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 Dynamics of ''Random''
Expanding networks Analysis,
modeling and simulation of Multi-Scale spatial exploration, spreading and morphogenesis under constraints. 

Studying the constrained hyphal growth in the filamentous fungus
''Podospora anserina''

=== Context ===

How fungi or plants invade a medium, how sexually transmitted diseases
spread over a population, how cancer tumors grow in human bodies, how communication routes densify,
are questions that may seem to refer to quite unrelated problems.
However, the structure, dynamics and shape of the underlying network
may rely on very similar models.

The nature of such networks is not uniquely defined: some examples are informational networks (of relation between individuals, citation graphs,...), technological (power grids, public transportation, computer network,...), or biological (vascular, biochemical, neural network,...). In all the aforementioned examples, transformation arises from individuals, be it the development of a new connection between existing entities, as it often appears in neurons, or the introduction of a new individual in the system.
All these contributions sum up to the evolution of the network as a unit on the macroscopic level.

Modeling of such intricate processes
ranges from simple explanatory toy-models to more realistic
approaches,
which need to be able to capture modifications at different scales.
This can be achieved by linking microscopic objects, which describe individuals, with
their collective mean behavior. Techniques borrowing from statistical physics for the
analysis of nonlinear, non-equilibrium physical systems in the study of such collective
behavior are of increasing use, in e.g. social, economical or biological systems.

The expansion of such networks may also be hindered by internal or
external constraints which can significantly affect the observed results and patterns.
When explicitly including the spatial dimension, the models considered
may provide a pertinent description of the interaction processes at
the small (micro) scale as well as the large (macro)scale featuring
the emerging behavior, possibly under the form of a (thin) propagating
front. The modeling and analysis of such dynamical processes within a
multi-scale framework, where the different granularities of the system are to be considered,
is a complex research field, that requires involving various disciplines.

In the DREAMS project, we want to specifically address the modeling and analysis of the expanding interconnected hyphal
network (the vegetative filaments produced to form the ''mycelium'')
of the fungus ''Podospora anserina''.

{| class="wikitable"
|[[File:Thalle2.png|210px]]
|[[File:imagecentrale2.png|250 px]]
|[[File:Petri2.png|240 px]]
|}
Left: Example of the reconstitution (as reconstructed by numerous pictures juxtaposition and conformal mapping, 112 tiles) of the complete ''thallus'' of ''P. anserina'', grown 23h on a Petri dish. The diameter is approx. 20 mm. Center: small-scale (approx. 1 mm) image processing and vectorization; raw data from experiment (at time t=18h) is superimposed to the output of the vectorization process. Right: observed macroscopic ''mycelium'' (diameter around 8 cm) after a 4 day-growth. Note the expanding front, represented by the hyphal concentration isovalues shown at different times (colored
lines).

=== A toy-model: ''Podospora Anserina'' ===
In real-life conditions, ''P. anserina''
is a coprophilous filamentous ascomycete that grows on herbivore dungs, a highly competitive habitat where several dozens of species are present and feed on partially degraded plant material. The success of the
filamentous fungi group in colonizing most natural environments (from
Antarctic ice to hot deserts and seawater) can be largely attributed
to hyphal growth and branching, allowing an efficient spatial
exploration and exploitation of the nutritive resources.
Some species, especially pathogens, present a finely tuned regulation
between a filamentous growth and a unicellular growth, the latter
property being essential for pathogenicity.

Within the Biology group at LIED Paris-Diderot, ''P. anserina'' is used as an efficient lab model because:
* it is very easy (and cheap!) to grow,
* the complete sexual cycle can be obtained in vitro in seven days, and yields to the production of sexual spores, named ''ascospores'',
* the availability of its genome sequence has enabled the development of several useful tools in molecular and cellular biology, as well as in cytology.

It hence represents a convenient lab-scale (toy)model for studying the development of filamentous fungi, or even more general
living systems networks. The efficient
growth of such filamentous fungi is adapted through a mycelial network, in particular in the presence of external constraints disturbing or impeding the
environmental exploration. Constraints can be of different nature:
e.g. i) chemical/physical like various carbon source, nutrient
deficiency/gradient, temperature gradient, hygrometry, electric
field, presence of a toxic chemical compound, ii) mechanical like the
avoiding of an obstacle or a labyrinthic geometry and also iii)
biological like the presence of another organism or the local deletion
of the hyphal network.

Note that the biological characterization of ''P. anserina
mutants available at LIED, affected in some key steps of their growth or development, is of interest per se, e.g. for the study of cell wall biogenesis, cellular polarization and branching process.

Developing quantitative tools, in collaboration with physicists, allows to determine the growth velocity of hyphae, to analyze the occurrence of branching and to measure hyphal density over time. Also note that the question of scales is indeed of paramount importance: the hypha is a few microns wide (typically 4 to 6), while the mycelial network can operate on scales ranging from a few square cm up to many square km.

=== DREAMS : an interdisciplinary project ===
In this interdisciplinary project, we wish to address the problem of the multi-scale
modeling and analysis of expanding dynamical networks under external
constraints both by analytical/numerical means and feed-backed lab-scale
experimental realizations. The main objectives of our collaboration can be broken down as follows:
* from a biological point of view, we wish to deepen the scientific knowledge of filamentous fungi biology and physiology, which indeed constitutes the main research topic of the B2C group at LIED;
* from a physics point of view, we might wish to try and build the thermodynamic formalism of the metabolism of growth; based on an already on-going collaboration on this topic between LJAD and the Physics group at LIED, we wish to derive from the force-speed relationship of energy conversion machines, such as a muscle, a high-level formalism dedicated to the production of matter and increase in complexity of the thallus;
* from a mathematical point of view, using statistical tools as well as probabilistic and SDE and PDE tools, we wish to build and assess robust and versatile models, analyze their mathematical properties as well as design (and also possibly analyze) adapted efficient numerical methods. We aim at both formal and (possibly) rigorous derivations of the models.

=== Participants ===
 
Yves D'Angelo (Scientific Coordinator), Rémi Catellier, Laurent Monasse, Claire Guerrier , Thierry Goudon (LJAD Nice)
 
Florence Chapeland-Leclerc, Gwenaël Ruprich-Robert, Eva Cabet (B2C Group, LIED)
 
Eric Herbert, Cécilia Bobée, Pascal David, Christophe Lalanne, (Physics Group, LIED)
 Amandine Véber (MAP5, Paris),
 Milica Tomasevic (CMAP, Ecole Polytechnique)
 
 Former participants
 Adélaïde Olivier (Math Lab at Orsay),
 Franco Flandoli (Scuola Normale di Pisa).
 
 PhD Students and Interns
 Clara Ledoux
 Moira Lampe
 Nicolas Fricker
 Sebastian Baudelet
 Thibault Chassereau
 Lena Kuwata
 Aurélien Corrado