# Quantifying the impact of different forms of stress on fungal growth: an inference method based on high-resolution pictures of the mycelial network

Lena Kuwata, Thibault Chassereau, Florence Chapeland-Leclerc, Pascal

David, Eric Herbert, Gwenaël Ruprich-Robert, Milica Tomašević, Amandine

Véber

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Lena Kuwata, Thibault Chassereau, Florence Chapeland-Leclerc, Pascal David, Eric Herbert, et al. Quantifying the impact of different forms of stress on fungal growth: an inference method based on high-resolution pictures of the mycelial network. 2025. hal-05149818v2

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Distributed under a Creative Commons Attribution 4.0 International License
Quantifying the impact of different forms of stress on fungal
growth: an inference method based on high-resolution pictures of
the mycelial network

Lena Kuwata¹, Thibault Chassereau², Florence Chapeland-Leclerc³, Pascal David², Eric Herbert², Gwenaël Ruprich-Robert³, Milica Tomašević⁴, and Amandine Véber*臺灣

¹Université Paris Cité, CNRS, MAP5, F-75006 Paris, France

²Université Paris Cité, CNRS, UMR 8236-LIED, F-75006 Paris, France

³Cibles Thérapeutiques et Conception de Médicaments (CiTCoM), UMR 8038, CNRS,
Faculté de Pharmacie de Paris, Université Paris Cité, F-75006 Paris, France

⁴CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau,
France

August 19, 2025

Abstract

In [5], a complete methodology for monitoring the growth of a filamentous fungus was in-
troduced, covering all aspects of this complex task going from the multi-scale imaging of the
network of filaments to the automated extraction of the graph structure and its key statis-
tics at regular time points. This methodology was applied to the fungus *Podospora anserina*
grown in the lab under various conditions. In parallel, a stochastic growth-fragmentation
model for the dynamics of such mycelial networks was introduced and studied in [25]. This
simple model depends on three parameters only: the elongation speed *v* of a single filament,
the branching rate *b*₁ of a filament at its open end, and the per unit length rate *b*₂ at which
a budding event happens, resulting in a new filament branching off from an existing one.
In this work, we develop a statistical inference method based on the large-time behaviour
of the growth-fragmentation model shown in [25], and on the high-resolution pictures of the
mycelial network obtained using the methodology described in [5], to reconstruct the param-
eters *v*, *b*₁ and *b*₂ from experimental data. We use this method to analyse the growth of *P.*
*anserina* observed under standard conditions and when several forms of stress are applied, in
order to quantify the effect of these stresses on the different mechanisms of fungal growth. By
comparing with the parameter estimates obtained from the dynamical tracking of individual
filaments, we show that reliable estimates of the individual elongation speed and branching
rates can be computed from the easily accessible data consisting in a single panorama of
the filament network pictured after several hours of growth and an empirical measure of the
exponential growth rate of the number of branch points and free extremities.

**Keywords:** fungal network, growth-fragmentation dynamics, parameter inference, asymptotic length distribution, impact of environmental conditions.

# 1 Introduction

The thallus of filamentous fungi is an intriguing case of branching and expanding networks. It consists of an interconnected network of filaments, the mycelium, whose construction is based upon fundamental cellular processes such as apical growth, branching and fusion (also called

<footnote>*Corresponding author: amandine.veber@parisdescartes.fr</footnote>
anastomosis). The growth dynamics of the mycelial network gives rise to an efficient spatial exploration and exploitation of the nutritive resources, and is robust enough to provide an effective response to external constraints disturbing the environmental exploration. This largely explains why filamentous fungi have colonised the entire terrestrial ecosystem [14]. In [5], an automated and reproducible experimental set-up to track the mycelial network construction was developed and applied to a filamentous fungus easily cultivable in the lab, *Podospora anserina*. This set-up uses the bright field technique to image the growth of networks which are physically constrained to remain two-dimensional, and allows us to obtain temporal series of panoramas of a fungal thallus, starting from a germinating ascospore and ending when the network reaches a width of a few centimeters. It comes with a robust image analysis routine which allows us to extract and precisely quantify the main statistics of the evolution of the mycelial network, such as the total length of the mycelium, the number of branching or fusion points (corresponding to the internal nodes of the graph representing the network) and the number of apexes (or open ends of a filament, corresponding to the leaves of the graph). Similar approaches have been developed for the monitoring of the growth of filamentous fungi [1, 23, 27] or of *Physarum* [13].

Extracting the behaviour of single filaments from the pictures obtained with this procedure is a difficult task, as it requires to align panoramas taken at different times and to identify the filament of interest in all of them. Recently, significant progress has been made in this direction [3, 4]. However, it remains very complex to follow the fine behaviour of a single apex and to estimate its elongation speed and its branching rates which can be heterogeneous in time and space, and even variable among apexes evolving in similar conditions. In addition, these results give us very interesting information about individual behaviours and potential interactions between neighbouring filaments, but this information is somehow too fine-grained to understand the overall growth of the network over large temporal and spatial scales, which may rather depend on summary statistics of the branching, elongation and fusion dynamics at the population level. The same question of what are the simplest and most accessible data that allow us to describe the structure and dynamics of a growing network arises in many other contexts, in which individual tracking tools, or even the high-resolution global dynamical monitoring tools used for lab-grown filamentous fungi, may simply not be available: for example, networks formed by leaf veins [6, 19], the morphogenesis of the gastrovascular system of jellyfish [24], viscous digitation [20, 26], or road networks [11]. As a consequence, there is a need for methods that are mostly based on “static” data, such as instant pictures of the network, and yet are able to extract key information to quantify the dynamics of fungal growth. We achieve this goal by using a growth-fragmentation model introduced in [25] to encode the dynamics of the network growth, and data consisting of a single panorama of the network taken after a sufficiently large time for the distribution in filament lengths to be “close to an equilibrium”, together with an empirical estimate of the growth rate of the number of internal nodes used as a measure of time. This is explained in more detail below.

The stochastic growth-fragmentation model we shall use is recalled in Section 2.2. It has only 3 parameters, supposed to be nonnegative constants: an elongation speed $v$, an *apical* branching rate $b_1$, and a *lateral* branching rate per unit length denoted by $b_2$. The network is represented as a population of “individuals”, each individual being a filament segment located between two branch points (such individuals are said to be *closed*, because they lie between two fixed points and their length is fixed throughout their life) or between a branch point and an apex (such individuals are said to be *open*, because they represent filament segments with an open end that continues to lengthen). From the modelling point of view, the main approximations made in this model are that the dynamics are homogeneous among all filament segments (no variability in branching rates or elongation speed), that the spatial structure of the network does not impact these dynamics (and therefore we can leave it aside and formulate a non-spatial model) and that there is no anastomosis. These assumptions and their relevance are discussed in [25]: the homogeneity assumption allows us to explore what type of population-
level summary statistics of individual dynamics are sufficient to quantify the growth of the network over relatively large scales; leaving out of the spatial structure answers the same need to understand how to summarise local interactions between filaments by a small number of simple descriptors; the absence of anastomosis will be justified by the fact that we shall mostly focus on data coming from a well-delimited ring in the exterior part of the network, where the filament density is not too high (and therefore, occasions for filament fusions are less frequent). What we show in this article is that the model, despite its assumptions which make it look very simplistic, is already sufficient to capture and quantify different aspects of the exponential growth of the mycelium. Furthermore, parameter inference in this model only requires a picture of the state of the network at a single large time $T$ and an estimate of the growth rate of the total number of internal nodes in the network. Indeed, for a given growth condition, we shall use the distribution in lengths “at stationarity” of a subset of the segments of filaments composing the network at time $T$ to infer the apical and lateral branching rates relative to the elongation speed, that is, to estimate $b_1/v$ and $b_2/v$. Then the ratio of the observed growth rate $\lambda_{obs}$ of the number of internal nodes to the relative growth rate $\lambda/v$ inferred from the model by using the estimates of $b_1/v$ and $b_2/v$ (and which corresponds to the growth rate of the number of internal nodes when we scale time so that filament elongation happens at speed is 1), assuming that $\lambda_{obs} \approx \lambda$, will give us an estimate of $v$. Using this methodology for parameter inference, we shall quantify the impact of various forms of stress by examining how $b_1$, $b_2$ and $v$ vary when compared to the benchmark growth conditions (M2) in which the fungus grows at optimal temperature and with unlimited nutrients. These results constitute the other main contribution of the present work.

To assess the reliability and robustness of our methodology, we shall compare the estimates we obtain to those derived from two other types of data. First, we shall use empirical estimates of $b_1$, $b_2$ and $v$ coming from the reconstruction of the continuous-time network growth history obtained using the dynamical monitoring pipeline with individual tracking described in [4]. Second, we shall apply our methodology to data that are directly simulated from the mathematical model. This will allow us to evaluate the precision of the reconstruction of the (known) parameters as a function of the number of filament segments present in the data at the time $T$ at which the state of the network is pictured. Note that, in this context where the empirical data are not necessarily numerous, classical cross-validation approaches are not easy to conduce since splitting the data set into two generally leads to performing parameter estimation based on too few of data. However, as our method is based on the length distribution of open segments only, a form of cross-validation can be obtained by simulating the stochastic growth process using the parameters estimated with experimental data, and then comparing the length distribution of the closed segments in the simulated data to the observed experimental distribution. We shall perform this comparison for a set of experiments, and obtain a very good fit between the simulated and empirical distributions for most of the growth conditions.

The inverse problem of reconstructing the operators characterising the growth-fragmentation dynamics from a set of observations is notoriously hard and is very dependent on the family of growth-fragmentation equations considered. A usual assumption made in this context is that the population of interest is very large, justifying the use of deterministic models [7, 15, 22]; most works consider fragmentation kernels that are supposed to be self-similar, so that the same characteristics are reconstructed from the observation of the smallest and largest individuals. The large-time behaviour of appropriate functionals of the size distribution is then used to identify the growth-fragmentation operators that best explain the data, either by analytical methods (as we do here in our parametric framework) or by numerical exploration. In [8], an original approach is developed in which the observation of the small time dynamics is used to reconstruct the fragmentation kernel via a power series expansion of the solution to the growth-fragmentation equation at small times and starting from a Dirac mass, assuming that the size-dependent fragmentation rate is polynomial (and known). Similarly, parametric and
nonparametric inference methods have been developed for continuous state branching processes, in order to reconstruct the state-dependent branching rates and offspring distributions from the observation of a sample of individuals (the “state” being the individual’s size, for instance). See, e.g., [10, 16]. The sample can be taken from the extant population at some large time $T$, or from the whole population up until time $T$. Again, the idea is that the empirical state distribution in the population converges to a limiting distribution as time goes to infinity, which can then be used to reconstruct the operators characterising the reproduction dynamics. All these approaches are reviewed in [9]. The specificity of our work is that we deal with a multi-type population, in which individuals of different types (open/closed) have different growth-fragmentation rules, but both sub-populations feed each other in such a way that their asymptotic behaviours are strongly coupled.

