2), a surface concentration lumping micro
The coupled set of dimensionless conservation equations in the solid medium, wall membrane, and inside the hypha, describing the mass transfer and apical growth of a germinated fungal spore before the branching process takes place, is given by the following set of parabolic partial differential equations of a moving boundary nature The solid substrate medium In the solid medium, the mass balance for substrate sm accounts for accumulation and diffusion, and it is given by CSm 8t
~2Sm )'sin 6312 •
(10)
Boundary conditions consider that at the edge of the solid medium of length Lm, there is an impermeable wall, i.e. at l = 0 Os,~/S1 = 0.
(11)
At the hyphal membrane, the diffusive flux of substrate is equal to its uptake by the carrier, i.e. at 1 = 1 ~Sm t31
@axSma Ka + a
(12)
The hyphal membrane Inside the hyphal membrane accumulation and diffusion processes take place; the mass balances for carrier a and carriersubstrate complex 0 are, respectively
8a
~2a
0 t = ~'"~~ 2
(13)
80 820 ~t = ) ' a s p ~
(14)
Pertinent boundary conditions show that the diffusive fluxes are equal to the uptake and release of substrate at both sides of the membrane; and are given by At ~ = 0 (outer membrane): ~a 8~ 00 a(
~laSma K.+a tPasSma K.+a"
(15)
(16)
Mass transfer and growth kinetics in filamentous fungi At ( = 1 (inner membrane): 
(~a/~(
2633
of consumption at the tip. According to this, at ~ = 1
=

~dO
 (?0/0( = ¢a~0.
(17)
Diffusion, convection and bioreaction inside the hypha (Model I) Dimensionless mass balances for substrate s and wall precursor p inside the hypha consider the moving boundary at the tip, i.e. at z = Lh(t), as fixed, by making the dimensionless axial coordinate = z/Lh(t) (Crank, 1984). The resulting set of equations are:
6qS=Daq(Kq~s)2
(25)
c~P_ D ( qP ~22 77 \ *7/ "
(26)
~
(18)
7s
The accumulation of q at the tip is determined by its rate of formation from s, minus its rate of consumption to form the new wall.
dq
(
s
)1
Daw(_ qp ~(l+q). (27)
a7 =
+ (
The dimensionless rate of tip enlargement is given by
\K. +
 D.p(

d;
J77
s~))t
(19)
\/
c3p f l O2p . 2 f ~2p± l_c3p']'~ =
+"'\
I. "kDap
,.
)J
2Jc~{
\K, + p/c=,
2.
(20)
The terms on the LHS ofeqs (19) and (20) describe the mass accumulations of s and p; the first terms on the RHS give the diffusive mass transport (axial and radial) inside the hypha, the second are convective terms which account for the moving boundary due to the hyphal growth at the tip and the convective flow; the third terms give the consumption of s to produce p inside the hypha. Reaction parameters D,p and D,w are Damk6hler numbers for the production of precursor p and wall synthesis at the tip. Parameter is a dimensionless flow velocity, accounting for the convective transport to the tip of wall precursor p and substrate s. Boundary conditions assume that the membrane at the wall is impermeable to p but not to s. Therefore At r = 0:
~?s/& = ~?p/c?r=
At r = 1:
t?p/& =
0
0
 ~?s/Or=  riO.
~.
(28)
Equations (26)(28) had been derived considering the definition of q [eq. (9)], and as a result of this, the wall precursor p shows a dependency with the hypal length (biomass) to the second power, and the dimensionless hyphal length 2 shows an exponential dependency with time which is in agreement with some experimental observations (Trinci, 1969, 1971). Parameters D,q is a Damk6hler numbers associated with the wall synthesis at the tip. Initial conditions at t = 0 assume that £h(0) = Dh, which is valid for moulds like A. niger, The density of biomass, Ph, is assumed to remain constant during hyphal growth. MODEL II
Equations for Model II are the same as for Model I, except for the bioreaction terms for consumption of substrate s and production of wall precursor p. The equivalent of the last term in the RHS of eq. (19) and (20) for model II is D
(29)
where D*p is a modified Damk6hler number for Model iI (see notation).
