Kinetics of mycelial growth and morphology

 A review by Liam McNulty

Contents

Introduction

Kinetic analyses show that fungal filamentous growth can be interpreted on the basis of a regular cell cycle, and therefore encourage the view that mycelial growth and morphology can be described mathematically. Here, we review published mathematical models that attempt to describe fungal growth and branching in the vegetative (mycelial) phase.

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Measurement methodologies

In order to describe and quantify hyphal growth and branching, measurement of the parameters hyphal diameter, h_d, and hyphal length, h_l, is essential. These allow hyphal volume, h_v, to be calculated, which when multiplied by the average density of the composite hyphal material, ρ, gives an estimate of biomass, X. If these measurements are taken over a series of time intervals it is possible to calculate hyphal extension rate, E, and thence rate of increase of biomass. Currently, automated image analysis systems permit real-time analysis of these microscopic parameters [1]. Some of these analyses suggest that hyphal tips grow in pulses [2], although this has been contested [3], particularly because the observations use video techniques and the pixelated image generated by both analogue and digital cameras will cause pulsation artefacts [4].

The most important macroscopic parameter is total biomass. Total hyphal length is proportional to total biomass, if h_d and ρ are assumed to be constant, but measurement can be difficult. Non-destructive direct mass measurement is rarely feasible, in most cases due to the technical difficulty (and sometimes impossibility) encountered in physically separating the mycelium from the substratum. Acuña et al. [5] developed a neural network that they trained to correlate colony radius with colony biomass. However, this relationship is only relevant to circular mycelia and measurements in two dimensions. More general relationships with biomass have been suggested for particular chemical compounds, the most promising of which is ergosterol, a sterol characteristic of fungal membranes [6, 7].  

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Describing hyphal branching

Detailed observation by Katz et al. [8] revealed that a germ-tube hypha will initially grow in length exponentially, at a rate that increases until a maximum, constant extension rate is reached, and that thereafter, it will increase in length linearly. The primary and subsequent branches will behave similarly. Thus, there develops a scenario in which individual hyphae extend linearly yet the biomass (or total length) of the whole mycelium increases exponentially. Trinci [9] offered a solution to this riddle by proposing that it was due to the exponential increase in tips due to branching.

In the same experiments, Katz et al. [8] studied the growth kinetics of Aspergillus nidulans on three different media, each with a distinct specific growth rate. From this they were able to proposed a number of general relationships that are conveyed in equation (1), elucidated by Steele & Trinci [10]:

where Ē is the mean tip extension rate, μ_max is the maximum specific growth rate, and G is the hyphal growth unit. G is defined as the average length of a hypha supporting a growing tip according to equation (2):

where L_t is total mycelial length, and N_t is the total number of tips. The hyphal growth unit is approximately equal to the width of the peripheral growth zone, which is a ring-shaped peripheral area of the mycelium that contributes to radial expansion of the colony [11, 12]. In a mycelium that is exploring the substratum, branching will be rare and thus G will be large. G is therefore an indicator of branching density; Katz et al. [8] postulated that a new branch is initiated when the capacity for a hypha to extend increases above Ē, thereby regulating G ≈ 1 unit, the value of which is indicative of the branching density of that species of fungi on that particular medium.

The next step is to use these kinetic laws in a model that describes the increase in tips and total mycelial growth over time. If this description is to be realistic it has to be based on sound biological premises.  One such premise is the ‘vesicle supply centre’ (VSC) [13], which proposes that tips extend by the incorporation into the tip membrane of new material that arrives packaged in vesicles supplied by the VSC. The concept is loosely based on the growth pattern described by Monod [14] who, whilst studying bacterial growth, developed mathematical expressions of how a finite growth resource is translated into biological growth.  

Prosser & Trinci (1979) used these premises in a model that successfully accounted for exponential growth and branching in fungal mycelia. The process was modelled in two steps:

In an extension of this model, Yang et al. [16] used the concepts of vesicle production and the duplication cycle to account for hyphal growth and septation, respectively, in the deterministic element of a two-part model. They also included a stochastic element that accounted for the branching process whereby the branching site and the direction of branch growth were generated by probability functions. This gave rise to a much more realistic mycelial shape without compromising the agreement with experimental data for the parameters that Prosser & Trinci [15] modelled. 