The rest of the article is organised as follows. In Section 2, we present the types of data we shall use for the inference procedure and its evaluation (Section 2.1), we recall the main features of our parametric growth-fragmentation model (Section 2.2), and we describe the full inference methodology (Section 2.3). In Section 3, we apply this methodology to data on *P. anserina* grown under six different conditions, and we analyse the impact of the corresponding forms of stress on the reconstructed elongation speed, and apical and lateral branching rates. In Section 4, we propose two evaluations of the quality of our inference method: in Section 4.1, the estimates obtained in Section 3 are compared to the estimates obtained by the dynamical tracking method of [4]; in Section 4.2, our inference method is applied to data simulated directly from the mathematical model, which allows us to evaluate the precision of the reconstruction of the (known) parameters as a function of the number of filament segments present in the data at the time $T$ at which the state of the network is pictured. All these results are discussed in Section 5.

# 2 Data, model and method

In this section, we first introduce the type of data that will be used in our inference method, consisting mainly of a single observation of the network at some time $T$. Next, we briefly present the more complex data that are now available thanks to recent developments in the treatment of dynamical observations of *P. anserina* fungal networks. This second type of data will be used to assess the reliability of our “static” inference method, since they provide different estimators for the three parameters in the growth-fragmentation model at the basis of our approach, which is recalled in Section 2.2. Finally, our inference methodology is expounded and its implementation based on the type of data presented in Section 2.1.1 is discussed in detail.

## 2.1 Static and dynamical observations of the mycelial network

### 2.1.1 Network panoramas and observations at a single time

The first motivation for this work was to identify which type of data with minimal complexity could be used to quantify the growth dynamics of the mycelial network over time. The individual unit in the network is the filament segment, i.e., a piece of filament located between two branch/fusion points (*closed* segment) or between a branch point and an apex (*open* segment). The approach we propose here is based on the observation (i) of the state of the distribution of the open segment lengths at some time $T$ (taken to be as large as is experimentally possible), and (ii) of the exponential growth rate $\lambda_{\text{obs}}$ of the total number of internal nodes present in the mycelium. The length distribution can be observed from a single panorama of the network at time $T$, while the exponential growth rate is a simple statistics which has to be estimated from the observation of the number of internal nodes at different times.

For each growth experiment, a temporal sequence of panoramas, ranging from the initial ascospore state until some reference time $T$, with a lag time $\tau = 18$ min between two im-
<figure><img src="image_2.png" /><figcaption>Figure 1: A) Initial germinating ascospore. B) Image of the mycelium after 9h of growth. C) Image of the mycelium after 20h of growth. D) Graph corresponding to a subpart of the mycelium shown in C); the elements of the graph are coloured according to their degree: apexes (nodes of degree 1) are in blue, the filament network skeleton (reconstructed nodes of degree 2) are in green, and branching and fusion points (nodes of degree 3) are in orange. Scale bar: 200µm.</figcaption></figure>

age acquisitions, is obtained. Several image pre-treatment processes are applied to the set of panoramas in order to digitise and identify the mycelial skeletal network. The resulting data is a graph embedded in $\mathbb{R}^2$ in which the nodes are of degree one, two or three and correspond to the positions of the apexes (degree one), branching and fusion points (degree three) and the skeleton of the mycelium (degree two). The graph is then pruned of all nodes of degree two, leaving only nodes of degrees one and three, while maintaining the connectivity of the network. Finally, each edge of the graph is given a weight which is equal to the sum of the lengths of the sub-segments (between degree 2 nodes) pruned along the path between the extremities of the segment. See Figure 1 for an illustration.

The data we shall use correspond to approximately $T \approx 20$ hours of growth, which is very close to the end time of the experiments. The network is naturally rooted at the node corresponding to the initial spore, roughly located in the middle of the Petri dish (and therefore of each panorama), and the distance we shall use later on is given by the edge weights (corresponding to the curvilinear lengths of the segments) measured in $\mu$m. We write $\Delta$ for the radius of the mycelium in any given experiment. Because the centre of the network becomes very dense after a few hours of growth, we shall have to restrict our attention to open segments whose open extremity is at a distance at least $r\Delta$ from the root of the graph, for some fixed $r \in [0, 1)$. The choice of $r$ will be made empirically and will depend on the growth conditions we explore. In other words, we shall focus on the apexes present in a ring of internal radius $r\Delta$, and of external radius $\Delta$. When several experiments are performed under the same conditions, the radius $\Delta$ may be different from one experiment to another, but the same proportion $r$ will be used to skim the data and keep only the outer part of the network. The data resulting from identical growth conditions will then be pooled to obtain a dataset of reasonably large size.

## 2.1.2 Dynamical individual tracking of apexes

In [4], a full pipeline is proposed to reconstruct the history of the network growth based on a series of panoramas, with a unique identification of each filament over time. This procedure yields a close to continuous time reconstruction of the network expansion, and therefore, gives access to natural estimators for the elongation speed and branching rates. Indeed, the trajectory of each apex is followed over time, which allows to classify the branch points into two categories, apical and lateral, depending on the chronology of their appearance: as a rule of thumb inspired by empirical evidence, apical branch points appear in the immediate vicinity of an apex, whereas the budding of lateral branches occurs far from the active apex of the filament. The resulting graph
is a 2-dimensional projection of the mycelial network in which a certain number of ambiguities need to be resolved to correctly identify the nature of the different nodes. We refer to [4] for a complete presentation and an analysis of the uncertainty associated with this reconstruction.

To a first order, the (discretized) time derivative of the number of apical branch points during a single experiment is given by $b_1$ times the current number of apexes. This property allows us to use a simple least square procedure to obtain an estimate of $b_1$ from the trajectories of the number of apical branch points and of the number of apexes in a single growth experiment. Likewise, the per-unit length lateral branching rate $b_2$ can be estimated using the property that the time derivative of the number of lateral branch points is given, to a first order, by the product of $b_2$ by the current total length of the mycelium. Finally, an estimation of the elongation speed can be obtained as the time derivative of the total length of the mycelium divided by the number of apexes at that time.

## 2.2 The stochastic growth-fragmentation model

In the modelling framework proposed in [25], the mycelium is modelled by a stochastic bi-type population model with a growth-fragmentation dynamics. Each individual in the population is represented by a pair $(\mathfrak{e}, x) \in S := (\{0, 1\} \times \mathbb{R}_+) \setminus \{(0, 0)\}$, where $\mathfrak{e} \in \{0, 1\}$ encodes whether the segment is open ($\mathfrak{e} = 1$) or closed ($\mathfrak{e} = 0$) and $x \ge 0$ is the current length of the corresponding segment. At every time $t \ge 0$, the set of all segments constituting the network is fully described by the following counting measure on $S$:

$$ \mathcal{Z}_t = \sum_{u \in V_t} \delta_{(\mathfrak{e}^u, x_t^u)}, $$

where $V_t$ denotes the index set of all segments present in the mycelium at time $t$ and $(\mathfrak{e}^u, x_t^u)$ denotes the characteristics at time $t$ of segment $u \in V_t$. With this representation, for any subset $S' \subset S$, the integral

$$ \langle \mathcal{Z}_t, \mathbb{1}_{S'} \rangle := \int_S \mathbb{1}_{S'}(\mathfrak{e}, x) \mathcal{Z}_t(d\mathfrak{e}, dx) = \sum_{u \in V_t} \mathbb{1}_{S'}(\mathfrak{e}^u, x_t^u) $$

counts how many segments present in the network at time $t$ have characteristics that belong to the subset $S'$.

The dynamics of the process $(\mathcal{Z}_t)_{t \ge 0}$ are as follows:

(a) **Elongation.** Open segments elongate deterministically at speed $v$, while the length of closed segments remains constant. More precisely, for every $s, t \ge 0$ and $u \in V_s$, conditionally on segment $u$ not being involved in a branching event during the time interval $[s, s+t)$, we have for every $r \in [0, t)$, $x_{s+r}^u = x_s^u + \mathfrak{e}^u v r$.

(b) **Apical branching.** Each open segment branches “at its apex” at rate $b_1$. That is, every segment $u$ such that $\mathfrak{e}^u = 1$, independently of each other and at rate $b_1$, is removed from the population and replaced by three new segments: a closed segment of the same length (therefore with characteristics $(0, x_{t-}^u)$), where $t$ is the time of the branching event), and two open segments of length 0 (and thus both with characteristics $(1, 0)$).

(c) **Lateral branching.** Every segment (open or closed) of length $x > 0$ branches “laterally” at rate $b_2x$ and the branch point is chosen uniformly at random along the segment. That is, for every $t \ge 0$ and every $u \in V_{t-}$ such that $x_{t-}^u > 0$, at instantaneous rate $b_2x_{t-}^u$ segment $u$ is removed from the population and is replaced by three new segments: a closed segment of length $\alpha^u x_{t-}^u$, a segment with first characteristics $\mathfrak{e}^u$ (i.e., open if the parent was open, or closed if the parent was closed) and length $(1 - \alpha^u)x_{t-}^u$, and finally an open segment of length 0, where $\alpha^u$ is an independent random variable with a uniform distribution over $[0, 1]$.
In what follows, we shall use the standard notation

$$ \langle \mathcal{Z}, f \rangle := \int_S f(\mathfrak{e}, x) \mathcal{Z}(d\mathfrak{e}, dx) $$

for the integral of the function $f$ with respect to the measure $\mathcal{Z}$. For every $t \ge 0$, let $n_t$ be the mean measure on $S$ defined by: for every bounded measurable function $f: S \to \mathbb{R}$,

$$ \langle n_t, f \rangle := \mathbb{E} [\langle \mathcal{Z}_t, f \rangle] = \mathbb{E} \left[ \sum_{u \in V_t} f(\mathfrak{e}^u, x_t^u) \right]. $$

The mean measure $n_t$ can be decomposed as follows: for every bounded measurable function $f: S \to \mathbb{R}$,

$$ \langle n_t, f \rangle = \int_{\mathbb{R}_+} f(1, x) n_t^1(dx) + \int_{\mathbb{R}_+^*} f(0, x) n_t^0(dx), $$

where $n_t^1$ and $n_t^0$ are the marginal length distributions for open and closed segments respectively. Under the assumption that $n_0^1$ and $n_0^0$ are absolutely continuous with respect to Lebesgue measure on $\mathbb{R}_+$ and $\mathbb{R}_+^*$ respectively, it was proved in [25, Proposition 1.2] that $n_t^1$ and $n_t^0$ are also absolutely continuous with respect to Lebesgue measure for every $t \ge 0$ and the pair of densities $(n_t^1, n_t^0)_{t \ge 0}$ is a weak solution to:

$$ \left\{ \begin{aligned} & \frac{\partial n_t^1}{\partial t}(x) + v \frac{\partial n_t^1}{\partial x}(x) + (b_1 + b_2x)n_t^1(x) = b_2 \int_x^{+\infty} n_t^1(y)dy, \\ & vn_t^1(0) = 2b_1 \int_0^{+\infty} n_t^1(y)dy + b_2 \int_0^{+\infty} y(n_t^1(y) + n_t^0(y))dy, \\ & \frac{\partial n_t^0}{\partial t}(x) + b_2xn_t^0(x) = b_1n_t^1(x) + 2b_2 \int_x^{+\infty} n_t^0(y)dy + b_2 \int_x^{+\infty} n_t^1(y)dy. \end{aligned} \right. \quad (1) $$

