Z. Angew. Math. Phys. (2017) 68:92
2017 Springer International Publishing AG
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
On a model for the evolution of morphogens in a growing tissue II: θ =log(2)case
G. M. Coclite and M. M. Coclite
Abstract. We analyze a model for the regulation of growth and patterning in developing tissues by diﬀusing morphogens. We
prove the well-posedness of the underlying systems of nonlinear PDEs. The key tool in the analysis is the transformation
of the underlying system to an equation with singular logarithmic diﬀusion.
Mathematics Subject Classiﬁcation. 35K60, 35Q92, 34B15.
Keywords. Logarithmic diﬀusion, Neumann boundary conditions, Existence, Uniqueness, Stability, Morphogen evolution.
In this paper, we continue the analysis started in  on the well-posedness of a model for morphogen
evolution proposed in . A morphogen is a substance that spreads from a localized source such that
its concentration declines in a continuous and predictable manner, providing a series of concentration
thresholds that control the behavior of surrounding cells as a function of their distance from the source,
see . They are involved in the growth processes of a single-celled embryo into a complex multicellular
organism, see [2,11,14,21]. Moreover, they play a key role in the process of patterning, namely the
diﬀerentiation of a naive ﬁeld of cells into speciﬁc cell types, based on the intra-cellular control and
signaling provided by an external substance [11,21].
One of the main open problems of developmental biology is the regulation of patterning and growth
(cellular proliferation) by morphogens. More precisely, issues like
• how a diﬀusing morphogen concentration can scale with growing tissue,
• how a graded (decaying) morphogen concentration proﬁle can lead to uniform proliferation rates in
a growing tissue,
• what determines the ﬁnite ﬁnal size of a tissue
still need to be completely understood.
Several mathematical models have been proposed to answer the previous questions. Let us brieﬂy
review some of them. The model proposed in  is based on the idea that the cellular proliferation
rate (growth rate) is based on the spatial gradient, and not on the absolute value, of the morphogen
concentration. It can partly explain uniform growth (if the morphogen concentration is approximately
linear) but fails to account for scaling and ﬁnite ﬁnal size of the tissue. Moreover, experiments  show
that a constant (in space) morphogen concentration can also lead to a growing tissue, thus negating this
model. An hybrid model, between growth depending on morphogen gradient in a part of the tissue and
absolute morphogen concentration in another part, has been proposed in . It explains uniform growth
but fails to account for scaling.
G. M. Coclite is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).