Cellular automata simulations of fungal growth on solid substrates

Bu~tech Adv Vol 11 pp 621-633. I tjtj3 Printed m Great Britain..MI Rights Reserved

0,.'r?.4-9750 , t~3 $24 t~.'l .r~. 1693 Pergamon Pre:,.z. Ltd

CELLULAR A U T O M A T A S I M U L A T I O N S O F F U N G A L G R O W T H O N SOLID SUBSTRATES JOSEPH A. LASZLO* and ROBERT W. SILMANt *Food Physical Chemistmt, ~'Ferrnentation Btochemistry Research, National Center for Agricultural UHlization Research, USDA-AR5 1815 N. Universtty St., Peoria,' IL 61604, U.S .4.

ABSTRACT Gro~th of filamentous fungi on the surface of cereal grains is a critical aspect of solid substrate fermentation (SSF). Numerous mathematical models have been developed to describe various aspects of fungal growth in SSF. These models consider hyphal geometD, and nutrient availability as determinants of colony morphology, and fungal physiological state. This work describes the use of cellular automata (CA) as an alternative method of modeling fungal growth. CA models reliant on a v e D' limited set of rules or "knowledge base" display a rich array of behaviors that mimic fungal growth. By incorporating probablistic growth rules into CA models, colony characteristics such as biomass accumulation rate, colony radial growth rate, mycelial density and fungal differentiation are readily generated. Key words: modeling: fungal growth; fermentation (solid-state). INTRODUCTION "To be or not to be ...", a question oft asked by anguished humans may also describe a central

issue of solid surface growth of fungi. Whether a fungal colony is growing on an agar plate, corn kernel, or shower curtain, the appearance and growth rate of the colony is determined largely by the availability of nutrients from the growth medium. Inversion of the anthropomorphic sense that fungal hyphae will extend into areas with available nutrients leads to a cellular automaton view of growth: "If I am unoccupied, have nutrients, and have fungal hyphae near me, I will (perhaps) become colonized." Thus, cellular automata (CA) models of fungal growth differ in perspective from traditional mathematical formulations. Available Matheozatical Models o f Fungal

ColonyGrowth

The gross morphological growth characteristics of filamentous fungi on solid surfaces, i.e., biomass accumulation rate, colony radial growth rate, hyphal tip growth rate, branching pattern, etc., have been studied and modeled in some detail. The observations by Trinci [1969; 1971] of colonies of Aspergillus and other species provide much of the experimental underpinnings of models developed by Trinci and many others. Prosser [1982] provides a

62l

n22

I. *. LASZLOand R. ~A'.SILMAN

comprehenc, ive o~.er'~ie,.,. ()f these "earlV' models. ]qle nlost readib, ob,,er',ed attribute of fungal grov, th on surfaces is that the colon} is circular. Thus. colon,, radial gro,>.th rate is a t~.idel) measured and modeled characteristic...M'ter an initial lag period and brief exp()nential gro~ tit phase, colony radius increases linearly ~ ith time [Trinci, 1071: Yarman & Ki)ak, 1900]. The linear radial growth rate of a fungal colon) results from limiting nutrient s u p p l ) o r available space constraints [Laukevics et al., 1985]. Hsphae v, ithin the interior t)f the colon} do not grm~. Onl.', hyphae on the colony periphery contribute to radial gro',,.th. For the ~ante reason, colon,, bioma,,,, does not increase e,~ponenttally with time. For simplicity's bake, fung.al bionta,;,; i,, usuall) as,~umed to be uniformly distributed throughout the colony. By further assuming a constant cohmy height (h). colon) biomass (m) is given b,. [Pirt 19751 m

=

,,rhp, 2

[1]

v. herep is biomass densit.,, and r i, c()lon.~ radius. Therefore the fungal bionlasb accumulati()n rate t d m . , d t ) is related to the colony radius, radial grov. th rate ( d r , ' d t ) and bi()nta% densit) by Equation 2. dm..dt