The large time behaviour of $(n_t^1, n_t^0)_{t \ge 0}$ was investigated in [25], where it was proved that there exists $\lambda > 0$, a function $\psi: S \to \mathbb{R}_+$, and two probability measures $N_1$ and $N_0$ (respectively on $\mathbb{R}_+$ and $\mathbb{R}_+^*$) such that, as $t \to \infty$, we have

$$ n_t^1 \sim \langle n_0, \psi \rangle e^{\lambda t} N_1 \quad \text{and} \quad n_t^0 \sim \langle n_0, \psi \rangle e^{\lambda t} N_0. \quad (2) $$

See [25, Theorem 4.3] for a more precise statement of this Perron-Frobenious-type convergence. An explicit expression for $\lambda$, $N_1$ and $N_0$ (and $\psi$, though we do not report it here) can be computed and their derivation is based on the following reasoning. For $t > 0$, let us formally replace $n_t^1$ and $n_t^0$ by $e^{\lambda t}N_1$ and $e^{\lambda t}N_0$, respectively (not yet knowing what these quantities are). We obtain a spectral problem, which captures the stationary profile $(N_1, N_0)$ of the marginal distributions associated to each possible eigenvalue $\lambda$:

$$ \left\{ \begin{aligned} & vN_1'(x) + (b_1 + b_2x + \lambda)N_1(x) = b_2 \int_x^{+\infty} N_1(y)dy, \\ & vN_1(0) = 2b_1 \int_0^{+\infty} N_1(y)dy + b_2 \int_0^{+\infty} y(N_1(y) + N_0(y))dy, \\ & (b_2x + \lambda)N_0(x) = b_1N_1(x) + 2b_2 \int_x^{+\infty} N_0(y)dy + b_2 \int_x^{+\infty} N_1(y)dy. \end{aligned} \right. \quad (3) $$

In principle, this system can be solved directly using the method expounded thereafter. However, all the components of the dynamics have linear rates, and therefore we can separate the effect of $v$ by dividing every equation in (3) by $v$. This corresponds to considering the asymptotic behaviour of the process $(n_t^1, n_t^0)_{t \ge 0}$, in which elongation happens at speed 1, apical branching at rate $b_1/v$ and lateral branching at rate $b_2/v$ per unit length. By linearity, the stationary distributions $N_1, N_0$ will not be affected by this scaling, but the corresponding eigenvalue will now be given by $\lambda/v$.
**Remark 2.1.** This scaling procedure may seem slightly artificial, but in fact, it will turn out to be key in our inference method: by observing the stationary distribution of open segments, supposed to be close to $N_1$ in some appropriate sense, we shall be able to estimate (only) the pair $(b_1/v, b_2/v)$ of relative branching rates. Although the elongation speed $v$ cannot be observed directly in experimental data, by comparing the observed growth rate $\lambda_{\text{obs}}$ to the inferred value for $\lambda/v$ (which is a function of $b_1/v$ and $b_2/v$ only), we shall obtain an estimate for $v$, and therefore for the unscaled rates $b_1$ and $b_2$.

Let us thus set the notation

$$ \tilde{b}_1 := \frac{b_1}{v}, \quad \tilde{b}_2 := \frac{b_2}{v}, \quad \tilde{\lambda} := \frac{\lambda}{v}. \qquad (4) $$

Dividing the three equations in (3) by $v$, we obtain the following system:

$$ \left\{ \begin{aligned} N'_{1}(x) + (\tilde{b}_{1} + \tilde{b}_{2}x + \tilde{\lambda})N_{1}(x) &= \tilde{b}_{2} \int_{x}^{+\infty} N_{1}(y)dy, \\ N_{1}(0) &= 2\tilde{b}_{1} \int_{0}^{+\infty} N_{1}(y)dy + \tilde{b}_{2} \int_{0}^{+\infty} y(N_{1}(y) + N_{0}(y))dy, \\ (\tilde{b}_{2}x + \tilde{\lambda})N_{0}(x) &= \tilde{b}_{1}N_{1}(x) + 2\tilde{b}_{2} \int_{x}^{+\infty} N_{0}(y)dy + \tilde{b}_{2} \int_{x}^{+\infty} N_{1}(y)dy. \end{aligned} \right. \qquad (5) $$

It is shown in Appendix A in [25] that an explicit solution to this system satisfying the normalising condition $\langle N_1 + N_0, 1 \rangle = 2$ can be found by first solving the equation satisfied by $N_1$ and then solving for $N_0$ (note that $N_0$ and $N_1$ necessarily have the same total mass, which is therefore equal to 1). We obtain that $\tilde{\lambda}$ is the positive solution to the equation

$$ \tilde{b}_1 + \frac{\tilde{b}_2}{\tilde{\lambda}} = \tilde{\lambda}, $$

that is,

$$ \tilde{\lambda} = \frac{\tilde{b}_1 + \sqrt{\tilde{b}_1^2 + 4\tilde{b}_2}}{2}, \qquad (6) $$

and furthermore we have for every $x \ge 0$,

$$ N_1(x) = (\tilde{b}_1 + \tilde{b}_2x + \tilde{\lambda}) \exp \left( - \int_0^x (\tilde{b}_1 + \tilde{b}_2y + \tilde{\lambda}) dy \right), \qquad (7) $$

and for every $x > 0$,

$$ N_0(x) = \frac{N_1(x)}{(\tilde{b}_2x + \tilde{\lambda})^2} (\tilde{b}_2 + \tilde{b}_1(\tilde{b}_2x + \tilde{\lambda})) + \frac{\tilde{b}_2 \exp\left(-\int_0^x (\tilde{b}_1 + \tilde{b}_2y + \tilde{\lambda})dy\right)}{(\tilde{b}_2x + \tilde{\lambda})^3} (2\tilde{b}_2 + \tilde{b}_1(\tilde{b}_2x + \tilde{\lambda})). \quad (8) $$

These expressions give us a detailed description of the long term behaviour of the mean measure $n_t$, corresponding to the average behaviour of the stochastic growth-fragmentation process $(Z_t)_{t \ge 0}$. To relate this average behaviour to the long term dynamics of the stochastic trajectories of $(Z_t)_{t \ge 0}$, the following functional law of large numbers was proved (see [25, Theorem 1.4]): for every measurable function $f: S \to \mathbb{R}$ such that $\sup_{(e,x) \in S} |f(e,x)|/(1+x) < \infty$, we have the convergence in probability

$$ \lim_{t \to \infty} \frac{\langle \mathcal{Z}_t, f \rangle}{\langle \mathcal{Z}_t, 1 \rangle} = \frac{\langle N_1, f(1, \cdot) \rangle + \langle N_0, f(0, \cdot) \rangle}{2}. \qquad (9) $$

As mentioned in Remark 2.1, the explicit expression (7) for the stationary profile of the average length distribution of open segments, along with the convergence result grossly stated in (2)
and the law of large number results recalled above, gives us an asymptotic expression for the empirical length distribution of open segments as time goes to infinity. We shall use this property to infer the relative growth parameters $\tilde{\lambda}$, $\tilde{b}_1$ and $\tilde{b}_2$. Indeed, using the convergence result (9) with functions $f$ of the form $f(e, x) = 1_{\{e=1\}}g(x)$, and noting that the ratio of the number of open segments $\langle \mathcal{Z}_t, 1_{\{e=1\}} \rangle$ to the total number of segments $\langle \mathcal{Z}_t, 1 \rangle$ goes to 1/2 as $t$ tends to infinity (as every branching event adds one open segment and one closed one to the process), we obtain that for every measurable function $g: [0, \infty) \to \mathbb{R}$ satisfying $\sup_{x \in [0, \infty)} |g(x)|/(1+x) < +\infty$, we have the convergence in probability:

$$ \lim_{t \to +\infty} \frac{\langle \mathcal{Z}_t, 1_{\{e=1\}} g(x) \rangle}{\langle \mathcal{Z}_t, 1_{\{e=1\}} \rangle} = \langle N_1, g \rangle. $$

This tells us that the empirical length distribution of open segments is close to the probability distribution $N_1$ when $t$ is large, and we can then infer the relative growth parameters $\tilde{\lambda}$, $\tilde{b}_1$ and $\tilde{b}_2$ by finding the values of $\tilde{b}_1$ and $\tilde{b}_2$ leading to the best fit between (the appropriate parts of) $N_1$ and the empirical distribution at some time $T$ large enough. The inference methodology is described in detail in the next section.

## 2.3 Inference method

In this section, we first present the theory behind our inference method based on a single panorama of the network structure at some large time $T$ coupled with an estimate of the empirical growth rate of the number of internal nodes (branching and fusion points). This is done in Section 2.3.1. Because it is based on the empirical length distribution at a single time $T$ (and not on the time dynamics of this distribution), we call this procedure a *static* inference method. This is a slight abuse of terminology, as determining the empirical growth rate of the number of internal nodes requires to observe the number of internal nodes at different time points. In Section 2.3.2, we present the implementation of this method and the various issues that need to be overcome. We apply it to experimental data on *P. anserina* growth under different environmental conditions in Section 3.

### 2.3.1 Static inference method

As briefly explained in Remark 2.1, our strategy is as follows. First, we use the fact that the empirical length distribution of open segments at some large time $T$ can be approximated by the probability distribution $N_1$ to obtain estimates of the relative branching rates $\tilde{b}_1$ and $\tilde{b}_2$ from a panorama of a mycelium taken at the end of a growth experiment. Using (6), we can deduce an estimate of the relative exponential growth rate $\tilde{\lambda}$. Once this first step is completed, we can then obtain an estimate of the elongation speed $v$ by comparing the observed growth rate of the number of internal nodes in our data, denoted by $\lambda_{\text{obs}}$, and the estimated value of the relative growth rate $\tilde{\lambda} = \lambda/v$. Indeed, assuming that $\lambda \simeq \lambda_{\text{obs}}$, a natural estimator for $v$ is the ratio

$$ \hat{v} := \frac{\lambda_{\text{obs}}}{\tilde{\lambda}}. \qquad (10) $$

Finally, multiplying the estimates of $\tilde{b}_1$ and $\tilde{b}_2$ by $\hat{v}$ yields the desired estimates for the branching rates in natural time units $b_1$ and $b_2$.

