Physical principles of morphogenesis in mushrooms

Mushroom species display distinctive morphogenetic features. For example, Amanita muscaria and Mycena chlorophos grow in a similar manner, their caps expanding outward quickly and then turning upward. However, only the latter finally develops a central depression in the cap. Here we use a mathematical approach unraveling the interplay between physics and biology driving the emergence of these two different morphologies. The proposed growth elastic model is solved analytically, mapping their shape evolution over time. Even if biological processes in both species make their caps grow turning upward, different physical factors result in different shapes. In fact, we show how for the relatively tall and big A. muscaria a central depression may be incompatible with the physical need to maintain stability against the wind. In contrast, the relatively short and small M. chlorophos is elastically stable with respect to environmental perturbations; thus, it may physically select a central depression to maximize the cap volume and the spore exposure. This work gives fully explicit analytic solutions highlighting the effect of the growth parameters on the morphological evolution, providing useful insights for novel bio-inspired material design.

Figure 1

The mushroom Amanita muscaria, also known as American fly agaric (yellow variant). Its cap morphological development is shown through (a)–(d): the cap gradually expands outward and becomes flat and finally turns slightly upward. Panels (a)–(d) are reconstructed from the video [4] (time around 2:26, 2:31, 2:34, and 2:37, respectively) .

Figure 2

The mushroom Mycena chlorophos, which is bioluminescent in the darkness, developing a central depression in the cap. For a better illustration of its growth process, the following two sources are used. Panels (a)–(c) are reconstructed from the video [5] (time around 1:16, 1:23, and 1:29, respectively) and (d) is reconstructed from Ref. [6] with permission.

Figure 3

The illustration of the multiplicative decomposition model $\mathbf{F}=\mathbf{A}\mathbf{G}$. The orthogonal basis vectors are shown in each configuration or state (initial configuration ${\mathcal{R}}_0$, virtual state ${\mathcal{R}}_g$, and current configuration ${\mathcal{R}}_t$), and $\mathbf{X}$ and $\mathbf{x}$ are position vectors in ${\mathcal{R}}_0$ and ${\mathcal{R}}_t$, respectively.

Figure 4

(a) Cross section of initial configuration and (b, c) two types of current configurations. (a) An elliptic strip with given geometric parameters $(a,b,h_b,{\beta }_0)$ where $a$ and $b$ are, respectively, the lengths of major semiaxis and minor semiaxis of the inner ellipse, and the vector $\stackrel{⟶}{BA}$ is transported to $\stackrel{⟶}{P_2{P}_1}$ in (b) and (c). (b) and (c) Circular strips with inner radius $r_i$, thickness $h$, and central angle $\beta$, and ${\alpha }^{*}$ is the positive azimuthal angle from the $z$-axis to the $r$-axis and $\gamma$ is the angle between $\stackrel{⟶}{P_2{P}_1}$ and positive $z$-axis.

Figure 5

Illustration of the morphology of A. muscaria in Fig. 1 by applying the analytic solutions (14, 15, 16, 17) together with parameters (21)–(22). Initial parameters are $a=0.78,b=0.54,h_b=0.41,{\beta }_0=0.64\pi , D_1=1.5,D_2=0$, and the growth parameters and position constant are $s=0.77, {\alpha }_0=0, c_2=0$ and for (b)–(d), respectively, $k=0.72,0.10,-0.16, d_1=0.82,1.8,2.4$, and $P_{1z}=6,8,8.5$.

Figure 6

Illustration of the morphology of M. chlorophos in Fig. 2 by applying the analytic solutions (14, 15, 16, 17) together with parameters (21)–(22). The initial parameters are $a=1.12,b=0.86,h_b=0.34,{\beta }_0=0.82\pi , D_1=3.6,D_2=0.1$, and the growth parameters and position constant are $s=0.94, P_{1r}=0.1$, and for (b)–(d), respectively, $\gamma =-{\alpha }_0=0.09\pi ,0.25\pi ,0.037\pi , k=0.93,0.73,-0.19, d_1=0.088,1.0,2.5$, and $P_{1z}=6,10,10.2$.

Figure 7

(a) The $z$-coordinate of the center of gravity as a function of $c_2$, given other parameters as in Fig. 5; (b) The volume $v$ as a function of $\gamma$, given other parameters as in Fig. 6.

Figure 8

Simulation of the growth of a tomato pericarp. (a, b) Respectively, the initial and current configurations. (c) The cross section of (b) and (d) the cross section of a tomato pericarp, which is adapted from Ref. [42].

Figure 9

Simulation of the development of a walnut shell. (a) The initial configuration and (b–e) the current configurations in different growth stages during which both the thickness and inner radius of the shell increase. The graph at the top is adapted from Ref. [43] under CC BY-NC-ND license [44].