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Describing the evolution of biomass

In many instances, models such as these are applied to describing the evolution of biomass by converting the total mycelial length to a weight of biomass, as suggested above. This is particularly useful when the fungus is grown in an industrial or biotechnological context. This area of research has therefore attracted much attention and some of the other attempts to model biomass evolution are now discussed. 

When grown on solid media, mycelial colonies are limited by space and by nutrient supply. Thus their exponential phase of growth is finite in length and biomass evolution can be compared to the growth of populations modelled using the logistic equation:

where α is any real number, N is the number of individuals and μ, the specific growth rate, which is defined as:

and decreases as a negative linear function of population size according to:

The curves generated by these equations are similar to the experimental curves of biomass evolution in batch cultures of filamentous fungi, where N would be replaced by the biomass, X.

Koch [17] modified the logistic equation so that it could be applied directly to mycelial growth by replacing N with the change in biomass with respect to volume, and changing the time variable to (T - t), where T is the time since the mycelium began growing, and t is the time when growth entered the unit volume under consideration. Thus the equation was rewritten, when integrated with respect to t, as:

where X_o is the initial biomass.

Equation (6) was further modified to describe the increase in biomass when the colony radius extended linearly. However, this modification makes two contentious assumptions: (a) it assumes that the colony is a regular, flat disc (which is unrealistic); (b) it relies on the relationship between radial extension rate and specific growth rate, which assumes that the peripheral growth zone, ω, is constant in the equation:

However, this parameter varies from species to species and under different environmental conditions, so the modified equation is effectively limited to regular disc-shaped colonies growing on uniform media.

In order to be predictive (as opposed to descriptive), a model must contain a microscopic parameter that accurately represents a biological parameter, and which can be changed to test how it affects overall growth and development.

Viniegra-Gonzalez et al. [18] developed the first model able to do this. It modelled the microscopic development of the mycelium based on the mathematics of symmetric binary trees. This was then used in a macroscopic description of the overall evolution of biomass, based on the logistic equation that accounts for the proportion of inactive biomass (equivalent to the proportion of dead hyphae) in the mycelium by varying the coefficient α in equation (3).

The microscopic component of the model redefined simple growth kinetic laws in terms of a branching level parameter, k. In a symmetric binary tree (Fig. 2) every branching level is associated with a number of tips, N_t, according to the equation:

The number of segments, each of average length L_av, can also be calculated:

(When considering a mycelium as a whole k is averaged;).

If

and

Then equation (2) can be rewritten as:

If a hypha branches at a critical biomass, X_c, the total biomass can be obtained from:

A time element can be introduced by the expression (Fig. 2):

Calculating μ_x according to equation (7) having substituted in equations (12), (13) and (14), and recalling that dX/dt=dX/dy.dy/dt and d(2^y)/dt = Ln2.2^y(dy/dt); where y = +1, dy/dt= Ē /L_av, we then obtain:

This expression allows the calculation of μ_x when the only parameter measured is the number of tips (provided the value of X_c is known).

When reformulated in terms of X to describe macroscopic growth, statistical analysis showed that the equation described the evolution of biomass of Aspergillus niger more closely than the logistic equation. Thus, macroscopic growth had been modelled with microscopic parameters.

Based on this approach, Ikasari & Mitchell [19] developed a two part empirical model that concentrated on describing the whole biomass evolution curve. They focussed on the transition from exponential to decelerating growth by studying the proportion of active biomass when mycelial colonies are grown in over-culture, thus emphasising this region of the curve. The exponential phase was modelled using the mathematics of binary symmetric trees, as above, to generate growing tips. In the deceleration phase 71-86% of these tips died instantaneously and they then continued to decline at an exponential rate thereafter. They were able to generate an even better fitting curve to their experimental data. However, the model is specific to Rhizopus oligosporus and represents a fine-tuned description of the growth curve of this species that is based on more general models of mycelial growth kinetics common to all species.

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Describing branching patterns

Having considered the kinetics of the branching process and its relationship to mycelial growth and biomass evolution, we now turn our attention to describing mycelial morphology. Once this has been reviewed, the mechanisms behind generating these morphologies are discussed.