**Estimation of $\tilde{b}_1$ and $\tilde{b}_2$.** Given a panorama of mycelium $\mathcal{P}_T$ taken at a large time $T$, we write $L_T$ for the random variable with values in $[0, \infty)$ whose distribution is given by the empirical length distribution of open segments in $\mathcal{P}_T$. Using the property that the law of $L_T$ is well-
approximated by $N_1$, we can write for $\ell \ge 0$

$$ \mathbb{P}(L_T \ge \ell) \approx N_1([\ell, \infty)) = \int_{\ell}^{\infty} (\tilde{b}_1 + \tilde{b}_2x + \tilde{\lambda}) \exp \left( - \int_{0}^{x} (\tilde{b}_1 + \tilde{b}_2y + \tilde{\lambda}) dy \right) dx \\ = \exp \left( - \int_{0}^{\ell} (\tilde{b}_1 + \tilde{b}_2x + \tilde{\lambda}) dx \right), \qquad (11) $$

so that

$$ - \frac{\log(\mathbb{P}(L_T \ge \ell))}{\ell} \approx \tilde{\lambda} + \tilde{b}_1 + \frac{\tilde{b}_2}{2}\ell. \qquad (12) $$

Recalling from (6) that $\tilde{\lambda} = (\tilde{b}_1 + \sqrt{\tilde{b}_1^2 + 4\tilde{b}_2})/2$, we obtain

$$ - \frac{\log(\mathbb{P}(L_T \ge \ell))}{\ell} \approx \frac{\tilde{b}_1 + \sqrt{\tilde{b}_1^2 + 4\tilde{b}_2}}{2} + \tilde{b}_1 + \frac{\tilde{b}_2}{2}\ell. $$

The function $(\ell \mapsto \mathbb{P}(L_T \ge \ell))$ is the empirical tail distribution in lengths of open segments, equal to

$$ \bar{F}(\ell) = \frac{\text{Number of open segments of length} \ge \ell}{\text{Total number of open segments}}, $$

so that we have

$$ - \frac{\log(\bar{F}(\ell))}{\ell} \approx \frac{\tilde{b}_1 + \sqrt{\tilde{b}_1^2 + 4\tilde{b}_2}}{2} + \tilde{b}_1 + \frac{\tilde{b}_2}{2}\ell. \qquad (13) $$

Then, we estimate $\tilde{b}_1$ and $\tilde{b}_2$ by finding the values that lead to the best fit between the functions of $\ell$ appearing on both sides of (13), and we use the estimates $\hat{b}_1$ and $\hat{b}_2$ thus obtained to compute an estimate of $\tilde{\lambda}$.

**Condition on the branching rates.** Notice that if $\tilde{b}_1 \ll \sqrt{\tilde{b}_2}$, then $\tilde{\lambda} \simeq \sqrt{\tilde{b}_2}$ and a small error in the estimate of $\tilde{b}_2$ will result in a large error in the estimate of $\tilde{b}_1$. Conversely, the typical length of an open segment is of the order of $1/\tilde{b}_1$ (since open segments branch apically at rate $b_1$), so that if $\tilde{b}_1 \gg \sqrt{\tilde{b}_2}$, we have $\tilde{\lambda} + \tilde{b}_1 \simeq 2\tilde{b}_1 \gg \tilde{b}_2\ell$. Consequently, a small error in the estimate of $\tilde{b}_1$ will result in a large error in the estimate of $\tilde{b}_2$. We thus need to have

$$ \tilde{b}_1 \asymp \sqrt{\tilde{b}_2} \qquad (14) $$

to obtain satisfactory joint estimates of $\tilde{b}_1$ and $\tilde{b}_2$. Condition (14) means that, in the stochastic process, apical and lateral branching occur at similar rates. Indeed, at time $t \ge 0$ apical branching occurs at rate $\tilde{b}_1$ times the number of apexes at time $t$ and lateral branching occurs at rate $\tilde{b}_2$ times the total length of the mycelium at time $t$. Using (9) we have the convergence in probability:

$$ \lim_{t \to \infty} \frac{\tilde{b}_1 \times \text{Number of open segments}}{\tilde{b}_2 \times \text{Total length of the mycelium}} = \frac{\tilde{b}_1 \tilde{\lambda}}{\tilde{b}_2}. $$

If $\tilde{b}_1$ and $\sqrt{\tilde{b}_2}$ are of the same order of magnitude, then $\tilde{\lambda}$ is of the same order of magnitude as $\tilde{b}_1$ and $\sqrt{\tilde{b}_2}$ so that $\tilde{b}_1 \tilde{\lambda}/\tilde{b}_2$ is of order 1.

Empirically, we observe that $\tilde{b}_1$ and $\sqrt{\tilde{b}_2}$ being of the same order is a reasonable assumption, at least for standard growth conditions. A typical sign that Condition (14) fails is that the standard deviation computed (e.g., by bootstrapping) for one of the relative branching rates is much larger than its estimated value. Now, even when Condition (14) fails, our method still yields interesting information on the network construction. Indeed, if the standard deviation of $\hat{b}_1$ is large and if $\hat{b}_2$ is much larger than $\tilde{b}_1^2$, then we can surmise that $\hat{b}_2$ has contributed the
majority of the total growth rate of the network and that the mycelium has therefore mainly branched laterally. This will be the case for two of the growth conditions explored in the next section. Conversely, if the standard deviation of $\hat{b}_2$ is much larger than its value and $\hat{b}_1^2$ is much larger than $\hat{b}_2$, then we can surmise that the mycelium has mainly branched apically.

**Validity of the approximation by the stationary distribution.** To apply our method, we shall need to make sure that the approximations that have been made in (11) and (13) are valid for the data we wish to use. These approximations come from a large time convergence theorem, but the complete statement of Theorem 1.3 in [25] shows that this convergence occurs exponentially fast (with a positive rate which cannot be computed explicitly). As this rate could be small, we verified that the convergence was indeed fast enough on simulated data (see Section 4.2).

An additional way to check the validity of the approximation is to use the following convergence in probability occurring in the growth-fragmentation model (also based on (9)):

$$ \lim_{t \to \infty} \frac{\text{Number of open segments at time } t}{\text{Total length of the mycelium at time } t} = \tilde{\lambda}. $$

We thus have two estimators for $\tilde{\lambda}$, both based on the large time behaviour of the stochastic growth-fragmentation model:

$$ \hat{\lambda}_1 = \frac{\hat{b}_1 + \sqrt{\hat{b}_1^2 + 4\hat{b}_2}}{2}, \qquad (15) $$

$$ \hat{\lambda}_2 = \frac{\text{Number of open segments at time } T}{\text{Total length of the mycelium at time } T}. \qquad (16) $$

If these two estimators yield different results, we can then conclude that the time $T$ at which the state of the mycelial network is considered is not large enough for the approximation (13) to hold and yield satisfactory estimates of $\tilde{b}_1$ and $\tilde{b}_2$.

### 2.3.2 Implementation on experimental data

We can now describe how we shall implement our approach for parameter inference on data obtained using the experimental setup described in [5] (see also Section 2.1.1). In this section, we describe the methodology; the results are presented in Section 3.

Data were obtained for the standard optimal conditions (hereafter referred to as M2) and for five different forms of stress: low nutrient medium (M0), high osmotic medium (using potassium chloride, KCl), strong lighting (LED), low and high temperatures (respectively referred to as T19 and T35). These conditions are summarised in Table 1 and more details can be found in [21]. Six growth experiments were realised for the standard conditions, and three experiments were realised for each form of stress. In each of these experiments, the panorama of the mycelial network obtained after a relatively large time $T$, of the order of 20 hours, was used in the procedure. Note that we do not need that these times should be identical between all growth experiments, we only assume that each terminal time is large enough for the length distribution of open segments to be close to its stationary distribution. Figure 2 displays the final state of the mycelial network in one experiment performed in each of these conditions.

Recall that the model neglects anastomosis (or filament fusions) and that the parameter estimation method hinges on several approximations, so that we need to verify the validity of our approach for this data.

**Limiting the impact of anastomosis.** Anastomosis may have stopped the growth of some of the observed segments and turned open segments into closed ones, distorting the stationary
<table><caption>Table 1: Conditions tested in the growth experiments.</caption><thead><tr><th>Label</th><th>Nature of the stress</th><th>Culture medium</th><th>Lighting</th><th>Temperature</th></tr></thead><tbody><tr><td>M2</td><td>reference</td><td>M2</td><td>neutral</td><td>27°C</td></tr><tr><td>M0</td><td>nutrient</td><td>M0</td><td>neutral</td><td>27°C</td></tr><tr><td>KCl</td><td>osmotic</td><td>M2 + KCl (0,3 M)</td><td>neutral</td><td>27°C</td></tr><tr><td>LED</td><td>intense lighting</td><td>M2</td><td>intense</td><td>27°C</td></tr><tr><td>T19</td><td>low temperature</td><td>M2</td><td>neutral</td><td>19°C</td></tr><tr><td>T35</td><td>high temperature</td><td>M2</td><td>neutral</td><td>35°C</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Figure 2: Mycelial skeletal networks obtained from panoramas taken at the end of growth experiments, under the different conditions listed in Table 1 (see subcaptions).</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Data set obtained for different values of *r* in the same experiment performed under the M2 condition (same experiment as in Figure 2(a)). The red lines delineate the open segments whose open end (or apex) is at distance at least *r*Δ from the network centre (Δ being the radius of the mycelium). These segments are those used to construct the empirical length distribution of open segments in our inference procedure. To visualise what proportion of open segments are left out by only keeping the external segments, all apexes in the network are marked with a blue dot.</figcaption></figure>

length profiles $N_0$ and $N_1$. To be able to neglect the effects of anastomosis on our data, we exclude the data from the central part of the mycelial network, which rapidly becomes very dense, thus increasing the probability of filament crossings and fusions. To do so, we only consider open segments with an apex which is at distance at least $r\Delta$ from the centre of the mycelium, where $\Delta$ is the radius of the mycelium and $r \in [0, 1]$ is a proportion which has to be determined empirically and which will depend on the growth conditions considered. See Figure 3 for an illustration. Indeed, in the external ring lying between distances $r\Delta$ and $\Delta$ of the network centre, anastomosis mainly happens to relatively long filaments, so we can expect that matching the data to the stationary distribution $N_1$ over an interval of lengths of the form $(0, L_{max})$ will allow us to get around the fact that fusions prevent some of the long segments to occur, thereby distorting the tail of the distribution.

**Quality of the approximation by the stationary behaviour.** To obtain the approximation (11), we have used the fact that the length distribution of open segments should be close to $N_1$ if the observation time $T$ is large enough, and so we need to verify that the time span of a growth experiment is long enough for the approximation to hold. In fact, it is more relevant to reason in terms of number of apexes. Indeed, due to the branching dynamics of the network and our assumption of homogeneous behaviour between segments, the number of apexes at time $T$ is an exponentially growing function of $T$ (with some variability due to stochasticity in the early phase of growth). The asymptotic of the branching system “at large times” is therefore equivalent to its asymptotic behaviour “at large numbers of individuals”. In addition, assuming that we have reached this asymptotic regime so that the length of any given open segment has a distribution which is close to $N_1$, we can expect the approximation of $N_1$ by the empirical length distribution of all open segments used in (13) to be all the better as the number of open segments present in the data is large (by a law of large number effect). This idea is reinforced by theoretical results obtained in other contexts [16]. Therefore, to check the quality of the approximations (11) and (13), we need to check that the number of open segments (or equivalently, the number of apexes) present in the data we use in the inference procedure is large enough.