= 27rHtp,lridt

[2]

\Vith a constant radial gro~th rate (K,-), total colony biomass a'; a function of time i,; given by Equation 3. m = ~ph(Krt

) 2 + 2,rphK.~rl)t + m o

[3]

m 0 and r u are colt)n} mass and radiub, re,,pectivel.,., after the initial exponential gro,.~.th phase and t i~ time from the ,;tart of linear grov, th. Equation 3 reflects the po,.~,er-law kineticg of fungal grov.th. Georeiou and Shuler [10St) l ha~e provided a detailed model of tungal biomass and col¢)n~ radial gro~th rates, including cell differentiation, based on substrate levels and Monod-t}pe kinetic,. Dif'.erentiation i,; induced by local depletion of nutrients. .-M a slight[', finer coh)n} morpholog} resolution, indMdual hyphal growth direction and branching patterns ha,.e been studied [Hutchinson et al.. 1980; Kotov & Reshemivov, 1990: Ritz & Crav, ford. 1091): Yang et al., 1992a,b]. Major trends suggested by these studieb include: • H)pha[ tips tend to gross at con,,tant rate and btraight ahead Izero degree of curvature from parent body), x~ilh Gaussian distributions for both measures [Hutchinson et al., 1980], ° Branching is nornml t i.e., at uq):) to the parent h vpha and has a Gaussian distributitm [Yang et :.d.. 1992a]. • F_,~tending tip,,, a~()id other h}phae [Hutchin,,on et al., 1980]. These indixidual hyphae characterisicb are sufficient to produce the ~+hole-colony grox~th patterns de,,cribed above. The stochastic nature of gro~th and branching is a critical aspect of models based on thebe characteri,~tic,, [Hutchinson et al., 1980; Kotov & Reshetnikov. 1990: Yang et al., 1992b].

CELLU1-AR AUTOMATA SIML~ATIONS

62£

Celhdar Automata

The concept of cellular automata is credited to John yon Neumann, the father of moderr digital computing [,.on Neumann, 1066]. CA are massivelly parallel, universal calculators They are used to model as diverse phenomena as the alignment of atomic dipole moments ir solids and the complexities of the immune system [Kaneko & Akutsu. 1986; Stauffer, 1991] The t'orld of an automaton is local, discrete, and built from a set of logic rules, The com. plexity and p o t e r of CA arise from the local interactions of individuals in a lattice-like array each acting on the same set of rules, exolving over time. The logic rules defined by the programmer constitute the CA model. The following nuts-and-bolts description of CA operation and logic applies to the 2dimensional models developed herein. Other CA types may and do use alternative geometries (1- and 3-dimensional arrays are common) operating with different constraint: (such as a relaxed definition of Iocalit) ). The fundamental building block of CA is the neighborhood. Individual sites (synonyms units, atoms, cells) are arranged in a square planar lattice. Each site is surrounded b,, eigh: neighbors (Figure 1). A site interacts directly only with members of its neighborhood. ThL' interaction consists of the exchange of information, one bit (0/1) of knowledge at a time Thus. using the terminolo~, defined in Figure 1, site Self can receive eight bits of knowledge simultaneousl,,, about its environment (neighborhood), one from each neighbor. Sites NE and SW, for instance, cannot interact directly. H o t e v e r , sites NE and SW can communicat~ by transferring information through other sites, such as via Self which is in the neighbor. hoods of both NE and SW. Bits of knm~ledge are the coinage of cellular automata.

