Size of the diatoms in a chainlike colony I
In this article we will discuss the sequence of diatoms sizes in such a culture and if one can observe them.
Note: 
A more detailed analysis of the size sequence of diatoms in chainshaped colonies is given in this publication:
On the size sequence of diatoms in clonal chains Harbich, Thomas (2020)
Submitted for: Diatom Morphogenesis [DIMO, Volume in the series: Diatoms: Biology & Applications, series editors: Richard Gordon & Joseph Seckbach, In preparation]. V. Annenkov, J. Seckbach and R. Gordon, (eds.) WileyScrivener, Beverly, MA, USA: In preparation.

As early as 1871, Pfitzer illustrated the successive reduction in size on the example of a short chain of diatoms of the genus Eunotia (see figure above). Specifically, attempts are being made to determine possible positions of a fragment of such a colony in the theoretical sequence. For this purpose, the sequence of sizes and orientations of the diatoms in a clonal chain must be known first. Ussing et al. (2005) have argued that the generation rule for this sequence can be described by a onedimensional Lindenmayer model (see Lindenmeyer, A. (1968)). This model has already been successfully used for various species (e.g. cyanobacterium Anabaena catenula).
Both the measurement of the size of diatoms as well as the assignment to the theoretical sequence prove to be no easy task. A closer look at the Lindenmayer system for chains of diatoms makes it possible to successfully analyze an example from nature. This is probably not the case for all species. Furthermore, one cannot assume that a found assignment is unique. In the following, the mathematical basics are explained and the practical challenges described. Then the mathematical method for analysis is presented and demonstrated by an example.
Description of the sequence of diatom sizes
Preliminary remark on asexual reproduction
The asexual reproduction of diatoms has already been briefly described in the introduction. Each cell division produces a diatom of the same size and a smaller one. In the following, the possible sizes of the larger valve are indexed by consecutive numbers k, where k = 0 corresponds to the largest possible diatom and k_{max} to the smallest. Starting from a cell of maximum size (generation 0) there exist in the generation n
diatoms of the size k (Animation in the introduction shows Pascal’s triangle). The applicability of the formula presumes the synchronicity of the divisions and applies only until the smallest possible cell is reached. If the results for a generation n are normalized to 1, the probability function for the binomial distribution is obtained (probability p = ½). In such an exemplary culture however, there is no probability distribution of the diatoms, but the number of diatoms of a certain size is deterministic. If samples are taken they are following a binomial distribution.
Lindenmayer system
Now the modeling of the colony by a Lindenmayer system is shown. A Lindenmayer system is a triple
D = (A, P, ω) consisting of an alphabet A, replacement rules P (also called productions) and an axiom (start word). The elements of the alphabet are intended to describe the orientation and size of the diatoms in a colony. A colony or a fragment thereof is characterized by a string of characters over the alphabet. The replacement rules P specify how this string changes from generation to generation. A starting condition ω, the so called axiom, determines with which string to start the calculation.
Alphabet A:
Let us imagine the colony as a sequence of characters (a string) written horizontally. The subsequently used notation for the characters of the string has been adopted from Ussing et al. (2005). A diatom whose left valve is larger than its right valve and whose larger valve is given by the size index k is called L^{k}.
Graphical representation of L^{k}:
Correspondingly, a diatom whose right valve is larger than its left valve and whose larger valve is given by the size index k is called R^{k}.
Graphical representation R^{k}:
L^{k} and R^{k} are mirrorsymmetrical. The alphabet consists of the union of sets { L^{k}  k = 0, 1, 2, …. k_{max} } and
{ R^{k}  k = 0, 1, 2, …. k_{max} }.
Production rules:
If a diatom that is characterized by the character L^{k} divides, two diatoms are generated in this arrangement:
Consequently the following replacement must be carried out:
L^{k} → L^{k} R^{k+1}
If the larger valve of the diatom is on its right side, it is only oriented differently to the observer:
This replacement rule results from mirroring:
R^{k} → L^{k+1} R^{k}
It is assumed that the cell divisions are synchronous. In the transition from one generation to the next generation, all elements must be replaced according to these rules so that the number of diatoms is doubled. The strings generated correspond to snapshots between the divisions.
With these replacement or production rules the Lindenmayer system is deterministic and contextfree and is called a D0Lsystem.