For each set of conditions listed in Table 1, several growth experiments were carried out (6 experiments for M2, 3 experiments for the other conditions). Since the length distribution of open segments obtained under similar growth conditions are supposed to be well-approximated by the same stationary profile $N_1$, a natural way of increasing the size of the data is to pool
the results of the experiments carried out under similar conditions. In principle, this pooling
strategy has a positive effect on the quality of the approximation of the theoretical distribution
$N_1$ by the empirical distribution of all open segment lengths and should allow us to perform
our inference procedure with a collection of experiments each having a relatively small set of
open segments. However, a relatively small number of open segments may be the sign that the
length distribution of open segments has not yet reached stationarity in a given experiment,
deteriorating the quality of the estimation. To identify the order of magnitude of the minimal
number of open segments necessary for the inference procedure to produce reliable estimates and
to determine whether reaching this number by pooling data coming from several experiments
is a relevant strategy, we use simulated data, for which we have access to the true length
distribution lying behind each experiment. This is carried out in Section 4.2.2, and we obtain
that the method yields satisfactory results even for a relatively low number of open segments
per experiment (approx. 50 segments) as long as the total number of open segments is sufficient
when pooling the three, or six, data sets from fungi grown under similar conditions (approx.
500 segments are needed in total).

Beside the issue of the convergence of the empirical measure to $N_1$, the fact that the data comes from a panorama taken at finite time $T$ also has an impact on the interval of lengths on which the approximation holds. Indeed, in theory segments can have grown at most to a length equal to $vT$, which distorts the upper part of the length distribution. In addition, very small open segments can only be observed directly after branching events, which happen at a finite rate, and over a finite time interval $[0, T]$ this randomness may also distort the length distribution close to 0. This mathematical issue adds up to the empirical issue that detecting very small segments in data is technically difficult, which introduces another bias in the empirical length distribution of open segments. Therefore, we shall have to match the data to the stationary distribution $N_1$ over an interval of the form $[L_{\min}, L_{\max}]$, with $L_{\min} > 0$.

**Implementation of the method.** Now that we have discussed the various issues we en-
counter with experimental data and how we can address them, let us gather all the above into
the description of the concrete implementation of our method on the data in which we are inter-
ested. First, we pool all the data coming from growth experiments carried out under the same
conditions. The lengths of open segments are the weights of the edges connecting the nodes of
degree one to their only adjacent node. To exclude the open segments from the centre of the
mycelium, we only keep the nodes of degree one that are at a distance at least $r\Delta$ from the
centre. Once the lengths of all the open segments considered are obtained, we can compute
the empirical tail distribution $\bar{F}$ and $-\log(\bar{F}(\ell))/\ell$. Note that anastomosis and crossing make
some segments appear shorter than they actually are, making $\bar{F}$ decrease faster for small values
of $\ell$ and slower for larger values of $\ell$, so that the slope of the function ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) can
become negative for larger values of $\ell$. We thus set the radius $r$ to be the smallest radius for
which the function ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) is linear and has a positive slope for a large enough
interval of lengths. We then choose the interval $[L_{\min}, L_{\max}]$ over which we shall match the
functions ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) and ($\ell \mapsto \tilde{\lambda} + \tilde{b}_1 + \tilde{b}_2\ell/2$): it is taken as the largest interval over
which the first function is indeed approximately linear and with a positive slope. See Figure 4
for an illustration of these steps.

Using the fact that $\tilde{\lambda}$ can be computed from knowing the values of $\tilde{b}_1$ and $\tilde{b}_2$, we can then
estimate the relative branching rates $\tilde{b}_1$ and $\tilde{b}_2$ by minimising the $L^2$ distance between the two
functions. We compute the standard deviation of the estimates $\hat{b}_1$ and $\hat{b}_2$ using a bootstrap
method. Note that for bootstrapping to be valid, we need the empirical distribution of the data
to be close to the real distribution, which is an assumption we already need for our method to be
valid. Once $\hat{b}_1$ and $\hat{b}_2$ have been thus obtained, we can compute $\hat{\lambda}_1$ thanks to the expression (15),
which we can compare to the estimator $\hat{\lambda}_2$ defined in (16) and obtained by dividing the number
of nodes of degree one (number of open segments) by the sum of the weights of all the edges in
<figure><img src="image_5.png" /><figcaption>Figure 4: Mapping of the function ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) for $r = 0$ and $r = 0.6$ for standard growth condition M2. In each graph, the vertical lines delimit the interval of lengths on which the function is well-approximated by a linear function with a strictly positive slope. Note that the curves are more and more irregular as we go towards larger values of $\ell$, due to the small number of long segments in the data.</figcaption></figure>

the graph (total length of the mycelium).

# 3 Quantifying the impact of stress on the mycelial growth

In this section, our inference method is used to analyse the growth characteristics of *P. anserina* mycelial networks under different growth conditions. In particular, we aim at quantifying the impact of several forms of stress on these characteristics. The implementation of our methodology requires an empirical calibration of several features such as the internal radius *r* of the ring delimiting the data considered, or the interval of lengths [*L*<sub>min</sub>, *L*<sub>max</sub>] used to compare the theoretical and empirical segment length distributions. This calibration and the robustness of our conclusions are discussed in Section 3.2.

## 3.1 Inference of $b_1$, $b_2$ and $v$

We applied the static inference method to the data described in Section 2.3.2, pooling the sets of segment lengths coming from experiments carried out under the same conditions. The estimates $\hat{b}_1$ and $\hat{b}_2$ of the relative branching rates $\tilde{b}_1$ and $\tilde{b}_2$ obtained for the six growth conditions can be found in Table 2 and the comparison of the two estimators $\hat{\lambda}_1$ and $\hat{\lambda}_2$ of the relative growth rate $\tilde{\lambda}$ defined in (15) and (16) can be found in Table 3.

We found that two growth conditions led to a violation of Condition (14). Indeed, for the intense lighting (LED) and low temperature (T19) conditions, the standard deviation obtained for $\hat{b}_1$ is larger than its value and $\hat{b}_2$ is much larger than $\hat{b}_1^2$. This suggests that under these two forms of stress, the lateral branching dominated over the apical branching in the construction of the mycelial network. The high variability of $\hat{b}_1$ and the large discrepancy between the two estimators $\tilde{\lambda}$ observed in the LED and T19 conditions also suggest that our inference method cannot be used reliably in these cases. Nevertheless, we carried out the entire procedure under these two growth conditions also, to illustrate its behaviour in cases where there are no theoretical guarantees, refraining from drawing biological conclusions on the basis of the parameter values reconstructed in these cases (LED and T19).

Based on the whole sequence of panoramas, it is possible to compute the observed growth rate $\lambda_{\text{obs}}$ of the number of internal nodes in our data, so that using our estimate $\hat{\lambda}_1$ of $\tilde{\lambda}$, we can obtain an estimate of the elongation speed $v$ as the ratio of $\lambda_{\text{obs}}$ to $\hat{\lambda}_1$, and compute estimates of the absolute branching rates $b_1$ and $b_2$ by multiplying $\hat{b}_1$ and $\hat{b}_2$ by the estimated value of $v$. The resulting values for $\lambda_{\text{obs}}$ and inferred values for $v$, $b_1$ and $b_2$ can be found in Table 4. To
<table><caption>Table 2: Estimates $\hat{b}_1$ and $\hat{b}_2$ of the relative branching rates $\tilde{b}_1$ and $\tilde{b}_2$, and total number of apexes (or open segments) used in the estimation procedure for the different growth conditions described in Table 1. Under the LED and T19 conditions, the variability in the estimation of $\tilde{b}_1$ is very high, and the estimated relative lateral branching rate is very large compared to the other scenarios. This invites us to be cautious with the results obtained in these two cases (in particular, it is easily checked that Condition (14) is not satisfied).</caption><thead><tr><th></th><th>β̂<sub>1</sub> × 10<sup>3</sup> [μm<sup>-1</sup>]</th><th>β̂<sub>2</sub> × 10<sup>6</sup> [μm<sup>-2</sup>]</th><th>Nb apexes</th></tr></thead><tbody><tr><td>M2</td><td>3.31 ± 0.38</td><td>7.03 ± 2.34</td><td>2585</td></tr><tr><td>M0</td><td>4.25 ± 0.38</td><td>4.30 ± 2.34</td><td>2768</td></tr><tr><td>KCl</td><td>4.06 ± 0.50</td><td>5.08 ± 3.13</td><td>1779</td></tr><tr><td>LED</td><td>1.31 ± 1.81</td><td>110 ± 33</td><td>497</td></tr><tr><td>T19</td><td>0.31 ± 0.94</td><td>43 ± 11</td><td>687</td></tr><tr><td>T35</td><td>5.06 ± 0.88</td><td>10.93 ± 8.98</td><td>1576</td></tr></tbody></table>

<table><caption>Table 3: Estimates of the relative growth rate of the number of internal nodes $\tilde{\lambda}$ for the different growth conditions described in Table 1, obtained using the two estimators $\tilde{\lambda}_1$ and $\tilde{\lambda}_2$ respectively defined in (15) and (16).</caption><thead><tr><th></th><th>λ̂<sub>1</sub> × 10<sup>3</sup> [μm<sup>-1</sup>]</th><th>λ̂<sub>2</sub> × 10<sup>3</sup> [μm<sup>-1</sup>]</th></tr></thead><tbody><tr><td>M2</td><td>4.75 ± 0.13</td><td>4.88 ± 0.34</td></tr><tr><td>M0</td><td>5.06 ± 0.13</td><td>4.50 ± 0.24</td></tr><tr><td>KCl</td><td>4.69 ± 0.19</td><td>4.88 ± 0.27</td></tr><tr><td>LED</td><td>11 ± 0.81</td><td>6.3 ± 0.75</td></tr><tr><td>T19</td><td>6.9 ± 0.49</td><td>4.2 ± 0.04</td></tr><tr><td>T35</td><td>6.88 ± 0.44</td><td>5.63 ± 0.21</td></tr></tbody></table>

account for the variability within and among experiments, the uncertainties on $\lambda_{\text{obs}}$ displayed in Table 4 are obtained by assuming Poissonian uncertainties on the count variables within each growth experiment, and then by taking the maximum between the standard deviation of the set of values of $\lambda_{\text{obs}}$ obtained in the different experiments, and the square root of the sum of the within-experiment variances (obtained by Poissonization) divided by the number of experiments.