NW

N

NE

W

Self

E

SW

S

SE

i

~

',

Figure 1. Depiction of a CA neighborhood with the central site (Self) surrounded by eight sites given compass designations (N = north, etc.). The information gleaned from neighboring sites may be acted upon (as determined by the logic rules) or stored in the site's memory, along with other information about the status o the site. A site has eight bits of memoD,, numbered 0 (lowest bit) to 7 (highest bit) by con vention. The lot' bit is the communication channel between neighbors. With eight bits o memory., or self-knowledge, each site has the potential of being in one of 256 different state~ (2 8 = states 0-255). The logic rules define the net, state of a site based on its current stat~ and the eight bits of information from the neighborhood. This leads to a great many possi bilities (2 ~° -- 65536). The state of all sites in the lattice are updated simultaneously ii discrete time units.

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I A. LASZLOand R. W. SILMAN

The logic rules defining the behavior of a cellular automaton can be as simple or complex as desired. Conway's "Game of Life" CA is a good example [Gardner, 1970, 1971]. With the low bit of a site determining whether it is dead or alive, the rules of Life are: 1.

If exactly two neighbor sites are alive, then Self does not change.

2.

If 3 neighbors are alive, then Self is alive.

3.

With all other combinations of dead/alive neighbors, Self is dead.

This seemingly trivial logic can produce incredibly complex and unpredictable behavior [Wolfram, 19841.

CA Logic atut Fungal Colony Growth Characteristics The CA models described below attempt to simulate colon,,' radial growth rates, biomass density (in a 2-dimensional plane), branching and differentiation patterns of filamentous fungi. Growth into unoccupied areas is dependent on nutrient availablity, which is affected by the initial amount of CA site "nutrient" and number of neighborhood occupied fungal sites. Similarly, fungal differentiation is driven by nutrient depletion, ~,hich is dependent on neighborhood occupancy and time..All growth and differentiation steps are given a stochastic nature by incorporating various probabilities into the CA rules. (For example, a site has a 50% probability of being occupied if it has tx~t~occupied neighbors.) This approach provides a simple yet flexible set of rules by which a diverse range of fungal colony growth traits can be simulated. MATERIALS AND METHODS

So]iware and Hardware CA models were executed using the CA LAB softaare package from Autodesk, Inc. (Sausalito, California), which runs on PC-compatible computers. Models and analysis programs were coded in Turbo Pascal (Version b. Borland International, Scotts Valley, California). The mention of firm names or trade products does not imply that they are endorsed or recommended by the U.S. Department of Agriculture over other firms or similar products not mentioned.

Analysis The CA LAB software creates a 320 x 200 (horizontal x vertical) lattice of sites. All fungal colonies developed from a single "spore" seeded at the center of the lattice. For the determination of colony growth rates, simulations were permitted to develop until the colony perimeter reached the vertical (north/south) edge. Colony radius was calculated by taking the mean of the distances (in integer units) from the colony seed site to the outermost occupied site in the horizontal, vertical and diagonal directions, a total of eight measures of colony radius. Occupied sites were assigned unit mass values. Colony mass density (number of occupied sites divided by total number of sites) was measured on colonies grown to confluence (i.e., covered the entire screen). Except for very early simulation times (less than 25 generations), colony characteristics for a given CA model instance varied little across multiple runs, therefore reported values are for a single run. The use of the term "generations" in the context of CA unit time steps should not be confused with the concept of doubling time in its conventional sense.

CELLULAR ~u-rOMATA SIMULATIONS

6,25

Model Nomenclantre CA models are designated and distinguished as follows: models with different logic rules are assigned unique integer names (i.e., Model 1, Model 2, etc.); models operating under the same logic rules but different probability assignments have letters appended to their model number (e.g., Model 2b). RESULTS The eight bits that constitute the state of a site in the fungal growth and development CA models are assigned the following significance: • bit 0: used to show presence of an occupied site (value = 1). This bit is seen by sites in the neighborhood. • bits 1-3: assigned various functions, such as keeping track of growth stages (up to eight), depending on the model, and in some cases are unused. • bits 4-6: random bit generators. At each generation (i.e., discrete time step) each bit is randomly assigned a zero or one value. Application of Boolean operators to combinations of these bits provides the set of probabilities that can be used by the logic rules. &side from the obvious probabilities of 0% and 100%. (no random bits required), the available probabilities are 12.5% (bit 4 AND bit 5 AND bit 6), 25%, 50,c'c, 75% and 87.5 ,~. • bit 7: used to keep track of whether the site is occupied (a fungal site). For all models discussed herein, the value of this bit is identical to the lowest bit (bit 0). This redundancy is maintained to pro'.ide for future model flexibility.