Axiom:
As an axiom (starting point) I choose a single cell of maximum size corresponding to the status after sexual reproduction. As the orientation of the cell is arbitrary, the axiom can be selected as ω = L^{0}. Ussing et al. (2005) use ω = R^{0}, which leads to mirrored, thus reversed chains. For practical reasons, which will be explained later, I prefer L^{0}. In connection with the properties of the chains produced, their dependence on the axiom will also be discussed.
Beginning with the axiom, the productions are applied iteratively to all elements of the string in parallel. This results in a sequence of strings G_{i} which describes the colony after the ith iteration, which is nothing but the ith generation.
G_{0} = ω = L^{0}
G_{1} = L^{0} R^{1}
G_{2} = L^{0} R^{1} L^{2} R^{1}
G_{3} = L^{0} R^{1} L^{2} R^{1} L^{2} R^{3} L^{2} R^{1}
etc.
In the following, I will term the string that is created after n iterations as "nth generation".
Observation and challenges
As the MacDonaldPfitzer rule has often been proved and the description of chainlike colonies is based solely on this rule, a proof of the sequence of sizes should not be difficult at first sight. Nevertheless, I had to realize that this is by no means the case. The following difficulties arise:
 Missing assignment of sizes to size index: When measuring a valve size, there is usually no way of assigning this value to the size index introduced above. In particular, the length of the largest possible valve is not known.
 Too short chains: Even if a fragment consisting of only a few diatoms can be measured well, a match with a theoretical sequence of sizes is only of limited value. It could be due to chance.
 Dead cells in the fragment.
 Small differences in the size of the valves of a diatom: The conception of valves which lie clearly inside one another, may not apply to many diatoms that form chains. The valve sizes appear to differ only by a fraction of their thickness. Natural fluctuations in the size of the valves could also play a role. A sufficiently accurate evaluation of the sequence of sizes of the Melosira colony which has been shown above was not possible to me.
The latter difficulty could possibly be overcome by the use of scanning electron microscopy. Even if all these challenges are mastered, the question remains whether the cell divisions that led to the whole fragment were really synchronized.
In the investigation of the lengths of Eunotia sp. the motility of the diatoms led to further problems. Fragments of several diatoms can separate from the colony. Individual diatoms often migrate away from the ends of the chain and can also detach and move away from the inside of the colony. Such a gap sometimes closes again as a result of expansion through cell division, so that the change remains undetected. The video below left shows several such events in 1500fold timelapse. Scenes further apart in time are interrupted by a dark pause. Surprisingly, it even happens that a diatom connects to the end of a chain, as can be seen in the video (1500x time lapse) at the bottom right (near the left edge of the frame). Individual free diatoms settle after some time and form a colony by dividing. It should be mentioned that these observations were made on cultures using the inverted microscope. The illumination (LED) of the microscope replaces daylight. In darkness the movement of the Eunotia comes to rest.
Ussing et al. (2005) discuss the sequence of size on the background of studies on Bacillaria paradoxa, but there is no indication as to whether this principle was observed and whether this was successful.
By chance I found on the internet at http://www.wunderkanone.de/ an excellent picture of a Fragilaria colony, which offers itself at first glance as an examination object. The next picture is shown by courtesy of Eckhard Völcker (see http://www.penard.de/):
Each of the diatoms shows significant differences in size between their valves, which is evident in the irregular upper and lower edges. The fragment contains 14 diatoms. A quite advanced breaking point is visible between the 6th and 7th diatoms from the left.
The agreement of the size sequence with a Lindenayer system could be shown.
LINDENMAYER, A. (1968a). Mathematical models for cellular interactions, in development
I. Filaments with onesided inputs. Journal of Theoretical Biology, 18, 290299
Pfitzer, E. (1871) Untersuchungen über Bau und Entwicklung der Bacillariaceen (Diatomeen).
Botanische Abhandlungen 2, 1–189.
USSING, A.P., GORDON, R., ECTOR, L., BUCZKO´ , K., DESNITSKIY, A.G. & VANLANDINGHAM, S.L. (2005). The colonial diatom ‘‘Bacillaria paradoxa’’: chaotic gliding motility, Lindemeyer Model of colonial morphogenesis, and bibliography, with translation of O.F. Müller (1783), “About a peculiar being in the beachwater”. Diatom Monographs, Vol. 5. Koeltz, Koenigstein, Germany.