The numerical solution
(22)
The numerical solution of the model was performed by using the methods of orthogonal collocation (Villadsen and Michelsen, 1978) to discretize the mass balance equations for the solid medium (three interior points), the hyphal membrane (one interior point) and the radial direction inside the hypha (one interior point). The method of global spline collocation (Villadsen and Michelsen, 1978), also known as orthogonal collocation on finite elements (Carey and Finlayson, 1975), was used to discretize the axial direction of the mass balance equations inside the hypha (two elements with three interior points each). Except for the radial direction inside the hypha where symmetry holds, for all the other coordinates, i.e. for (,
(23)
At ~ = 0: O.
qp
(21)
It can be seen that eq. (23) relates the diffusive mass flux of substrate at the inner part of the membrane with their release at the inner part of the hypha.
8s/8{ = ap/O~ =
(
d}= D,w \ K ~  ~ }
(24)
At the tip, substrate s is transformed into q (eqs (25) and (27)), which then reacts with wall precursor p to form a new wall [Eq. (26)], producing the enlargement of the tip. The fluxes of s and p are equal to their rates
2634
F. L6pezIsunza et al.
and l, the interior collocation points are the zeroes of P~'P)(~) = 0 with 7 = f l = 0 , i.e. Legendre polynomials (Villadsen and Michelsen, 1977). The resulting coupled set of nonlinear algebraicdifferential equations were solved by the methods of Broyden and a fourthorder RungeKutta. Although other numerical methods are better suited to solve more efficiently the nonlinear algebraicdifferential system of equations, no numerical instability was observed for the set of transport and kinetic parameters used in the simulations. The computer run for the reference case lasted 25 min and 49 s of CPU time using an Indigo2/R8000 from Silicon Graphics.
moreover, they describe the typical response of a tubular catalytic reactor with diffusion and reaction along the germ tube, and the depletion of all species due to its consumption at the (moving boundary) tip. In all simulations the same parameter values were subsequently employed with both models, including those at different levels of glucose concentration. It is important to mention that the only information sup
1.2 14h o
RESULTS
AND
0.8
DISCUSSION
Transport and kinetic parameters of Models I and II Model simulations were performed to predict the experimental observations of the growth of Aspergillus niger on agar plates using glucose as a carbon substrate (LarraldeCorona, 1996) with Models I and II. The estimation of transport and kinetic parameters was performed by model simulations, and using data from the literature. Experimental data obtained via image analysis techniques for the growth of Aspergillus niger on a medium with 40 g/1 of glucose (LarraldeCorona, 1996), was used as the reference case. Figure 2 compares predictions using models I and II with the experimental observations of hyphal growth for this case, in the absence of convective flow. Figures 3 and 4 show the corresponding dimensionless concentration profiles of substrate s and precursor p inside the hypha for Models I and II respectively, using the set of parameter values given in Table 2 and in the notation. It can be observed from Fig. 2 that predictions with model I give a slightly better description of the evolution of the hyphal length than Model II. It is important to note that only few parameters were different for the two simulations shown in Fig. 2 (see Table 2 and notation). Profiles shown in Figs 3 and 4 are different, as expected, and
E o
~ e
0,4
14h
0.2 8
h
~
p
0 0
0.2 0.4 0.6 0.8 1 Hyphal Length (dimensionless)
1.2
Fig. 3. Predicted internal profiles of dimensionlesssubstrate (s) and wall precursor (p) concentrations inside the hypha, at two different times during hyphal growth using model I.
8h 0.8 c
"3o
0.6 O
",~ 0.4 cO
350
8h
0.6
8h
0.2
P t
0 0
300
I
I
I
I
0.2 0.4 0.6 0.8 1 Hyphal Length (dimensionless)
e~
E
o
2 ,..2
250
Fig. 4. Predicted internal profiles of dimensionlesssubstrate (s) and wall precursor (p) concentrations inside the hypha, at two different times during hyphal growth using model II.
200 150
model I
I
/
el00
Table 2. Kinetic parameters used in Fig. 2
50 0 5
10 Time (hours)
15
20
Fig. 2. Comparison of experimental vs predicted hyphal growth for the reference case using Models I and II.
Parameters
Model I
Model II
k, kp kq K's K~
0.55 × 10 3 0.45 × 108 50 2.5 × 10 7 5x10 8
0.80 x 10 3 0.60 x 103 52 2.5 × 10 7 5x10 7
Mass transfer and growth kinetics in filamentous fungi plied to the models were the experimentally measured hyphal diameter (Dh) and length (Lc) of germ tube attained before branching occurs, and the initial glucose concentration of the solid medium.