Leopold [20] examined the generality of natural branching systems in trees and streams. Based on the classification system of Horton [21], she labelled each branch of a tree or river network depending on how many tributary branches it supported. Hence, first-order branches have no tributaries; second-order branches support only first-order branches; third-order branches support only first and second-order branches; etc. She also measured the lengths of each branch to obtain an average value for each order of branching (the length of an n-order branch includes the length of its longest (n-1)-order tributary ). She found that straight-line plots resulted when branch order was plotted against the logarithm of (i) the number of branches of a given order, and (ii) the average length of a branch of a given order. The gradient of these lines was interpreted as (i) the branching ratio (BR = the average number of n-order branches for each (n+1)-order branch); and (ii) the length ratio (LR = the average length of each n-order branch as a multiple of the average length of each (n-1)-order branch). Observations suggest that the values of these ratios showed little variation over a range of tree species (BR = 4.7 - 6.5; LR = 2.5 - 3.6) and river networks (BR=3.5; LR=2.3).

Gull [22] applied this analysis to the mycelial branching characteristics of the filamentous fungus, Thamnidium elegans, and observed branching and length ratios of 3.8 and 4.0, respectively, for a third-order system and 2.6 and 2.7 for a fourth-order system. Though it gave no biological insight into the mechanisms of branching, Gull’s work demonstrated that mycelia employ branching as a strategy for colonising the maximum area of space using the minimum total mycelial length, and indicate that the values obtained can be interpreted as a quantification of branching frequency.

Another approach to quantifying branching frequency relies on the mathematics of fractal geometry. In the box-counting method of fractal analysis a grid of boxes, each with side length ε, is placed over the pattern, and the number of boxes, N_box, that are intersected by the pattern is counted. If a pattern is fractal, it will be ‘self-similar’ at all scales. This means that a true fractal pattern has an infinite length. However, the geometry of the pattern limits the degree to which it can fill the plane. This is quantified in terms of the fractal dimension, D, according to the formula:

where C is a constant. A straight line has a fractal dimension, D = 1, and a completely filled plane has D = 2.

By looking at higher and higher resolutions (i.e. in the limit ε ® 0) the repeating pattern will be revealed to cover a limited proportion of the plane. When the logarithm of N_box is plotted against the logarithm of (1/ε), a straight line is obtained with gradient equal to D. When applied to fungal mycelia, ε is limited by the hyphal diameter microscopically and by the mycelial diameter macroscopically. Thus, mycelia are not true fractals. However, this range is sufficient to allow reasonably accurate regression analysis for D, and thus quantification of the space filling capacity, or branching frequency, of mycelia can be obtained.

Obert et al. [23] applied this method to mycelia of Ashbya gossypii. They found that mycelia did indeed behave as fractals, and calculated a fractal dimension, D = 1.94. Such a high value for D indicates a mature mycelium whose centre has been almost homogenously filled by branching hyphae. For the edge of mycelia they calculated D = 1.45. Thus, as a mycelium develops its fractal dimension converges towards 2 when the whole mycelium is considered and towards 1.5 when only the edge is considered. Ritz & Crawford [24] and Jones et al. [25] corroborated these findings.

Matsuura & Miyazima [26] used a different form of fractal analysis to quantify the ‘roughness’ of the edges of mycelia grown at different temperatures and on different media. Unfavourable conditions (e.g. low temperature, low nutrient concentration or stiff media) were found to produce rough edges corresponding to a lower branching frequency.

However, these applications are only able to quantify branching, and offer no insight into the mechanisms behind generating the patterns. Mycelia growing on flat, uniform surfaces almost invariably grow into circular colonies. For this reason, much work has been invested in designing models that can produce this morphology. Such models are more complex and have developed alongside the development of computers and computer programming.

Cohen [27] pioneered computer analysis by devising a program that was able to generate a range of branching patterns found in the natural world from a set of simple growth and branching rules. In his model growth occurred only at the tip and branching was only initiated behind the tip. Thus, it is directly applicable to mycelial growth of most fungi. Growth proceeded with respect to local density fields, calculated with reference to 36 sample points spaced 10° apart around the circumference of a circle centred on a growing tip, and quantifying the pattern density in the locality. Local density minima were key parameters that directed growth into unoccupied space. Branching probability was also made a function of local density minima. A random trial incorporated into the program decided, independently, if branch initiation should occur. Finally, the direction of both growth and branching were dependent on a ‘persistence factor’ that quantified to what degree they continued in the same direction in spite of gradients in the density field. When these rules were iterated, with the persistence factor for growth nullified (i.e. growing tips proceeded in a straight line), a circular branching pattern emerged.