Vegetatit'e Growth Models Maximum growth rates. The square planar world of a CA simulation is somewhat problematic when applied to a circular phenomenon such as colony radial growth. Square sites must be converted to round sites mathematically to give them a "radius". With the radius of a single site, R, defined as V"l/rr (approx. 0.56419), the site will have unit area (i.e.. 7rR 2 = 1). For the CA models described herein, unoccupied sites may become occupied only after a neighborhood site is occupied first - growth along contiguous sites is required. The maximum growth rate achievable under such a constraint would be obtained by having a site become occupied whenever a neighbor is occupied. Under these conditions, colony mass accumulates at a simple geometric rate (m = 4(t-0.5)2), forming a solid square block, and colony radius increases linearly at a rate of 2R (approx. 1.13). (A hypothetical, radialgeometry CA with circular sites of unit diameter (r = 0.5) would have a radial growth rate of 1.0.) These conditions do not define a compelling fungal growth model, but, rather, set the upper boundary for the growth parameters. Model 1. The logic of Model 1 is quite simple. If a site in the neighborhood is occupied then Self (refer to Fig. 1) will become occupied with a probability defined by the number of neighbors. There are no other conditions or constraints. For Model la these probabilities are: 1 neighbor, 12.5,t'Yc; 2 neighbors, 25,~; 3 neighbors, 50%: 4-8 neighbors, 0%, following the intuitive sense that having more occupied neighbors (up to a limit) should result in a higher probability of becoming occupied. Figure 2 shows the fungal colon)' produced using Model la probabilities. The colon)' is roughly circular and about two-thirds of the sites are occupied. Model la colony mass and radius as a function of time are shown in Fig. 3. Colony radius increases linearly with time. In fact, this is a characteristic common to all models described herein. A summary, of growth traits (colony biomass density and radial extension rate) for all vegetative-growth-only models is given in Table I.

62o

i. AmLASZLO,rod R. W. SILMAN

Table I. Summary of colon)' characteristics generated by CA vegetative growth models. Model Grm~ th Probabilitiest

Biomass Density (C~c)

Radial G r o ~ th Rate

la

L: 12.5: 2:25 3: 50; 4-8: ()

~8.8

11.57

1b

l: 50; 2 : 2 5 3: 12.5: 4-s: ~)

58.1

1.02

lc

I:

12.5" 2 - , 8 : 0

38.1

1/.38

2a

1: 12.5; 2-S: o

35.4

1/.35

3a

I: 51): 2 - 8 : 0

37.0

0.20

3b

1: 5fl: ."-. . . ":, 3: 12.5: 4 - 8 : 0

50.4

0.23

CA Model

] Format: occupied neighbors: prnhabilip, (G-I. 100

16000

80 12000 /I ~'

O'

60

O O

o r',,"

•

O 2~

•

8000

,O 0 0 CO

40

O C)

• O /

•

•

•

O ,o

• .o

20 JO • O" ,•

4OOO

•

,m

•' • n

m.m 'm

0 I •'•'ia'l 20 40

I

60

80

c O

I

I

I

100

120

140

0 60

Generation (Time) Figure 3. Model la colon)' radius and b i o m a s s as a function of time.

CELLULAR .,~LJOM,gTA SIMULATIONS

Figure 2. Model la colony after 160 generations. ~t

Figure 4. Model lc colony after 250 generations.