The effect of substrate concentration in the medium (Models I and 1I). An important test for any model of this kind is the ability to predict the hyphal growth for different substrate concentrations in the medium. It has been observed from experiments with A. niger that, as the glucose concentration is increased, the length (Lc) and diameter (Dh) of the germ tube are reduced (LarraldeCorona, 1996). Figures 5 and 6 show predictions with Models I and II for glucose concentrations: 10, 40, 70, 120, and 300 g/l, using the corresponding measured
500
g
400
•
300
~
.
lOg/1 40 7O
d~ e~
200
f
~
120
¢e~
100 300
0 ~ 0
t 5
. . . 10 Time (hours)
15
20
Fig. 5. Comparison of experimental and predicted hyphal growth for different levels of substrate concentration in the solid medium. (0) 10 g/1 glucose; (A) 40 g/l glucose; (0) 70 g/l glucose; (11) 120 g/1 glucose; (T) 300 g/1 glucose.
2635
hyphal diameter (Dh) and length (Lc) given in Table 1, as input data. Although it can be observed from Figs 5 and 6 that predictions agree qualitatively well with the experimental data, predictions with Model I give a better description of the evolution of the hyphal length than Model II, suggesting that the use of the empirical relation given by Eq. (6) allows for a better description, although it has not been derived from first principles. This shows a better capability of Model I to describe qualitatively the observations. Model simulations have shown the limitations of both models: they cannot predict a priori the effects due to different concentrations of glucose in the medium, unless key parameters like D h and Lc are supplied as data to the model. It could be expected that in the case of high concentrations of the carbon source, the inclusion of phenomena like enzyme excretion for the assimilation of substrate from the medium, water activity and internal osmotic pressure (Jennings, 1976), could improve the predictions, showing other important effects.
Diffusion, convection and growth kinetics inside the hyphae (Model I) An important aspect of these models, in comparison with others previously published, is that they include mass transfer by diffusion and convection. Therefore, it is important to assess the relative contribution of both mechanisms to hyphal growth. Jennings (1976) has reported measured values of nuclei movement for different fungi, in the range 0.3 to 2/am/s. Figure 7 shows the predicted hyphal growth with Model I, for the reference case, using values of flow velocity uo: 0.05, 0.1 and 0.3 pm/s. It can be observed that when convective flow exists inside the hypha, there is an increase in their length of 17.4 and 28.4% for the first two values, respectively. However when Uo = 0.3, there is an increase of only 8.3% with respect to the reference case, showing a decrease with respect to the previous two values of Uo. This may be due to shortcomings in the numerical solution; for
50O
400
•
10g/1 400
0.1 0.05 0.3 Uo=0
350 '.~
30O
//122
~
200
*
.o
300 250 200 150
100
100 0
5
10 Time (hours)
15
20
Fig. 6. Comparison of experimental and predicted hyphal growth for different levels of substrate concentration in the solid medium. (0) 10 g/l glucose, (A) 40 g/1 glucose, (0) 70 g/1 glucose, (11) 120 g/1 glucose, (V) 300 g/1 glucose.
50
oL
.
5
I
.
.
10 Time (hours)
I
15
20
Fig. 7. Comparison of mass transport by diffusion vs diffusionconvection inside the hypha.
F. L6pezIsunza et al.
2636
larger values of u0 ~> 0.5, model predictions were not satisfactory, indicating that the amount of finite elements in the axial direction of the hypha need to be increased, to improve the numerical solution and to describe adequately sharper profiles inside the germ tube.
Diffusional effects in the solid media surrounding the hypha (Model 1) It has been shown experimentally (Robson et al., 1987) and theoretically (Georgio and Shuler, 1986), that there are important substrate concentration gradients in the solid medium during the hyphal growth of a colony. In this respect, it was important to perform model simulations to estimate the magnitude of these gradients, considering a diffusional path Lm = 100L~. Figure 8 shows the concentration profiles of substrate in the solid medium for different values of the glucose diffusion coefficient in the medium, D~m, and Table 3 gives the corresponding values for the maximum predicted hyphal length after 14.4 h. It can be seen that the magnitude of these microgradients is mainly determined by the relative rates of diffusive mass flux and uptake rate of substrate at the wall membrane. F o r substrates with low diffusion
aa=35
coefficients like big organic molecules, these microgradients may become very large and they may start to control the growth process along the time, which may also force the mould to branch out. It could be said that when external diffusional mass transfer limitations does not exist in the solid medium, the rate of hyphal tip enlargement is mainly controlled by the couple mass transportgrowth kinetics inside the hypha, as seen from the previous numerical results.
