Outline of the mathematics on which the Neighbour-Sensing model depends

Simulating the growth patterns of fungal tissues

Most mathematical models of hyphal growth published so far simulate growth of mycelia on a single plane; they are concerned primarily with reproducing the circularity of Petri-dish style fungal colonies. However, two dimensional space has some specific peculiarities that can affect the conclusions: forbidding crossings between hyphae in the Indermitte et al. models³ being a case in point. In a real three dimensional world a large number of points can be connected without the need for the connection paths to cross; whereas the number of such points is limited on a flat plane. The need to cross will also have effect on models where patterning is based on a hyphal density field, generated by all parts of the growing mycelium, as suggested by Cohen¹. Growth in this case is regulated by the absolute value of this field and is directed by its gradient (equivalent to negative autotropism). In two dimensional space turning up or down is not an option, so a tip approaching an existing hypha must go across the latter, moving against a large (possibly infinite, as the distance approaches zero) value of the density field. Cohen’s original model produced polarized tree-like structures, quite different from the typical spherical fungal colonies, and while the Indermitte et al. models³ succeeded in forming circular colonies, their analysis remained in two dimensions. Consequently, knowing how the circular colony arises on a flat plane is not enough; it is crucial to understand the formation of a spherical colony in three dimensional space.

Our purpose here is to suggest a model, which we call the Neighbour-Sensing model, that, whilst being as simple as possible, is able to simulate formation of a spherical, uniformly dense fungal colony in a visualisation in three dimensional space. Following Indermitte et al.³ we gauge our success on the basis that our model successfully imitates the three branching strategies of fungal mycelia illustrated by Nils Fries in 1943².

Verbal description of the Neighbour-Sensing Model

The process of simulation is defined as a closed loop. This loop is performed for each currently existing hyphal tip of the mycelium and the algorithm:

1. Finds the number of neighbouring segments of mycelium (N). A segment is counted as neighbouring if it is closer than the given critical distance (R). In the simplest case we did not use the concept of the density field, preferring a more general formulation about the number of the neighbouring tips.

2. If N<N_branch (the given number of neighbours required to suppress branching), there is a certain given probability (P_branch) that the tip will branch. If the generated random number (0..1) is less than this probability, the new branch is created and the branching angle takes a random value. The location of the new tip initially coincides with the current tip. This stochastic branch generation model is similar overall to earlier ones in which distance between branches and branching angles followed experimentally measured statistical distributions. This, however, was not required to reach the desired shape of the colony in our model. Rather, we use a uniform distribution, as did Indermitte et al.³

We assumed that all hyphal tips in mycelia grow at constant speed. This assumption was sufficient to get the desired shape and structure of the colony.

In the simplest version, the growth direction is defined during branching and is not altered subsequently. In other words, the initial model does not implement tropic reactions (to test the kind of morphogenesis that might arise without this component).  Later versions of the model tested how implementation of the density field hypothesis would affect colony growth. The density field features were made analogous to an electrical field as these are well understood.

Implementation of a negative autotropic reaction requires the concept of the density field, as the growth must be directed by the gradient of this field. We also implemented the suggestion¹ that the tip should change direction gradually (the so-called persistence factor). In our implementation, the growth speed remained constant and the density gradient alters only the growth direction. Otherwise, a high gradient, if formed accidentally, would cause unreliably fast growth in some parts of the mycelium.

With low values of the persistence factor the model is able to form small linear structures. This is because, with such a parameter set, immediately after branching the hyphal density field tends to orient the new tip strictly in the opposite growth direction from the old tip. That is, the new hypha is directed to grow parallel with the old hypha but in the opposite direction. If we suppose that the hyphal density field is generated just by tips and branch points, this direction remains optimal until the tip goes sufficiently far from the branch point to start interacting with other hyphae. Changing parameters while the colony is still nearly linear can produce ellipsoidal or tubular structures.

We have experimented with a variety of extensions of the model (see hyperlinks to our publications, below): for example, growth being suppressed by a high number of neighbouring tips; or allowing the growing tip only to be active for a fixed time before it stops growing and branching. Such changes can result in more optimal packing of the hyphae, but are not required to form a spherical colony. Real fungal colonies are rarely uniform in structure, so the question arises whether any smaller new structures can form in a virtual colony growing in accordance with this model. We found that this could happen following abrupt changes of the model parameter set (especially R and N_branch).