Hutchinson et al. [28] developed this work further by applying it directly to mycelial colonies of Mucor hiemalis. They determined the variability of tip growth rate, distance between branches, and branching angle throughout the colonies. They were then able to fit these data to known distribution curves with defined probability density functions. Tip growth rate was found to follow a half-normal distribution, distance between branches followed a gamma distribution, and branching angle followed a normal distribution. This formed the basis for a model in which values for the three specified variables were generated from the respective probability density functions over a series of time intervals.

Although this model includes no allowances for tropic interactions between hyphae, it generates a circular mycelium. This came as something of a surprise because it had been assumed that the fact that growing hyphae actively avoid each other played a role in determining spatial organisation in mycelia. Such avoidance strategies, known as negative autotropism, have been observed experimentally in several independent studies [28-31]. Indermitte et al. [32] constructed a similar random growth model that also generated circular colonies. They further tested how colony morphology and growth efficiency (the ratio of biomass used to area of medium covered) were affected by tropisms. Their results indicated that tropism increased the growth efficiency, but a circular colony was generated in all cases.

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Tropisms and Hyphal Interactions

While it is significant that a purely stochastic approach can generate realistic colony morphology, this does not mean that tropism and hyphal interactions are irrelevant to modelling hyphal growth. Hyphae certainly do use autotropic behaviour (positive and negative) to control spatial organisation in particular regions of the mycelium. Where the hyphal density is high, as in fungal tissues, interactions are inevitable.

Edelstein [33] was the first to consider such interactions. Her approach differed in that instead of considering the mechanisms of growth in discrete hyphae, she looked at the mycelium as a whole. She assumed that growth occurred at a constant rate throughout the mycelium. This she set at μ_max and so also limited her model to a tangential abstraction of the growth curve. Her model owes something to Cohen [27] in that it considers the density of the mycelium with respect to space as a key feature. Two density parameters were defined:

p = p(x,t) the hyphal density per unit area

n = n(x,t) the tip density per unit area

and the model was then based on two partial differential equations:

where δ = δ(p) is the rate of hyphal death, σ = σ(n,p) is the rate of tip creation, and can be considered as tip flux.

Edelstein [33] also defined, in mathematical terms, all the hyphal interactions that affect the parameter n, and which are contained in the function σ. These included both branching mechanisms, as well as tip death and tip-tip and tip-hypha anastomoses. 

She then used phase plane analysis to determine which of various combinations of hyphal interactions, expressed mathematically in the function σ, had bounded non-negative solutions of equations (17) and (18). These represented combinations that yielded spatially propagating colonies. Her results showed that when δ = 0, only colonies that branched dichotomously and formed tip-hypha anastomosis could propagate. However, when δ > 0, most combinations of hyphal interactions yielded propagating colonies. Thus, hyphal death was an important feature of mycelial growth.

Ferret et al. [34] adopted a similar approach to Edelstein [33], using two partial differential equations that considered parameters defined in dimensions of density. However, they sought to apply the model macroscopically by adjusting Ē (mean tip extension rate ) with respect to X. This was done by collecting data that quantified how E varied when two hyphae came into close proximity with each other. This effect was incorporated into the differential equation concerned with the rate of change of biomass (proportional to hyphal-) density so that it was more likely to be applied in regions where the density of biomass was high, and had a greater effect on regions where the density of tips was high. Thus, Ē decreased as the mycelium grew and biomass increased.

This approach provides an alternative to incorporating hyphal death into the model that has the advantage of also affecting Ē, thus limiting growth in a manner typical of batch culture. This limiting mechanism is an approximation however, as the true limiting parameter for biomass evolution is specific growth rate (see equation (5)). In reality, μ can decrease as hyphae die, whereas the mean extension rate of the remaining hyphae can be maintained despite overall biomass evolution slowing.

The rate of change of tip density was determined with respect to a diffusion coefficient that was dependent on temperature and medium conditions. This offered a means of modelling the affect of changing environmental conditions on macroscopic growth kinetics when only microscopic parameters had been measured. The value of this parameter was estimated from experimental data and gave comparable results when used in the mathematical model. This was validated further by the agreement between experimental and simulated data on the width of the peripheral growth zone, even though this parameter was not included in the initial design of the model.

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Opinions

Applying a mathematical description to fungal growth serves two main purposes. Firstly, it is able to validate the biological knowledge on which the mathematical descriptions are based. Secondly, it is able to quantify certain growth parameters. This feature is particularly relevant to optimisation strategies when the fungus is used in an industrial or biotechnological context.