## 3.2 Robustness of the estimates

**Validity of the large time assumption.** For every growth condition in which Condition (14) is satisfied, the two estimators $\tilde{\lambda}_1$ and $\tilde{\lambda}_2$ for the exponential growth rate $\tilde{\lambda}$, defined respectively in (15) and (16), yield similar results. This strongly suggests that the time span of the exper-

<table><caption>Table 4: Observed growth rate $\lambda_{\text{obs}}$ of the number of internal nodes, estimate of the elongation speed $v$ obtained as the ratio of $\lambda_{\text{obs}}$ to $\tilde{\lambda}_1$, and estimates of the absolute branching rates $b_1$ and $b_2$ obtained by multiplying $\hat{b}_1$ and $\hat{b}_2$ by the estimated value of $v$.</caption><thead><tr><th></th><th>λ<sub>obs</sub> [h<sup>-1</sup>]</th><th>v [μm.h<sup>-1</sup>]</th><th>b<sub>1</sub> [h<sup>-1</sup>]</th><th>b<sub>2</sub> × 10<sup>4</sup> [μm<sup>-1</sup>.h<sup>-1</sup>]</th></tr></thead><tbody><tr><td>M2</td><td>0.39 ± 0.05</td><td>82 ± 10</td><td>0.27 ± 0.05</td><td>6 ± 2</td></tr><tr><td>M0</td><td>0.31 ± 0.02</td><td>61 ± 5</td><td>0.26 ± 0.03</td><td>3 ± 1</td></tr><tr><td>KCl</td><td>0.28 ± 0.02</td><td>59 ± 6</td><td>0.24 ± 0.04</td><td>3 ± 2</td></tr><tr><td>LED</td><td>0.22 ± 0.01</td><td>20 ± 2</td><td>0.03 ± 0.04</td><td>22 ± 7</td></tr><tr><td>T19</td><td>0.25 ± 0.02</td><td>36 ± 3</td><td>0.01 ± 0.03</td><td>16 ± 4</td></tr><tr><td>T35</td><td>0.37 ± 0.15</td><td>54 ± 22</td><td>0.27 ± 0.12</td><td>6 ± 5</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Figure 5: Mapping of the function ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) for different values of $r$ ranging from 0 to 0.7, obtained using the pooled data corresponding to the 6 experiments performed under the standard growth conditions M2.</figcaption></figure>

iments may indeed be sufficient to estimate the growth parameters. To further verify that the growth parameters obtained correspond to the observed data, we simulated the process using the estimates of $\tilde{b}_1$ and $\tilde{b}_2$, and we compared the simulated length distribution of open and closed segments to the experimental distributions (see Section 4.2.3). The fit obtained is indeed very good for nearly all growth conditions.

**Choice of r.** Recall that to obtain these estimates, we have only kept the apexes that are at a distance at least $r\Delta$ from the centre of the mycelium. The segments at the centre of the mycelium tend to be smaller so that, when taking these out from the data, the tail distribution $\bar{F}$ increases for small values of $\ell$ while remaining similar for larger values of $\ell$. This means that as the radius $r$ increases, the slope of the function ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) will increase (and so will $b_2$), and its intercept will decrease. Figure 5 shows an illustration of these concepts based on the data corresponding to the M2 growth conditions. The intercept is equal to $\tilde{\lambda} + \tilde{b}_1$, and so the increase in $\tilde{\lambda}$ as $\tilde{b}_2$ increases implies that $\tilde{b}_1$ decreases. This is indeed what we observe when estimating $\tilde{b}_1$ and $\tilde{b}_2$ for different choices of $r$ for the M0 growth conditions (see Table 5). Choosing the smallest $r$ for which the function ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) is approximately linear with a positive slope on a large interval of lengths allows us to take out some of the segments that appear shorter than they are due to anastomosis or crossing while keeping enough segments to obtain satisfactory estimates of $\tilde{b}_1$ and $\tilde{b}_2$ and also have a large enough interval over which to match the functions.

<table><caption>Table 5: Estimates $\hat{b}_1$ and $\hat{b}_2$ of $\tilde{b}_1$ and $\tilde{b}_2$ obtained for different choices of $r$, based on the pooled data corresponding to the 3 experiments performed under the M0 growth conditions.</caption><thead><tr><th>r</th><th>hat_1 &times; 10<sup>3</sup> [&mu;m<sup>-1</sup>]</th><th>hat_2 &times; 10<sup>6</sup> [&mu;m<sup>-2</sup>]</th><th>Nb apexes</th></tr></thead><tbody><tr><td>0</td><td>4.25 &plusmn; 0.38</td><td>4.30 &plusmn; 2.34</td><td>2768</td></tr><tr><td>0.1</td><td>4.19 &plusmn; 0.38</td><td>4.30 &plusmn; 2.34</td><td>2693</td></tr><tr><td>0.2</td><td>4.13 &plusmn; 0.38</td><td>3.91 &plusmn; 2.34</td><td>2512</td></tr><tr><td>0.3</td><td>4.06 &plusmn; 0.44</td><td>3.91 &plusmn; 2.73</td><td>2216</td></tr><tr><td>0.4</td><td>3.81 &plusmn; 0.38</td><td>4.30 &plusmn; 2.34</td><td>1861</td></tr><tr><td>0.5</td><td>3.06 &plusmn; 0.50</td><td>7.03 &plusmn; 2.73</td><td>1409</td></tr><tr><td>0.6</td><td>2.31 &plusmn; 0.50</td><td>10.94 &plusmn; 3.13</td><td>939</td></tr><tr><td>0.7</td><td>1.69 &plusmn; 0.63</td><td>11.72 &plusmn; 3.91</td><td>523</td></tr></tbody></table>
<table><caption>Table 6: Estimates $\hat{b}_1$ and $\hat{b}_2$ of $\tilde{b}_1$ and $\tilde{b}_2$ obtained with $r = 0$ and total number of apexes (or open segments) used in the estimation procedure for conditions M2 and KCl. These values should be compared to the estimates displayed in Table 2, obtained using $r = 0.6$ for M2 and $r = 0.4$ for KCl.</caption><thead><tr><th></th><th>β̂<sub>1</sub> × 10<sup>3</sup> [μm<sup>-1</sup>]</th><th>β̂<sub>2</sub> × 10<sup>6</sup> [μm<sup>-2</sup>]</th><th>Nb apexes</th></tr></thead><tbody><tr><td>M2</td><td>5.44 ± 0.58</td><td>6.25 ± 5.86</td><td>7939</td></tr><tr><td>KCl</td><td>4.25 ± 1.0</td><td>1.41 ± 1.13</td><td>2739</td></tr></tbody></table>

For conditions M2, M0 and KCl, we choose $r$ equal to 0.6, 0 and 0.4 respectively. For conditions LED, T19 and T35, the total number of segments at the observation time $T$ is too small to allow us to ignore data at the centre of the network, and so we have to take $r = 0$ and match the two functions on a smaller interval. In fact, taking $r = 0$ for conditions M2 and KCl and matching the two functions on the small interval on which $(\ell \mapsto -\log(\bar{F}(\ell))/\ell)$ is linear with a positive slope leads to estimates of $\tilde{b}_1$ and $\tilde{b}_2$ which are of the same order of magnitude as the ones obtained with $r = 0.6$ and $r = 0.4$ respectively. Matching the two functions on a smaller interval will, however, result in a larger standard deviation for the estimates (see Table 6).

**Choice of the interval** [$L_{\min}$, $L_{\max}$]. Once $r$ has been chosen, we choose the interval of lengths [$L_{\min}$, $L_{\max}$] over which to match the empirical distribution and the stationary profile as the largest interval over which $(\ell \mapsto -\log(\bar{F}(\ell))/\ell)$ is indeed linear and with a positive slope. If $r$ has been chosen such that $(\ell \mapsto -\log(\bar{F}(\ell))/\ell)$ is linear with a positive slope on a large interval, we observe empirically that the precise choice of this interval does not have a clear impact on the estimates of $b_1$ and $b_2$ (data not shown). If $r$ is chosen smaller, so that we have to match the two functions on a smaller interval, the choice of the interval will have a larger impact on the estimates.

# 4 Validation of the static inference approach through a comparison with dynamically collected and simulated data

In order to validate our approach based on a single panorama of the mycelial network at a large time $T$ coupled with an estimate of the growth rate of the number of internal nodes in this network, we use two types of comparisons. First, we compare the estimates of $b_1$, $b_2$ and $v$ displayed in Table 4 to measures obtained via the dynamical reconstruction of the network growth with individual apex tracking expounded in Section 2.1.2. Next, we use data simulated from the stochastic growth-fragmentation model to investigate the potential issues discussed in Section 2.3.2.

## 4.1 Estimates based on dynamical apex tracking data

As explained in Section 2.1.2, the reconstruction of the mycelium network as a dynamical graph allows us to estimate $b_1$, $b_2$ and $v$ directly from the observed dynamics. The means and standard deviations of the numerical values of $v^{\text{exp}}$, $b_1^{\text{exp}}$ and $b_2^{\text{exp}}$, obtained by least square fits, are shown in Table 7 for the different growth conditions.

In this approach, data from different experiments cannot be pooled and the measurements of $v^{\text{exp}}$, $b_1^{\text{exp}}$ and $b_2^{\text{exp}}$ are done for each of the three or six growth experiments corresponding a given set of conditions separately. In Figure 6, the resulting values are compared to the estimates obtained by the static inference method and displayed in Table 4, with standard deviations obtained by considering a Poissonian uncertainty on each count data. Overall, the agreement between the values of $v$, $b_1$ and $b_2$ reconstructed thanks to the two methods is very satisfactory in all conditions. The elongation speeds $v$ are underestimated by the static method in all cases. This can be explained by the fact that anastomoses are not included in the model.
<table><caption>Table 7: Mean ± standard deviation experimental measurements of $v^{\text{exp}}$, $b_1^{\text{exp}}$ and $b_2^{\text{exp}}$ for the different growth conditions.</caption><thead><tr><th>Condition</th><th><i>v</i><sup>exp</sup> [µm.h<sup>-1</sup>]</th><th><i>b</i><sub>1</sub><sup>exp</sup> [h<sup>-1</sup>]</th><th><i>b</i><sub>2</sub><sup>exp</sup> × 10<sup>4</sup> [µm<sup>-1</sup>.h<sup>-1</sup>]</th></tr></thead><tbody><tr><td>M2</td><td>122 ± 11</td><td>0.28 ± 0.05</td><td>7 ± 2</td></tr><tr><td>M0</td><td>102 ± 2</td><td>0.25 ± 0.04</td><td>3.5 ± 0.8</td></tr><tr><td>KCl</td><td>97 ± 11</td><td>0.20 ± 0.03</td><td>6.0 ± 0.8</td></tr><tr><td>LED</td><td>34 ± 1</td><td>0.09 ± 0.04</td><td>10 ± 2</td></tr><tr><td>T19</td><td>73 ± 16</td><td>0.13 ± 0.04</td><td>5 ± 1</td></tr><tr><td>T35</td><td>68 ± 6</td><td>0.20 ± 0.07</td><td>8 ± 3</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Figure 6: Experimental measurements of $v^{\text{exp}}$, $b_1^{\text{exp}}$ and $b_2^{\text{exp}}$ (squares) and comparison with the estimates of $v$, $b_1$ and $b_2$ obtained by the static inference method (triangles, see Table 4) for different growth conditions. Standard deviations per growth condition are indicated by vertical lines. Recall that Condition (14) is not satisfied under the LED and T19 conditions, and so there are no theoretical guarantees on the estimates obtained with the static inference method.</figcaption></figure>
Without these fusions, the filaments never stop growing, which logically implies a lower elongation speed per apex for the same level of total growth. Nevertheless, estimates of the apical and lateral branching rates are consistent across conditions and the reconstructed elongation speeds obtained with the two methods have similar orders of magnitudes.

## 4.2 Simulation of the stochastic growth-fragmentation process

### 4.2.1 Simulation algorithm

The mycelium corresponds to a collection of pairs $(\epsilon^u, x_t^u) \in S$ encoding each segment. At each step of the simulation, we randomly draw the waiting time to the next branching event and the index of the branching segment, which is then replaced by three new segments, as described in the model. Between two branching events, the lengths of the open segments $(\epsilon = 1)$ grow at speed $v = 1$ and the lengths of the closed segments $(\epsilon = 0)$ remain the same.