~2T

r,2,~

I.A. LASZLOand R. W. SILb.I.A~N

Model lb inverts the probability logic of Model la, i.e.. the fewer the occupied neighbor sites, the higher the probability of becoming occupied. Growth probabilities for Model lb are: 1 neighbor, 50c~c: 2 neighbors, 25%; 3 neighbors, 12.5,~; 4-8 neighbors. 0,tTe. These probabilities produce a colony that grows more rapidly and is less dense than Model la (Table I). Model lc takes this logic to its extreme: 1 neighbor. 12.5q2 probability, 2-8 neighbors. 0cI' probabilit), of being occupied. This produces a slow growing, sparse colony (Table I. Fig. 4). Model 2. Model 2 adds a slight t~ist to the logic of Model 1, nameb, unoccupied sites are allov, ed to "age" probablistically when neighborhood sites are occupied. AJ'ter a fe~ aging steps unoecupied sites become unavailable for growth. Aging of unoccupied sites could be considered the result of nutrient depletion (diffusion to occupied sites) or accumulation of inhibitor3. substances. Model 2a uses the same grov, th probabilitites as Model lc (I neighbor, 12.5cL all others 0%). but unoccupied sites also sequence through four age-stages, after which they are inactive. The neighbor-dependent aging probabilities are: l, 50.~: 2, 75%: 38. 100c~e for each stage (the probabilities could be stage-dependent as well, if desired). The colony produced b.', Model 2a gro~s slightly more slov, ly and less densely than Model lc ITable I). It also creates noticeable ~oid areas in which potential growth sites are surrounded b.~ inactive, unoccupied sites (Fig. 51). The size and/or frequency of void areas is alterable ,.ia changes to the gro~th and aging probabilities in the e,~pected manner - slower growth and faster aging produce m~re ~oMs I not sho~n). Model 3. Models presented so far have allo~ed smlultaneous growth into multiple empty sites, an act of dubious biological significance. To avoid this potential shortcoming, Model 3 uses a technique adapted from particle physics CA models [Toffoli & Margolus, 1987]. Unoccupied sites c).cle s).nchronously through eight stages (using hits 1-3). Each stage is associated with a specific gro~th direction and the site can only become occupied ~hen the stage growth direction coincides with an occupied neighbor. The sequence in which the growth directions are staged impacts colony appearance. Clockwise and counterclockwise protocols produce growing ends that tend to curl upon themselves (not shown). Therefore, a staggered sequence of gro~th directions ~as employed (i.e., sequentially N, SW, E, NW, S, NE, W, SE). ('Spiral' growth of h.~phae on solid media is obsep,'ed for many species [Trinci, 1984].) Model 3a uses a simple neighbor-dependent probability scheme: one occupied neighbor, 50~ probabilit.~ of being occupied: 2-8 neighbors, 0% probability. Implicit in these gro~',th probabilitites is that all branching events are dichotomous. Model 3a produces an extremel,, slo~ growing colon)', ~ith a biomass density slightly higher than the theoretical lower limit for Model 3 of one-in-three sites filled (Fig. 6, Table I). Model 3b uses the same growth probabilities as Model lb. Colony biomass density is similar for the two models, but Model 3b',~ gro~th is substantial[',, slo~er (Table l'J.

Growth with Differemiation ,'~L~xh'ls. Model 4. Whereas the logic of vegetauve growth CA models logic resides entire b in the unoccupied sites, Model 4 adds a developmental logic to occupied fungal sites. Model 4 defines three stages, ~egetative grov, th, differentiating mycelia, and conidiation/spore formation, using bits 1-3. (There could be as man) as eight different stages, but this model assigns four. three and one bit-states, respectively, to the three growth stages.) Reasoning from the observation that fungal differentiation results from nutrient depletion [Georgiou & Shuler, 1986], the probabilities for transition from one stage to the next were defined to be dependent on the number of occupied neighbor sites. Model 4a uses the vegetati~,e growth conditions and probabilities of Model 2a. Funga[ sites developed with neighbor-dependent probabilities for all stages of 0Q for 0-2 neighbors and 12.5c~-. for 3-8 neighbors produces a colony with an appearance (,Fig. 7. see color plate) reminiscent of the concentric ring structure of natural fungal colonies. Obviously. numerous combinations of vegetative growth models and differentiation probabilities are possible, permitting great lattitude in fungal growth simulation.

