A geometrical approach explains Lake Ball (Marimo) formations in the green alga, Aegagropila linnaei

An extremely rare alga, Aegagropila linnaei, is known for its beautiful spherical filamentous aggregations called Lake Ball (Marimo). It has long been a mystery in biology as to why this species forms 3D ball-like aggregations. This alga also forms two-dimensional mat-like aggregations. Here we show that forming ball-like aggregations is an adaptive strategy to increase biomass in the extremely limited environments suitable for growth of this alga. We estimate the maximum biomass attained by ball colonies and compare it to that attained by mat colonies. As a result, a ball colony can become larger in areal biomass than the mat colony. In the two large ball colonies studied so far, they actually have larger biomasses than the mat colonies. The uniqueness of Lake Balls in nature seems to be due to the rarity of such environmental conditions. This implies that the conservation of this alga is difficult, but important.


Results
We compare the maximum biomass of the ball and mat colonies to find the optimal solution to maximize the biomass. Fig. 2 shows the comparison between the biomass of a closely-packed ball colony that consists of equal ball-like aggregations with the maximum biomass of the mat colony under the same environmental condition as the ball colony per cm 2 . For a single layer of ball-like aggregations, our analysis shows that the mat colony has a greater biomass if the radius of each ball is under 4.13 cm (Fig. 2a). When the radius increases beyond 4.13 cm, however, the ball colony is predicted to have a greater biomass, and presumably, higher productivity. Note that ball-like aggregations become hollow if the radius is larger than 5 cm. When the number of layers (n) is increased, this threshold radius for greater biomass becomes smaller (Fig. 2b). For example, it is 2.07 cm for two layers and 1.38 cm for three layers. This means that two or more layers are necessary to have a greater biomass for the ball colony that consists of solid ball-like aggregations with much less than 5 cm in radius.
We analyse the field data on size classes of ball-like aggregations, the number of ball-like aggregations in each class and the total area of the colonies in two stable Lake Ball congregations in Churui bay and Kinetanpe bay of Lake Akan 10 . Using these data, we estimated the biomass of a ball colony and the maximum biomass of the mat colony that has the same area under the same environmental condition as   the ball colony. Note that the maximum biomass of a mat colony is always 5 cm 3 /cm 2 because the maximum thickness of aggregations of filaments is 5 cm, as described in the introduction. Fig. 3 compares the biomass between a ball colony and the mat colony. In Churui bay, the total biomass of the ball colony is 10.2 3 10 9 cm 3 and the maximum total biomass of the mat colony is 3.96 3 10 9 cm 3 . Similarly, in Kinetanpe bay, the total biomass of the ball colony is 10.3 3 10 9 cm 3 and the maximum total biomass of the mat colony is 2.28 3 10 9 cm 3 . www.nature.com/scientificreports Therefore, in Churui bay, the biomass of the ball colony per cm 2 is 12.9 cm 3 because the total area of this colony is 7.92 3 10 8 cm 2 , implying an increase of 12.9 2 5 5 7.9 cm 3 /cm 2 by forming ball-like aggregations. Similarly, in Kinetanpe bay, the biomass of the ball colony per cm 2 is 22.6 cm 3 because the total area of this colony is 4.56 3 10 8 cm 2 , implying an increase of 22.6 2 5 5 17.6 cm 3 /cm 2 by forming ball-like aggregations.

Discussion
A. linnaei shows a continuous increase in the biomass of filaments. In many other holocarpic annual algae, all gametophytes and sporophytes disappear once a year after investing all cell materials to produce gametes and zoospores, respectively. In contrast, swimming cells (gametes or zoospores) are rarely produced by this alga, although they have been observed 11,12 . Therefore, this alga does not appear to depend much on sexual reproduction by gametes and asexual reproduction by zoospores in nature. Furthermore, this alga is perennial: the filaments do not disappear throughout the year 8 . A ball colony can have a larger biomass than the mat colony, with the same area under the same environmental condition as the ball colony (Fig. 2). Specifically, closely packed ball colonies always have a larger biomass than the mat colonies when single-layer ball-like aggregations have a radius greater than 4.13 cm (Fig. 2a). When the spheres are not so large, as in the beginning stages of the ball formation, ball colonies can have a larger biomass than the mat colonies by piling up ball-like aggregations in layers (Fig. 2b). Note that small ball-like aggregations with a radius less than 5 cm actually have a much higher density than larger ones (see the legend of Fig. 3). Thus, A. linnaei can efficiently increase biomass by forming ball-like aggregations. We suggest that this is a significant reason why 1) filaments of this alga form ball-like aggregations, 2) the balls become large, and 3) the density of the spheres is remarkably high in nature. Field data support our theory well (Fig. 3). In the stable ball colonies in Lake Akan (Kinetanpe and Churui bays), the biomass of the balllike aggregation is much higher than the maximum biomass of the mat colony. A. linnaei forms ball-like aggregations only in environments suitable for their growth (see also below for more details), and increases the biomass. In worse conditions, however, they remain a mat-like aggregation. It might be because they cannot increase the biomass even if they form ball-like aggregations (Fig. 2).
Lake balls buried under layers of other balls, and filaments in the dark side can also receive light and survive. First, ball-like aggregations are often nearly strictly spherical in shape (e.g. in Churui bay, the mean ratio of the minor axis to the major axis is 0.9) 13 . As they increase the size, they become more strict spheres. Second, lake balls are buoyant because of oxygen bubbles produced by photosynthesis 7 . They sometimes rise even to the surface of the lake 2 . Finally, wave action positively rotates lake balls 2 . Ball-like aggregations are actually formed in suitable places for their photosynthesis, where there are continuous waves (i.e. in shallow waters, see also below for more details). Thus, lake balls frequently rotate and change their relative positions.
The environmental conditions suitable for a ball colony seem to be actually more severe than those for a mat colony. For example, the ball colonies disappeared before mat colonies in Lake Zeller (Zellersee) in Salzburg, Austria, after the water became polluted 6 . In Lake Akan, while both ball and mat colonies still remain, they are separated 2 . Ball-like aggregations are found in shallow waters (ca. 2 m in depth) near the shore, while the mats are found in deeper waters away from the shore 2 . Therefore, there is greater light available to ball colonies, and thus, for algal growth, than to mat colonies 8 . Finally, ball-like aggregations appear to be formed in oligotrophic waters that spring out from the bottom of the lake 6 . Lake Akan is not a representative oligotrophic lake. However, in the northern part of the lake where there are stable ball colonies, Biochemical Oxygen Demand (BOD) is 0.5-1.0 mg/L and a Total Nitrogen/Total Phosphorus ratio is less than 10 (TN/TP) 14,15 .
Forming ball-like aggregations can be a mechanism to maximize the biomass in the very limited conditions suitable for their growth. It is working well in the stable ball colonies. Lake Akan and Lake Myvatn are two of the very few lakes in the world that harbor such habitats as needed for Lake Balls. Therefore, these lakes need to be carefully protected and monitored.