This review has focussed on the vegetative phase of mycelial growth. Yet, this is in many respects the least interesting growth form. It is the ‘default’ growth mode of the fungal cell and any changes that occur in it are imposed by external forces (nutrients, environmental conditions, etc.). Of much greater biological interest is the way in which the ‘default’ growth mode might be altered by internal controls to generate the numerous differentiated cells that hyphae can produce and the native interactions between hyphae that cause them to co-operate and co-ordinate in the morphogenesis of fungal tissues. Although some attempt has been made to extend the vesicle supply centre model of apical growth [13] into 2-dimensional and 3-dimensional models of apical growth and differentiation [35, 36], we are not aware of any kinetic analysis of fungal tissue morphogenesis. It is certain, though, that the equation E = μG (equation 1) is fundamental to understanding branching kinetics and that the ratio E/μ can tell us a lot about mycelial morphology, as it relates to the hyphal growth unit length, G, which can also be expressed as a volume [37]. Observation has shown that temperature increases do not affect G in some species. However, paramorphogens have been identified that do alter this ratio and hence G and morphology [38].

In our vector based Neighbour-Sensing mathematical model, which is introduced in the “Computer Models” section of this website, the inclusion of certain tropism vectors is also able to alter this ratio by affecting the parameter E and results in a striking array of different morphologies, some of which seem to suggest a morphogenetic process that goes beyond mycelial growth and towards differentiated tissues.

The Neighbour-Sensing model is able to simulate mycelial growth and branching in three dimensions. Furthermore, it is able to implement negative autotropism and gravitropism. Experiments have shown that by adjusting key parameters before or during the simulation it is capable of producing a wide range of morphologies, ranging from simple circular colonies to cup-shaped structures resembling fruit bodies. The model and a discussion of these results can be found by following this link (Computer Models).

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References

1.         Adams, H.L. and C.R. Thomas, The use of image analysis for morphological measurements on filamentous organisms. Biotechnology and Bioengineering, 1988. 32: p. 707-712.

2.         Money, N.P., The pulse of the machine - reevaluating tip-growth methodology. New Phytologist, 2001. 151: p. 553-555.

3.         Jackson, S.L., Do hyphae pulse as they grow? New Phytologist, 2001. 151: p. 556-560.

4.         Hammad, F., R. Watling, and D. Moore, Artifacts in video measurements cause growth curves to advance in steps. Journal of Microbiological Methods, 1993. 18: p. 113-117.

5.         Acuña, G., et al., A neural network estimator for total biomass of filamentous fungi growing on two dimensional solid substrate. Biotechnology Techniques, 1998. 17(7): p. 515-519.

6.         Desgranges, C., et al., Biomass estimation in solid state fermentation. I. Manual biochemical methods. Applied Microbiology and Biotechnology, 1991. 35(2): p. 200-205.

7.         Desgranges, C., et al., Biomass estimation in solid state fermentation. II. On-line measurements. Applied Microbiology and Biotechnology, 1991. 35(2): p. 206-209.

8.         Katz, D., D. Goldstein, and R.F. Rosenberger, Model for branch initiation in Aspergillus nidulans based on measurement of growth parameters. Journal of Bacteriology, 1972. 109: p. 1097-1100.

9.         Trinci, A.P.J., A study of the kinetics of hyphal extension and branch initiation of fungal mycelia. Journal of General Microbiology, 1974. 81: p. 225-236.

10.       Steele, G.C. and A.P.J. Trinci, The extension zone of mycelial hyphae. New Phytologist, 1975. 75: p. 583-587.

11.       Trinci, A.P.J., Influence of the peripheral growth zone on the radial growth rate of fungal colonies. Journal of General Microbiology, 1971. 67: p. 325-344.

12.       Pirt, S.J., A kinetic study of the mode of growth surface colonies of bacteria and fungi. Journal of General Microbiology, 1967. 47: p. 181-197.

13.       Bartnicki-Garcia, S., Fundamental aspects of hyphal morphogenesis. Symposia of the Society of General Microbiology, 1973. 23: p. 245-267.

14.       Monod, J., The growth of bacterial cultures. Annual Review of Microbiology, 1949. 3: p. 371-394.

15.       Prosser, J.I. and A.P.J. Trinci, A model for hyphal growth and branching. Journal of General Microbiology, 1979. 111: p. 153-164.