Mathematical description

Let, at the time tÎZ₊, the mycelium contain n growing hyphal tips. Let y_i and g_i be position and growth vectors, respectively, of the i-th growing hyphal tip at time tÎZ₊. Let Y be a set, containing other points of the mycelium that are sensed as neighbouring tips and/or branch points. Now, let

where F is a Heaviside function.

Let

where α_t, β_t and γ_t all form sequences of independent, uniformly distributed random variables over the range [0..1[.

Now compute an array b, containing all the values of i that satisfy the condition N_i < N_branch and dt < P_branch. Here dt forms a sequence of independent, uniformly distributed values over the range [0..1[, and P_branch is the model parameter. Let m be the length of this array. For each k[1..m], define:

Finally, define y′_i = y_i + ag_i, g′_i = g _i and n′ = n+m (the model parameter a determines the tip growth rate in length units per defined iteration period). Define Y′ = Y+"y_k:kÎb. Now we have y′, g′, n′ and Y′ defining the state of the colony after one iteration of the model algorithm.

This basic algorithm can be extended by assuming that the tip can be active only for a fixed time (S_max iterations) and stops growing after its length reaches L_max length units. Also, it could be assumed that the growth is suppressed if N_i >N_growth. To implement these extensions, let us define the age array S(S′_i = S_i+1, S′_n+k = 0) and the length array L (L′_i = L′_i+a, L′_n+k = 0). Then y′_i must be re-defined as ag_iF(S_i-S_max)F(L_i-L_max)F(N_growth-N_i) and the condition for the value, i, to join the array b must be extended to S_i < S_max. In the density field version of the model, (1) must be replaced by (4):

Also, N_max changes the biological meaning to the maximal value of the density field.

Negative autotropism was implemented using Cohen’s approach¹. In this case, v_i should be replaced by:

where

.

Again, (1) must be replaced by (4). In equation (5), the parameter k is a model parameter, defining a particular coefficient of persistence, which is used to ensure that branches change direction gradually; and it operates on the previous growth vector g_i. The derivatives are computed by numeric differentiation. The function norm(x) ensures that the density gradient alters the direction but not the speed of the growth.

Implementation

Both versions of the model were implemented in Java™ together with the simple visualiser:

y_i and Y being contained in a tree-like data structure. Interactive adjustment of  [0..¥[, and  [-π..π] enabled experimental observation of the growing colony and visual appreciation of its shape. To permit examination of the internal structure of the colony, the application will display a slice of chosen thickness across the colony. This complete interactive application is available for personal experimentation elsewhere on this CD.

Experimental use of the model

The Neighbour-Sensing model is an experimental tool. It is not a game or a painting program. Rather, it provides the user with a way of experimenting with features that may regulate hyphal growth patterns to arrive at suggestions that could be tested with live fungi.

Initial experiments show that a random growth and branching model (i.e. one that does not include the local hyphal tip density field effect) is sufficient to form a spherical colony. The colony formed by such a model is more densely branched in the centre and sparser at the border; a feature observed in living mycelia. Our conclusion is that there is no magic in a spherical colony - the shape is reached by the overwhelming majority of parameter settings.

Models incorporating local hyphal tip density field to affect patterning produced the most regular spherical colonies. As with the random growth models, making branching sensitive to the number of neighbouring tips forms a colony in which a near uniformly dense, essentially spherical, core is surrounded by a thin layer of slightly less dense mycelium. Using the branching types discussed by Fries² as our paradigm, the morphology of virtual colonies produced when branching (but not growth vector) was made sensitive to the number of neighbouring tips was closest to the so-called Boletus type.

This suggests that the Boletus type branching strategy does not use tropic reactions to determine patterning, nor some pre-defined branching algorithm. Following Occam’s rule that a simpler model must be preferred if it explains the experimental data equally well, we conclude that hyphal tropisms are not always required to explain “circular” (= spherical) mycelia.