CELLULAR ALrTOMATA SIMLrLA11ONS

629

Figure 5. Model 2a colony after 275 generations.

=.

Figure 6. Model 3a colony after 500 generations.

630

I A. L A S Z L O a n d

R. I,~'. SILM~M"~

DISCUSSION

All of the presented CA models produce a colony with approximately circular shape and whose radius grows linearly ~ith time, fulfilling basic requirements of any fungal gro~th simulation. The linear growth rates of the CA colonies can be related directb to growth parameters defined by Trinci [1971]. Trinci's colony growth zone width ("9 corresponds to a CA site length of 2R. The radial growth rates (Kr) given in Table I are con',ertible to specific growth rates (a =K,-/2R. ~ith units of time t ) and mean doubling times (Td = l n ( 2 ) / a ) . which should be directly compatible with experimental observations after appropriate scaling of CA generation time. The CA models confirm Trinci's contention that colony radial gro~th rate is an inadequate measure when comparing different growth conditions [Trinci, 1969, 1'-)71]. For instance. Model 3b has a radial growth rate 15% greater than Mode[ 3a, but 52c~ greater biomass density (Table I), resulting in a much higher biomass accumulation rate for Model 3b. Trinci [1969] observed that h.~phal density v, as a function of growth medium depth. This trait is readily accommodated in the CA ,,egetati,,e growth models b.,, adjusting either gro~th probabilities (Model 1) or ~ite aging rates (Model 2). Awa.~ from the gro~ing periphery, the CA colonies had uniform mas~ densities. While constant mass density is commonly assumed for modeling purpose~, the ~alidit). of the,, assumption deserves closer experimental scrutim. CA models can be ale',eloped that ha~e a time-dependent mass densit',, if this ~ere a desired characteristic. The model-generated low biomass density colonies gre~ in a manner consistent with the concepts of negative autotropism (h~phal tips tend to gro~ a~,av from each other) and negative chemotropism (aging of unoccupied sites corresponding to the accumulation of to\ic factors elaborated from nearby occupied sites). The models of Georgiou and Shuler [1980]. Yang et al. [1992a, 1992b]. and Kotov and Reshemikov [1990] do not accommodate these "crowding" effects. Although hyphal growth direction and branching characteristics ~ere not explicitly treated in the present ~ork, the CA growth patterns can be examined in a manner similar to the other growth traits. CA site length (2R) can be scaled to correspond with an experimentally determined average distance bet'.~een branching points on h.~phae. The use of cellular automata to model fungal grov, th differs radically from traditional approaches based on mathematical formulations. CA models are driven entireb by empirically derived rules of logic. Apart from the computational simplicity of CA, the plasticity of the system is unrivaled by traditional methods, allowing rapid development, testing and modification of models. The presented models do not represent an exhaustive examination of the potential CA to model fungal growth on solid substrates. [n particular, the selected examples of differentiating mycelia (Model 41 ~ere chosen largely because of their aesthetic appeal rather than based on experimental evidence. Other aspects of solid-substrate fermentation, such as oxygen and nutrient diffusion and consumption, heat generation, and growth in highly constrained ~olumes. can be incorporated into CA models, to provide insight for real-world problems, in each instance, the need to await appropriate data generation for model testing is a severe limitation not restricted to CA. Models, whether logic rules or mathematical formulations, provide a means of concisely defining complex behavior and may prove to be as valuable in the initial formulation of testable hypotheses as they are describing fully delineated phenomenon.