Consider some initial set of segments containing $A_0$ apexes and having a total length $L_0$. Denote the instants of the first $k$ branching events by $T_1, \dots, T_k$ and suppose that we have simulated the process up to time $T_k$. Let $A_{T_k}$ be the number of open segments at time $T_k$ and $L_{T_k}$ be the total length of the mycelium at time $T_k$. The next branching event will occur at time $T_{k+1} = T_{k+1}^a \wedge T_{k+1}^\ell$ where $T_{k+1}^a$ is the instant of the first apical branching event after $T_k$ and $T_{k+1}^\ell$ is the instant of the first lateral branching event after $T_k$. Each open segment branches apically at rate $b_1$ so that the whole mycelium will branch apically at rate $A_{T_k}b_1$, and the waiting time $T_{k+1}^a - T_k$ follows an exponential distribution with parameter $A_{T_k}b_1$. Each segment branches laterally at instantaneous rate $b_2x_t$ so that the whole mycelium branches laterally at instantaneous rate $b_2L_t$, where $L_t = L_{T_k} + A_{T_k}(t - T_k)$. We thus have that the cumulative distribution function (cdf) of the waiting time $T_{k+1}^\ell - T_k$ is equal to

$$ \mathbb{P} (T_{k+1}^{\ell} - T_k \le t) = 1 - \exp \left( -b_2 \int_{T_k}^{T_k+t} L_s ds \right) = 1 - \exp \left( -b_2 \left( L_{T_k}t + A_{T_k} \frac{t^2}{2} \right) \right). $$

We can then randomly draw the waiting time $T_{k+1}^\ell - T_k$ by using the inverse-cdf method, that is to say by drawing $u$ uniformly on $(0, 1)$ and computing the smallest time $t$ for which

$$ \mathbb{P} (T_{k+1}^{\ell} - T_k \le t) = 1 - \exp \left( -b_2 \left( L_{T_k}t + A_{T_k} \frac{t^2}{2} \right) \right) = u. $$

We have

$$ \frac{b_2 A_{T_k} t^2}{2} + b_2 L_{T_k} t + \log(1-u) = 0, $$

and $t$ is the positive solution to this second order polynomial equation, equal to

$$ -\frac{L_{T_k}}{A_{T_k}} + \frac{1}{b_2 A_{T_k}} \sqrt{b_2^2 L_{T_k}^2 - 2b_2 A_{T_k} \log(1-u)}. $$

Once $T_{k+1}$ has been drawn, we can then update the lengths of the segments up to time $T_{k+1}$ and compute $A_{T_{k+1}} = A_{T_k} + 1$ and $L_{T_{k+1}} = L_{T_k} + A_{T_k}(T_{k+1} - T_k)$. If $T_{k+1}^a \le T_{k+1}^\ell$, then apical branching will occur at time $T_{k+1}$ and the index of the open segment that branches apically is chosen uniformly at random (since each open segment branches apically at the same rate $b_1$). If $T_{k+1}^\ell \le T_{k+1}^a$, then lateral branching will occur at time $T_{k+1}$ and the segment of index $u$ will branch with probability $x_{T_{k+1}}^u / L_{T_{k+1}}$. We can then iterate this step until we have the desired number of segments or until we reach some stopping time $T$.

### 4.2.2 Estimation based on simulated data

We want to check that our estimators give satisfactory results based on data simulated from our model and obtained from the state of our process at some time $T$ not too large. Recall
that to obtain reliable estimates of $\tilde{b}_1$ and $\tilde{b}_2$, $\tilde{b}_1$ and $\sqrt{\tilde{b}_2}$ need to be of the same order. In the simulation procedure described in Section 4.2.1, the elongation speed $v$ is set to 1, and so we have $\tilde{b}_1 = b_1$ and $\tilde{b}_2 = b_2$. Let $b_1 \ge 0$ and $b_2 \asymp b_1^2$. If we scale all lengths by a factor $b_1$, elongation now happens at speed $b_1$ and lateral branching occurs at rate $b_2/b_1$ and if we scale time by a factor $1/b_1$ so that elongation happens once again at speed 1, apical branching now occurs at rate $1/b_2$ and lateral branching at rate $b_2/b_1^2 \asymp 1$. This means that for different values of the parameters such that $b_2 \asymp b_1^2$ the number of apexes needed to obtain satisfactory estimates and the relative error on the estimates of the branching rates is the same as in the situation where the apical branching rate is 1 and the lateral branching rate is $b_2/b_1^2$.

We now simulate the process for $b_1 = 1$ and $b_2 = 1$ and estimate $b_1$ and $b_2$ based on the simulated data. In our experimental data, the number of apexes per experiment ranges from about 150 (for intense lighting) to about 1500 (for the standard condition) and we have three replicas of each experimental condition except for standard condition M2 (for which we have six), so that the total number of apexes ranges from about 500 to about 8000. Note that these figures differ from those displayed in Table 4 as in the latter, only the apexes being at distance at least $r\Delta$ from the network centre are counted. We test our inference method for different numbers of apexes per simulations and using data from different numbers of simulations. The number of apexes $N_{apex}$ ranges from 10 to 1000 and the number of simulations $N_{sim}$ is taken so that the total number of open segments $N_{total} = N_{apex} \times N_{sim}$ used to estimate $b_1$ and $b_2$ range from 500 to 10000. For each pair $(N_{apex}, N_{sim})$, we do 100 repetitions and use the same interval of lengths to estimate $b_1$ and $b_2$. The estimates of $b_1$, $b_2$ and of $\lambda$ using (16) for the different combinations $(N_{apex}, N_{sim})$ can be found in Table 8. We find that even for low numbers of apexes per simulation, we can estimate the parameters $b_1$ and $b_2$ if the total number of apexes (pooled across all replicates) used for the estimation is sufficient. We also find that the error on the estimates depends mostly on the total number of open segments used for the estimation, rather than on the number of apexes in each simulation.

### 4.2.3 Comparison with the experimental data

We now use the estimates of the parameters we have obtained to verify that the simulated length distributions of open and closed segments, obtained at a time $T$ approximately equal to the duration of the growth experiments, are indeed close to the experimental ones. In order for elongation to happen at speed 1 as in the simulated process, we need to scale time by a factor $1/v$ and consider the process $(Z_{t/v})_{t \ge 0}$, so that apical branching occurs at rate $\tilde{b}_1$ and lateral branching at rate $\tilde{b}_2$. Therefore, to compare the simulations to the experiments, we need to simulate the process with parameter $\hat{b}_1$ and $\hat{b}_2$ up to time $vT$. We can then compute the empirical tail distributions in length of open and closed segments $\bar{F}_1$ and $\bar{F}_0$ for the simulated process and compare them with the experimental empirical tail distributions. For all four growth conditions for which Condition 14 holds and we have estimates of $\tilde{b}_1$ and $\tilde{b}_2$, the empirical tail distributions in length of open segments for the simulated processes and for the experimental data are indeed close on the interval over which we have matched $(\ell \mapsto -\log(\bar{F}(\ell))/\ell)$ and $(\ell \mapsto \tilde{b}_1 + \tilde{\lambda} + \tilde{b}_2\ell/2)$ (see Figure 7). The fit is also reasonably good for the distribution in lengths of closed segments, apart from the T35 conditions, for which the empirical tail distribution for the simulated process and for the experimental data differ considerably over a large part of the range of possible values for $\ell$ (see Figure 8). This may be due to the fact that we have chosen $r=0$ to estimate the parameters, and by keeping the segments present in the denser part of the network, anastomosis and crossings may have a strong impact on the empirical tail distribution obtained from the experimental data.
<table><caption>Table 8: Estimates of $b_1$, $b_2$ and $\lambda$ (computed using (16)) for different numbers $N_{apex}$ of apexes per simulation and different numbers $N_{sim}$ of simulations. The total number of apexes used to compute the estimates is given by $N_{total} = N_{apex} \times N_{sim}$. True parameter values are $b_1 = 1 = b_2$ and $\lambda = (1 + \sqrt{5})/2 \approx 1.62$.</caption><thead><tr><th>N<sub>apex</sub></th><th>N<sub>sim</sub></th><th>N<sub>total</sub></th><th>b<sub>1</sub></th><th>b<sub>2</sub></th><th>&lambda;</th></tr></thead><tbody><tr><td>10</td><td>50</td><td>500</td><td>0.83 &plusmn; 0.31</td><td>1.24 &plusmn; 0.70</td><td>2.06 &plusmn; 0.09</td></tr><tr><td>50</td><td>10</td><td>500</td><td>0.95 &plusmn; 0.27</td><td>1.10 &plusmn; 0.64</td><td>1.67 &plusmn; 0.06</td></tr><tr><td>100</td><td>5</td><td>500</td><td>0.99 &plusmn; 0.29</td><td>1.04 &plusmn; 0.64</td><td>1.64 &plusmn; 0.05</td></tr><tr><td>500</td><td>1</td><td>500</td><td>1.16 &plusmn; 0.32</td><td>1.16 &plusmn; 0.71</td><td>1.62 &plusmn; 0.05</td></tr><tr><td>10</td><td>100</td><td>1000</td><td>0.87 &plusmn; 0.20</td><td>1.13 &plusmn; 0.45</td><td>2.05 &plusmn; 0.05</td></tr><tr><td>50</td><td>20</td><td>1000</td><td>0.99 &plusmn; 0.20</td><td>1.02 &plusmn; 0.46</td><td>1.68 &plusmn; 0.04</td></tr><tr><td>100</td><td>10</td><td>1000</td><td>0.99 &plusmn; 0.21</td><td>1.02 &plusmn; 0.46</td><td>1.66 &plusmn; 0.04</td></tr><tr><td>500</td><td>2</td><td>1000</td><td>1.00 &plusmn; 0.21</td><td>1.03 &plusmn; 0.47</td><td>1.64 &plusmn; 0.04</td></tr><tr><td>1000</td><td>1</td><td>1000</td><td>0.95 &plusmn; 0.21</td><td>1.14 &plusmn; 0.46</td><td>1.62 &plusmn; 0.04</td></tr><tr><td>10</td><td>200</td><td>2000</td><td>0.87 &plusmn; 0.15</td><td>1.12 &plusmn; 0.33</td><td>2.05 &plusmn; 0.04</td></tr><tr><td>50</td><td>40</td><td>2000</td><td>1.00 &plusmn; 0.16</td><td>0.94 &plusmn; 0.32</td><td>1.68 &plusmn; 0.03</td></tr><tr><td>100</td><td>20</td><td>2000</td><td>0.97 &plusmn; 0.13</td><td>1.08 &plusmn; 0.33</td><td>1.65 &plusmn; 0.03</td></tr><tr><td>500</td><td>4</td><td>2000</td><td>0.98 &plusmn; 0.13</td><td>1.03 &plusmn; 0.30</td><td>1.63 &plusmn; 0.03</td></tr><tr><td>1000</td><td>2</td><td>2000</td><td>1.01 &plusmn; 0.14</td><td>0.99 &plusmn; 0.31</td><td>1.62 &plusmn; 0.03</td></tr><tr><td>2000</td><td>1</td><td>2000</td><td>0.99 &plusmn; 0.16</td><td>1.02 &plusmn; 0.35</td><td>1.62 &plusmn; 0.03</td></tr><tr><td>10</td><td>500</td><td>5000</td><td>0.86 &plusmn; 0.09</td><td>1.14 &plusmn; 0.22</td><td>2.05 &plusmn; 0.03</td></tr><tr><td>50</td><td>100</td><td>5000</td><td>1.00 &plusmn; 0.09</td><td>0.97 &plusmn; 0.19</td><td>1.68 &plusmn; 0.02</td></tr><tr><td>100</td><td>50</td><td>5000</td><td>1.00 &plusmn; 0.10</td><td>0.96 &plusmn; 0.20</td><td>1.65 &plusmn; 0.02</td></tr><tr><td>500</td><td>10</td><td>5000</td><td>0.99 &plusmn; 0.10</td><td>1.02 &plusmn; 0.21</td><td>1.62 &plusmn; 0.02</td></tr><tr><td>1000</td><td>5</td><td>5000</td><td>0.99 &plusmn; 0.10</td><td>1.03 &plusmn; 0.22</td><td>1.62 &plusmn; 0.02</td></tr><tr><td>10</td><td>1000</td><td>10000</td><td>0.86 &plusmn; 0.07</td><td>1.12 &plusmn; 0.15</td><td>2.05 &plusmn; 0.02</td></tr><tr><td>50</td><td>200</td><td>10000</td><td>1.00 &plusmn; 0.07</td><td>0.96 &plusmn; 0.15</td><td>1.68 &plusmn; 0.01</td></tr><tr><td>100</td><td>100</td><td>10000</td><td>0.99 &plusmn; 0.06</td><td>1.00 &plusmn; 0.13</td><td>1.65 &plusmn; 0.01</td></tr><tr><td>500</td><td>20</td><td>10000</td><td>1.00 &plusmn; 0.07</td><td>1.00 &plusmn; 0.15</td><td>1.62 &plusmn; 0.01</td></tr></tbody></table>