When our model implements the negative autotropism of hyphae, a spherical, near uniformly dense colony is also formed, but branching is still regulated by the number of neighbouring tips (not by the density field). However the structure of such a colony is different from the previously mentioned Boletus type, being more similar to the Amanita rubescens type², characterised by a certain degree of differentiation between hyphae. First rank hyphae tend to grow away from the centre of the colony; second rank hyphae grow less regularly, and fill-in the remaining space. In the early stages of development such a colony is more star-like than spherical. We wish to emphasise that this remarkable differentiation of hyphae emerges in the visualisation even though the program settings used do not include routines implementing differences in hyphal behaviour. In the mathematical model all virtual hyphae are driven by the same algorithm. By altering the persistence factor, it is possible to generate the whole range of intermediate forms between Boletus and Amanita types.

Finally, when both autotropic reaction and branching are regulated by the hyphal density field, a spherical, uniformly dense colony is also formed. However, the structure is different again, such a colony being similar to the Tricholoma type illustrated by Fries². This type has the appearance of a dichotomous branching pattern, but it is not a true dichotomy. Rather the new branch, being very close to its parent, generates a strong density field that turns the older tip. In the previous model a tip nearby has no stronger effect than a more distant tip as long as they are both closer than R.

Hence, experimentation with the Neighbour-Sensing model allows us to suggest that the Amanita rubescens and Tricholoma branching strategies may be based on a negative autotropic reaction of the growing hyphae while the Boletus strategy may be based on the absence of such a reaction, relying only on density-dependent branching. Differences between Amanita and Tricholoma in the way that the growing tip senses its neighbours may be obscured in life. In Amanita and Boletus types, the tip may sense the number of other tips in its immediate surroundings. In the Tricholoma type, the tip may sense all other parts of the mycelium, but the local segments have the greatest impact.

These simple initial experiments with our model show that the broadly different types of branching observed in the fungal mycelium are likely to be based on differential expression of relatively simple control mechanisms. We presume that the 'rules' governing branch patterning (that is, the mechanisms causing the patterning) are likely to change in the life of a mycelium, as both intracellular and extracellular conditions alter. We have imitated some of these changes by making alterations to particular model parameters during the course of a simulation. Some of the results show that the Neighbour-Sensing model is capable of generating a range of morphologies in its virtual mycelia which are reminiscent of fungal tissues. These experiments make it evident that it is not necessary to impose complex spatial controls over development of the mycelium to achieve particular geometrical forms. Rather, geometrical form of the mycelium emerges as a consequence of the operation of specific locally-effective hyphal tip interactions. We hope that further experimentation with the model will enable us to predict how tissue branching patterns are established in real life.

The above is a brief outline of the kernel of the Neighbour-Sensing mathematical model. All the details about the original model and the enhancements that have now been completed have been published in our recent papers, together with an extensive collection of examples and experimental results. You can download reprints of these publications from this CD collection as PDF files from the hyperlinks immediately below. We recommend that you read these in this order:

(i)    Moore, D., McNulty L. J. & Meškauskas, A. (2004). Branching in fungal hyphae and fungal tissues: growing mycelia in a desktop computer. An invited chapter for the book Branching Morphogenesis, ed. J. Davies published by Landes Bioscience Publishing/Eurekah.com [published on line at http://www.eurekah.com/isbn.php?isbn=1-58706-257-7&chapid=1849&bookid=125&catid=20]. DOWNLOAD a PDF file of the published paper using this hyperlink.

(ii)    Meškauskas, A., McNulty, L. J. & Moore, D. (2004). Concerted regulation of all hyphal tips generates fungal fruit body structures: experiments with computer visualisations produced by a new mathematical model of hyphal growth. Mycological Research 108, 341-353. DOWNLOAD a PDF file of the published paper using this hyperlink.

(iii)    Meškauskas, A., Fricker, M. D. & Moore, D. (2004). Simulating colonial growth of fungi with the Neighbour-Sensing model of hyphal growth. Mycological Research 108, 1241-1256. DOWNLOAD a PDF file of the published paper using this hyperlink.

1. Cohen, D (1967) Computer simulation of biological pattern generation processes Nature 216, 246-248.

2. Fries, N (1943) Untersuchungen über Sporenkeimung und Mycelentwicklung bodenbewohneneder Hymenomyceten. Symb.Bot.Upsal. 6, 4-4.

3. Indermitte, C, Liebling, T. M and Clemencon, H (1994) Culture analysis and external interaction models of mycelial growth Bulletin of mathematical biology 56, 633-664.