The UGI number assures a unique identifier for any existing hierarchical gating strategy based on the Prime Population System (PPS) and Gödel Numbers. Similar to the PPS that identifies cell types, the UGI is unique for every existing gating strategy that has been drawn in the past and will be drawn in the future. Considering the present manuscript and the previous description of the PPS (2), Figure 2 schematizes the relationships among the objects of arithmetization, the system's rules, and the prime numbers. Prime numbers and the fundamental theorem of arithmetic represents the basis of the system. The markers, cell types, gates, and sequence of gates are the objects of the arithmetization whereas the PPS, the modified Gödel Numbers and the UGI are the rules governing the relationship among the objects.

The UGI numbers can be used in databases, manuscripts, and cytometry software to exchange information about gating strategies. Since the UGI is linked to the identified cell type via the PPS, the capacity of the two systems allows the integration of information generated in different experimental settings. It is essential to note that, leveraging the PPS, the UGI number connects the gating strategy with the biology of the identified cell type since cell type and gating strategy are coded in the same system (Fig. 2).

Gating-ML 2.0 (6) is a standard supported by the International Society for the Advancement of Cytometry (ISAC) that unambiguously describes gates and data transformation to exchange this information among different software tools. It handles information about type and sequence of conventional gates, compensation, scale transformation, and custom metadata. However, the standard does not propose a unique way to identify a gating strategy in relation to the biological markers and, therefore, it is limited to the experimental domain. Indeed, there is no convention to name the markers inside Gating-ML. However, UGI numbers can be added inside the metadata section of Gating-ML at the gate level and render the standard able to build connections among different experiments.

Automated gating methods have been developed to reduce user-to-user variability and increase the speed of analysis (7). In the cluster analysis framework, when hundreds of clusters are generated in every experiment, it is impossible to name them via a meaningful biological string. The PPS system will be useful to label with a unique tag every cluster. The information can then be compared to any other data set using the PPS, and when the dataset is also analyzed by a conventional hierarchical gating, the gating strategy can be retrieved via the UGI.

A limitation of the present standard is that cleaning gates, such as the one used to identify doublets in flow or in mass cytometry (8), are not considered. Since cleaning gates are not based on biological markers, they cannot be included in the system and must be represented in a separate format. At the moment this part of the gating strategy more technical and not linked to the biology is out of the scope of the UGI.

The idea behind the arithmetization of cytometry starting from prime numbers is to build a precise framework to code for cell types and gating strategies and, in the future, for other characteristics of the cells, such as the expression profiles, cell-to-cell interactions, and spatial relationships. Every characteristic will be precisely linked to the original prime and successively to the one-to-one relationship with the UniProt database. The PPS and UGI recast the relationship among markers, cell types, and gating strategy in terms of the fundamental concept of number theory. Thus building a base for future implementations.

The system I describe has vocation to work in the foreground of software, database, and repositories. This manuscript aims to propose a theoretical system to the cytometry community; any implementation will be useful only when integrated with a cytometry software complete with all the conventional functions.

Existing software tools can easily manage numbers generated by the PPS (2). However, the product of sequential primes raised to the PPS will generate numbers that will be difficult to manage. UGI may be handled, for instance, as vectors of bases and powers. I imagine every software company willing to implement the arithmetization will choose the best solution for their environment. This is primarily engineering and out of the scope of the present manuscript. The unique identifier concept holds independently from the way everyone will process the UGI expressions. For example, the number 5491 can be written as a product of primes: 172 x 19, hexadecimal: 1573, or in scientific notation, 5*1000+4*100+9*10+1*1. The underlying meaning will remain the same: a cell type negative for marker F and positive for marker E. It remains a unique identifier. The PPS and UGI represent a common language to share information and keep a meaningful link to biology independently of how they are represented inside a software package. For instance, number theory holds independently of the numerals used to describe it.

Finally PPS and UGI code for cell types and gating strategies using the same language, and this represents an enormous advantage in terms of interoperability and information sharing among software. Since the proposed arithmetization provides an unambiguous link to biology, the system may be expanded to encode additional cell characteristics in a mathematical language.