# 5 Discussion

In this work, we aimed at quantifying the growth dynamics of the mycelial network and the impact of different forms of stress using data with minimal complexity. To this effect, a relatively simple growth-fragmentation model was introduced in [25], in which there are three parameters: an elongation speed $v$, an apical branching rate $b_1$ and a lateral branching rate per unit length $b_2$. An explicit expression for the distribution in length of open segments was obtained, which we have used to estimate the relative branching rates $\tilde{b}_1 = b_1/v$ and $\tilde{b}_2 = b_2/v$ based on data from a single panorama of the state of the mycelium at some time $T$ large enough. We have then used these estimates to obtain an estimate of the relative exponential growth rate $\tilde{\lambda} = \lambda/v$, which is a known function of $\tilde{b}_1$ and $\tilde{b}_2$. Additionally, using an estimate of the growth rate of the total number of internal nodes in the network $\lambda_{obs}$, we have obtained an estimate of the elongation speed as the ratio of $\lambda_{obs}$ to $\tilde{\lambda}$, which then enabled us to infer the absolute branching rates $b_1$ and $b_2$. By simulating the process, we have determined the number of segments needed for the method to yield satisfactory results. Based on this investigation, we have verified that the time $T$ at which the panoramas we use in the subsequent analysis were taken was large enough by verifying that the number of segments it contained was sufficient (as it is an exponentially growing function of $T$). Next, to assess the robustness of the estimates we have obtained, we have compared them to estimates obtained via the dynamical tracking of individual apexes and have found a very satisfactory overall agreement between the estimates. We have also simulated the process using the estimated parameters and compared the tail distributions in lengths of
<figure><img src="image_8.png" /><figcaption>Figure 7: Mapping of the function ($\ell \mapsto -\log(\bar{F}_1(\ell))/\ell$), where $\bar{F}_1$ is the empirical tail distribution in lengths of open segments, for simulated data (orange curve) obtained using the estimates $\hat{b}_1$ and $\hat{b}_2$ and experimental data (blue curve), and mapping of ($\ell \mapsto \tilde{b}_1 + \lambda_1 + \tilde{b}_2\ell/2$) (grey curve) for different growth conditions.</figcaption></figure>

open and closed segments for the experimental and the simulated data and have also found a good fit between the simulated and empirical distributions for most of the growth conditions. This method may thus provide estimates of branching rates and elongation speed for networks for which the dynamical tracking of the growth may not be feasible over long periods of time. Interesting examples for which data are available are pathogenic forms of the yeast *Candida albicans* [18], the gastrovascular canal network in *Aurelia* jellyfish [24], and *Arabidopsis* root systems [12].

We have also shown that the model introduced in [25], despite its simplicity, is sufficient to capture and quantify the exponential growth of the mycelium and the impact of several forms of stress on the elongation and branching dynamics. Moreover, this can be done using only a picture of the state of the network at a single large time $T$ and an estimate of the growth rate of the total number of internal nodes in the network. However, the simplicity of the model has also led to various issues when implementing the inference method.

To start with, to obtain satisfactory joint estimates of $\tilde{b}_1$ and $\tilde{b}_2$, we have found that $\tilde{b}_1$ and $\sqrt{\tilde{b}_2}$ need to be of the same order of magnitude, which means that apical and lateral branching need to occur at similar rates. This was the case for four out of the six growth conditions considered in this work. However, we have also seen that if the condition $\tilde{b}_1 \simeq \sqrt{\tilde{b}_2}$ was violated, we could still obtain some information on the impact of stress on the growth of the mycelium, as we can then deduce which of the two types of branching is favoured by the mycelium, giving us some insights into how the mycelium adapts its growth. We can also extend the method to the case where $b_1 = 0$ or $b_2 = 0$, that is to say, where only one type of branching occurs, by matching ($\ell \mapsto -\log(\bar{F}(\ell))/\ell$) to either ($\ell \mapsto \tilde{b}_2 + \tilde{b}_2\ell/2$) or ($\ell \mapsto 2\tilde{b}_1$) instead of ($\ell \mapsto \tilde{\lambda} + \tilde{b}_1 + \tilde{b}_2\ell/2$). In
<figure><img src="image_9.png" /><figcaption>Figure 8: Mapping of the function ($\ell \mapsto -\log(\bar{F}_0(\ell))/\ell$), where $\bar{F}_0$ is the empirical tail distribution in lengths of closed segments, for simulated data (orange curve) obtained using the estimates $\hat{b}_1$ and $\hat{b}_2$ and experimental data (blue curve), and mapping of ($\ell \mapsto -\log(N_0(\ell))/\ell$ (grey curve) for different growth conditions.</figcaption></figure>

the case where one of the two types of branching occurs at a much higher rate than the other, this method might allow us to quantify the impact of different forms of stress on the elongation speed and on the rate at which the dominant type of branching occurs. However, it will give no information on the variation of the rate of the other type of branching.

When implementing the inference method, most of the difficulties stemmed from the fact that anastomosis was not taken into account in the model we have used. In order to minimise the effect of anastomosis on our data, we have excluded the data from the central part of the mycelial network, which rapidly becomes very dense so that segments are more likely to cross and anastomosis is more likely to occur. We have thus only considered open segments with an apex which is at distance at least $r\Delta \ge 0$ from the centre of the mycelium, where $\Delta$ is the radius of the mycelium and $r \in [0, 1]$ is a proportion which was determined empirically for each growth condition considered.

As a consequence, we have less data to infer the branching rates, so that we have had to pool data from several experiment under the same growth conditions, to make sure that the number of segments used is sufficient to have a satisfactory approximation of the length distribution of open segments. However, in cases where taking out the data from the denser part at the centre of the mycelium does not leave enough data even after pooling, we have noticed that, taking $r = 0$, we can still obtain satisfactory estimates of the branching rates, as we still had a satisfactory agreement with the estimates obtained via the dynamical tracking of apexes. Furthermore, taking $r = 0$ for the other growth conditions, we find that the estimates obtained remain of a similar order of magnitude, albeit with larger standard deviations. Nevertheless, when comparing the experimental data to data simulated using the estimated parameters, we
have found that the empirical tail distributions in lengths of closed segments for the simulated
process and for the experimental data differ considerably for the smaller lengths, which may
be due to the effects of anastomosis and crossings on the empirical tail distribution for the
experimental data.

Choosing $r > 0$ also distorts the length distribution of the open segments. Indeed, the
segments at the centre of the mycelium tend to be smaller. As a consequence, even though
taking out the apexes at distance less than $r\Delta$ from the centre of the mycelium allows us to
take out some of the segments that appear shorter than they are due to anastomosis or crossing,
we also disproportionately take out smaller segments from the data. We have found that as the
proportion $r$ increases, the estimate of $\tilde{b}_1$ decreases while the estimate of $\tilde{b}_2$ increases (both of
them keeping the same orders of magnitude). The impact of the choice of $r$ could not be precisely
quantified and differed between growth conditions, and so it remains difficult to compare the
parameter estimates obtained under different conditions and to draw precise conclusions on how
the mycelium has adapted its growth. We have found, however, that while the estimates remain
of the same order of magnitude for the four growth condition for which the condition $\tilde{b}_1 \simeq \sqrt{\tilde{b}_2}$ is
satisfied, we not only have a good overall agreement with the estimates obtained via dynamical
tracking but we also have, for the different forms of stress, similar ratios of the estimates to the
ones obtained for the standard condition.

The reconstruction algorithm described in [4] allows us to distinguish between branching,
overlap and anastomosis so that we could take out only the segments that have been affected
by anastomosis, which would limit the distortion of the distribution in length of open segments
resulting from the choice of $r$. However, this algorithm requires tracking the dynamics of indi-
vidual apexes, so that the data needed would no longer be of minimal complexity. It should also
be noted that, in the absence of anastomosis, the segments that have been taken out would have
continued growing and branching so that, especially in the case where anastomosis is frequent,
the distribution might still be distorted from the theoretical distribution.

Anastomosis being neglected in the model used in the inference method comes from the fact
that the model does not take into account the spatial structure of the mycelium so that the
crossing and potential subsequent merging of the hyphae cannot be modelled. Thus, a first step
towards an inference method taking anastomosis into account would be to introduce spatially
explicit models for mycelial growth. This has been done in [2] and [17]. Further study of these
models is needed in order to obtain a method allowing us to infer the different growth parameters
in these much more complex models.

**Code availability:** The scripts used in the inference procedures and examples of fungal network
data sets are available on GitHub: github.com/amandineveber/FungInfer.

**Acknowledgements:** This research was supported by the French National Research Agency, through the ANR grant NEMATIC (ANR-21-CE40010-01). LK, MT and AV acknowledge partial support from the *chaire Modélisation Mathématique et Biodiversité* of Veolia Environnement-École Polytechnique-Museum National d’Histoire Naturelle-Fondation X. All the authors thank Christophe Lalanne and Clara Ledoux for insightful discussions and for sharing some of the data and scripts, the entire NEMATIC consortium for many inspiring exchanges, the participants to the 2025 Colloquium on “Branching networks: From experiments to theory” in Paris for interesting discussions and presentations, and Jules Depersin and Antoine Caillebotte for their work on earlier parts of the project.

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