The role of kinetic and transport parameters inside the hypha (Model I) In order to assess the role of various kinetic parameters in model I, like rate constants k,, kp, kq and kw, model simulations were performed; they are shown in Fig. 9, and relative changes with respect to the reference case are given in Table 4. On the other hand, simulations varying saturation kinetic parameters K'~, K's, K'p and K'q are shown in Fig. 10, and relative changes with respect to the reference case are given in Table 5. Finally, to assess for diffusional mass transfer limitations inside the hypha, when convective flow is neglected, a simulation was performed reducing the values of the difussion coefficients of wall precursor p and substrate s (Degp from 6 to 2; De,, from 8 to 3). Figure 11 shows that these changes reduces the maximum germ tube length attained at 14.4 h from 307.6 to 245.6 p.m, indicating the important effects of diffusion inside the hypha, as expected.
0.8
~"~ o= ~
0.6 0.4
kp
400
3.5
300
0.2 0.35 0 0
I I I I 0.2 0.4 0.6 0.8 Length of Solid Medium (dimensionless)
Fig. 8. Dimensionless substrate concentration profiles in the solid medium for different values of the effective diffusion coefficient of substrate after 14.4 h.
.3
200 100
0 0
Table 3. Maximum length attained by the hypha, Lh(t), for different values of the diffusion coefficient of substrate in the medium, D,m
5
10 Time (hours)
15
20
Fig. 9. Contribution of kinetic parameters to hyphal growth.
Lh (t) (microns) after D~m(cm2/s) x 10 s
14.4 b.
0.35 3.50 35.00
58.2 294.3 307.6
Note: The measured value for 40 g/l of glucose was Lc = 311.5 gin after 14 h (LarraldeCorona, 1996).
Table 4. Kinetic parameter values used in Fig. 9 Parameter
Reference case (l/s)
New value (I/s)
k.
0.55 × 10  3
0.715 x 10  3
kp kq kw
0.45×10 a 50 450
0.9x10 s 65 585
Mass transfer and growth kinetics in filamentous fungi 400
Kp///•
350 =
/
300
g
Ks Kq I ref. case
~
250
K
a
200 150 ~:~ ~z
100 50 0 0
5
10 Time (hours)
15
20
Fig. 10. Contribution of saturation kinetic parameters to hyphal growth.
behaviour of hyphal growth at different substrate concentration levels. Detailed measurements of substrate concentration profiles performed in the model solid media to estimate microgradients, with and without the presence of fungal growth, may be very difficult, but they are needed to understand more about the branching process; these will also allow to estimate, independently, the diffusion coefficient of the carbon sources and the kinetic parameters for substrate uptake at the wall. Micrometric measurements during hyphal growth with labelled compounds may help in the estimation of mass transport and growth kinetic parameters to validate mechanistic models like the ones presented in this work.
Acknowledgements The authors are grateful to CONACYT (Consejo, Nacional de Ciencia y Tecnologia, M6xico) for the financial support.
Table 5. Saturation Kinetic parameter values used in Fig. 10 a Parameter
Reference case (g/l)
New value (g/l)
K'~ K~
4 250 500 50
10 50 100 10
K 'v g'q
at A D ap
D*, Daq
350 ref. case
300
Daw
250
DeG Defts =3
e.
ro e" d=
200
effp:2
Deft,
150 100
Dh DA
5O
DAs
0 0
5
10 Time (hours)
15
20
Fig. 11. Contribution of internal diffusional mass transfer to hyphal growth, in the absence of convective flow.