16.       Yang, H., et al., Mathematical model for apical growth, septation, and branching of mycelial microorganisms. Biotechnology and Bioengineering, 1992. 39: p. 49-58.

17.       Koch, K.L., The kinetics of mycelial growth. Journal of General Microbiology, 1975. 89: p. 209-216.

18.       Viniegra-Gonzalez, G., et al., Symmetric branching model for the kinetics of mycelial growth. Biotechnology and Bioengineering, 1993. 42: p. 1-10.

19.       Ikasari, L. and D.A. Mitchell, Two-phase model of the kinetics of growth of Rhizopus oligosporus in membrane culture. Biotechnology and Bioengineering, 2000. 68(6): p. 619-627.

20.       Leopold, L.B., Trees and streams: the efficiency of branching patterns. Journal of Theoretical Biology, 1971. 31: p. 339-354.

21.       Horton, R.E., Erosional development of streams and their drainage basins: hydrophysical approach to quantitative morphometry. Bulletin of the Geographical Society of America, 1945. 56: p. 275-370.

22.       Gull, K., Mycelium branch patterns of Thamnidium elegans. Transactions of the British Mycological Society, 1975. 64: p. 321-324.

23.       Obert, M., P. Pfeifer, and M. Sernetz, Microbial growth patterns described by fractal geometry. Journal of Bacteriology, 1990. 172: p. 1180-1185.

24.       Ritz, K. and J. Crawford, Quantification of the fractal nature of colonies of Trichoderma viride. Mycological Research, 1990. 94: p. 1138-1152.

25.       Jones, C.L., G.T. Lonergan, and D.E. Mainwaring, A rapid method for the fractal analysis of fungal colony growth using image processing. Binary, 1993. 5: p. 171-180.

26.       Matsuura, S. and S. Miyazima, Colony of the fungus Aspergillus oryzae and self-affine fractal geometry of growth fronts. Fractals, 1993. 1: p. 11-19.

27.       Cohen, D., Computer simulation of biological pattern generation processes. Nature, 1967. 216: p. 246-248.

28.       Hutchinson, S.A., et al., Control of hyphal orientation in colonies of Mucor hiemalis. Transactions of the British Mycological Society, 1980. 75: p. 177-191.

29.       Robinson, P.M., Chemotropism in fungi. Transactions of the British Mycological Society, 1973. 61: p. 303-313.

30.       Robinson, P.M., Autotropism in fungal spores and hyphae. Botanical Review, 1973. 39: p. 367-384.

31.       Trinci, A.P.J., et al., Spiral growth of mycelial and reproductive hyphae. Transactions of the British Mycological Society, 1979. 73: p. 283-292.

32.       Indermitte, C., T.M. Liebling, and H. Clémençon, Culture analysis and external interaction models of mycelial growth. Bulletin of Mathematical Biology, 1994. 56(4): p. 633-664.

33.       Edelstein, L., The propagation of fungal colonies: a model for tissue growth. Journal of Theoretical Biology, 1982. 98: p. 679-701.

34.       Ferret, E., et al., Macroscopic growth of filamentous fungi on solid substrate explained by a microscopic approach. Biotechnology and Bioengineering, 1999. 65(5): p. 512-522.

35.       Bartnicki-Garcia, S., F. Hergert, and G. Gierz, Computer simulation of fungal morphogenesis and the mathematical basis of hyphal (tip) growth. Protoplasma, 1989. 153: p. 46-57.

36.       Gierz, G. and S. Bartnicki-Garcia, A three-dimensional model of fungal morphogenesis based on the vesicle supply center concept. Journal of Theoretical Biology, 2001. 208(2): p. 151-164.

37.       Trinci, A.P.J., Regulation of hyphal branching and hyphal orientation, in Ecology and Physiology of the Fungal Mycelium, D.H. Jennings and A.D.M. Rayner, Editors. 1984, Cambridge University Press: Cambridge, UK. p. 23-52.

38.       Trinci, A.P.J., M.G. Wiebe, and G.D. Robson, The mycelium as an integrated entity (The Mycota, vol. 1), in Growth, Differentiation and Sexuality, J.G.H. Wessels and F. Meinhardt, Editors. 1994, Springer-Verlag: Berlin, Heidelberg. p. 175-193.

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