CELLULAR ALITOMATA SIMULATIONS

631

Figure 7. Model 4a colony after 275 generations. Black represents vegetative growth, yellow sites are differentiating mycelia, and conidia are red.

CELLULAR A L ~ O M A T A SIMLILAT[ONS

e,33

REFERENCES

Gardner, M. (1970) The fantastic combinations of John Conway's ne`'~ solitaire game "Life'. ScL Amer. 223 (4). 120-123. Gardner, M. (1971) On cellular automata, self-reproduction, the Garden of Eden and the game "Life'. ScL ,4mer. 224, 112-117. Georgiou, G., and Shuler, M. L. (1980) A computer model for the growth and differentiation of a funga[ colon}' on solid substrate. Biotech. Bioeng. 28, 41)5-416. Hutchinson, S. A., Sharma. P., Clarke, K. R., and MacDonald. I. (1980) Control of h}phal orientation in colonies of Mucor hiemali~. Trans. Br. Mycol. Soc. 75. 177-191. Kotov, V., and Reshetnikov, S. V. (1940) A stochastic model for early m}celial gro~th. M)'col. Res. 5, 577-586. Kunihiko, K., and Akutsu. Y. (1086) Phase transitions in two-dimensitmal stochastic cellular automata. J. Phys. A: AhTth. Gen. 19, 440-446. Lauke`"ics. J. J., Apsite, A. F., Viesturs, U. S., and Tengerdy, R. P. (1985) Steric hinderance of growth of filamentous fungi in solid substrate fermentation of ,,,,heat straw. Biotech. Bioeng. 27, 1687-1691. Pitt, S. J. (1075) Principles ol Microbe and Cell Cultivation, Black,veil Scientific Publ., Oxford, pp. 234-242. Prosser, J. I. (1982) Gro,,vth of fungi. In: Microbial Population Dvmmffc.~ (Bazin, M. J., ed.), CRC Press, Boca Raton, Florida, pp. 125-1o0. Ritz, K., and Cra`'`'ford, J. (1990) Quantification of the fractal nature of colonies of Trichoderma uiride. Mycol. Res. 94, 1138- 1152. Stauffer, D. (1991)Computer simulations of cellular automata. Z Phys. A: Math. Gen. 24. 909-927. Toffoli, T., and Margolus. N. (1987) Cellular ,4utomata 3h~chine.~ - A New Environment .l~r Modelhzg, MIT Press, Cambridge, MA, pp. 119-138. Trinci, A. P. J. (1969) A kinetic stud~ of the gro~th of,qpser~illus nidulans and other fungi. J. Gen. MicrobioL 57, 11-24. Trinci, A. P. J. (1971) Influence of the width of the peripheral growth zone on the radial growth rate of fungal colonies on solid media. J. Gen. MicrobioL 07, 325-344. Trinci, A. P. J. (1984) Regulation of hyphal branching and hyphal orientation, In: The Ecolog)' attd PIo'siologv of the Fungal Myceliutn (Jennings. D. H.. and Rayner, A. D. M., eds.). Cambridge Universit} Press, Cambridge, pp. 23-52. Von Neumann, J. (1966) Theory o/Self-Reproducing Automata (Burks, A. W., ed.). Uni`'ersity of Illinois Press, Urbana, Illinois. Wolfram, S. (1984) Cellular automata as models of comp[exit}. Nature 311, 410-424. Yang, H., King, R., Reichl, U., and Gilles, E. D. (1492a~ Mathematical model for apical gro~ th, septation, and branching of mycelial microorganisms. Biotech. Bioeng. 39, 49-58. Yang, H., Reichl, U., King, R., and Gilles, E. D. (1992b) Measurement and simulation of the morphological development of filamentous microorganisms. Biotech. Bioeng. 39, 44-48. Yarman, T., and Kiyak, N. (1990) A simple mathematical modeling of the gro~th of microorganism colonies. J. BioL Ph~. 17, 265-269.