CONCLUSION A mechanistic model has been derived from first principles considering mass transfer and hyphal growth in three regions: solid substrate medium, wall membrane and inside the hypha. However, the use of empirical relations Ieq. (6)] have shown its usefulness in the description of the observed behaviour during hyphal growth of A. niger on a solid substrate. Model simulations have shown the relative contribution of mass transport and kinetic parameters, as well as the
2637
Dsm Ev Eq Ew k, kd kp
kq kw
NOTATION dimensionless mass concentration of carrier at the wall membrane aspect ratio (114.52)(= 2Lc/Dh) carrier concentration Damk6hler number for production of wall precursor P (0.5269) ( = nkpD~L3/4Deff,) modified Damk6hler number for production of wall precursor P in model II ( = kpEpL2/phDeG) Damk6hler number for production of Q (1.18) (= kqEqLc/phOeG) Damk6hler number for wall production (2.183) ( = kwEwL~/Deft,) effective diffusion coefficient of P inside the hypha ( = 6 x 10 8), cmZ/s effective diffusion coefficient of S inside the hypha (= 8 × 108), cmZ/s diameter of the hypha (= 5.44),/~m effective diffusion coefficient of carrier A inside the membrane ( = 5 × 108), cm2/s effective diffusion coefficient of carriersubstrate complex 05 inside the membrane ( = 4 x 108), cm2/s effective diffusion coefficient of substrate in the solid medium ( = 35 x 10s), cm2/s concentration of enzyme Ep. defined in Eq. (6) for model I; otherwise for model II (3 x 10 7) concentration of enzyme Eq (= 5 X 10101 concentration of enzyme Ew ( = 3 x 10 7) MichaelisMenten rate constant for uptake kinetics ( = 0.55 x 103), 1/s MichaelisMenten rate constant for release kinetics ( = 0.5), 1/s MichaelisMenten rate constant for production of P ( = 0.45x 108), 1/s MichaelisMenten rate constant for production of Q ( = 50), l/s Michaelis Menten rate constant for wall production (= 450), 1/s
F. L6pezlsunza et al.
2638 K~
Kb K~ K~ K~ K~ Kq Ks l L~
Lh P P q
MichaelisMenten saturation rate constant (= 4), g/1 MichaelisMenten saturation rate constant (= 500), g/1 MichaelisMenten saturation rate constant (= 50), g/1 Michaelis Menten saturation rate constant (= 250), g/1 dimensionless MichaelisMenten constant (= K'/Smo) dimensionless MichaelisMenten constant (= K'ffph) dimensionless MichaelisMenten constant (= K'q/ph) dimensionless MichaelisMenten constant (= K'/ph) dimensionless coordinate in the membrane (for L,, = 100L~) (= 1/Lm) maximum length of germ tube or main hypha (pm) at 14h (= 311.5) length of the hypha at time t, pm (t) dimensionless concentration of precursor P (= V/ph) mass concentration of wall precursor P, g/1 dimensionless concentration of Q, (= Q/
(p 0 0~
length of wall membrane (= 0.5) #m dimensionless concentration of carriersubstrate complex (= Os/ph) concentration of carriersubstrate complex,
g/1 2
dimensionless hyphal length at time t (= Ln(t)/Lc) dimensionless hyphal coordinate [ = z /
Ph
mass density of the hypha (= 1100), g/l characteristic time, s (= L~/D~,) dimensionless rate constant for substrate uptake at the wall membranesolid medium (18.31) (= kaDnPnLm/4SmoDsm) dimensionless rate constant for substrate uptake at the membrane (0.748× 10 4 )
Ln(t)] z
~.~ ~ga
(= ka~eDh/4Da) ~]as
~d
Ods
dimensionless rate constant for carriersubstrate formation at the membrane (0.935 x 10 4) (= k.~eDh/4Das) dimensionless rate constant for carriersubstrate desorption (0.068) (= k,~eDh/4Da) dimensionless rate constant for carriersubstrate desorption (0.85) (= kd~eDff4Das)
phLh(t))
Q r' r
Rh S Sm
S Sm Smo t' t Uo z
concentration of species Q, g/1 hyphal radial coordinate dimensionless hyphal radial coordinate (= r'/Rh) radius of the hypha (= Dh/2), #m dimensionless substrate concentration inside the hypha (= S/ph) dimensionless substrate concentration in solid medium (= Sm/S~o) substrate concentration inside the hypha substrate concentration in solid medium, g/1 initial substrate concentration in the solid medium (= 40), g/1 time, s dimensionless time (= t'/z) axial flow velocity inside the hypha, pm/s hyphal axial coordinate
Greek letters
fl
Tap ])asp
dimensionless axial flow velocity inside the hypha (= uoLc/De~,) dimensionless rate parameter for substrate release inside the hypha (0.2312) (= kdD2/8 D~fr~) dimensionless diffusion coefficient (2.234 x 105) (= DaL2c/Deff,~) dimensionless diffusion coefficient (2.588 x
10 5) (= DAsLc2/ D o.,(.)2 7~
7sm
dimensionless diffusion coefficient (1.333) (= D~frff D~ff,) dimensionless diffusion coefficient (0.5833 x
(
dimensionless wall membrane coordinate
10  3 ) (= Dsm/Deff,)
(= ~'/~.) ('
wall membrane coordinate
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