Scientists have created a growth model for aquatic plants, which has made it possible to clarify the formation of leaves depending on whether they are lying on water or suspended in the air. As a result, the variety of leaf morphologies was explained theoretically and also confirmed experimentally. The results could be useful in developing deployable biomimetic structures, the authors write in Physical Review Letters.
Many organs and tissues of living organisms acquire complex shapes or colors as they grow. The regularities of the appearance of such features can be described using mathematical models, sometimes without even going into the biological details of the ongoing processes. A classic example of the success of this approach is the work of Alan Turing in 1952, in which a mathematical model was obtained that could reproduce drawings on animal skins.
Another situation from biology, in which tissues of the same internal properties are outwardly different, is the growth of leaves of some plants, such as lilies and lotuses. If the leaves of these species grow on the surface of the water, then they have a flat shape with fine tortuosity around the perimeter. However, in the same plant, some of the leaves can hang over the water, and then they acquire a curved bowl shape with smooth waviness along the edge.
The key work on the growth of soft tissues and the resulting organ shape is the 1994 article, which builds a general mathematical formalism for describing such a problem, it boils down to the simultaneous description of tissue proliferation and the stretches and displacements that occur in it.
Fan Xu and his colleagues at Fudan University completed an earlier model of leaf growth that correctly described the emerging shape. Two key innovations turned out to be taking into account the possible mechanical support from the water surface and the possibility of an inhomogeneous growth rate, due to which, in particular, general leaf bending is possible, since the side opposite to the Sun can grow faster.
Numerical simulations in the new model were able to correctly reproduce the variety of lotus leaf shapes. The scientists also decided to experimentally test the conclusions, for which they made artificial leaves from a material that swells when in contact with water. Experiments with selective wetting of places of the most intensive growth or observation of a change in the shape of a leaf substitute lying on the water showed agreement with theoretical estimates and real plants.
According to theoretical estimates, in all cases, the shape of the leaf is explained by minimizing the energy of the leaf, the edges of which do not grow fast enough in comparison with its main part, due to which the ratio of the perimeter to the area should fall. If the sheet lies on the water, then it actually sticks to it, and the bending of the edge also leads to the rise of water - in such a situation it is energetically beneficial to form a lot of irregularities of small amplitude. At the same time, the growth of a leaf in the air is less limited, which is why large fluctuations are available, which turn out to be less energy-intensive in the absence of water. At the same time, the mechanical properties of the tissue can affect this process: leaves with more rigid veins are bent weaker.
Previously, scientists found that the color of lizards can be described using a von Neumann cellular automaton, and the key factor that determines the size of the leaves is the likelihood of freezing at night.