A symmetry model for genetic coding via a wallpaper group composed of the traditional four bases and an imaginary base E: towards category theory-like systematization of molecular/genetic biology

Several supplementary concepts for applied category-theoretical states over an extended Petri net using an example relating to genetic coding: Toward an abstract algebraic formulation of molecular/genetic biology

algebraic concepts such as category are considered cornerstones on which logical consistency relies in any sophisticated study of natural phenomena. However, to the best of our knowledge, in molecular/genetic biology, their application is still severely limited because they capture neither the dynamics nor provide a visual form. The Petri net (PN) has often been used to illustrate visually parallel, asynchronous dynamic events in small data systems. A prototypal hybrid model combining both category theory and extended PNs may instead be indispensable for that purpose. This hybrid model incorporates 1) token-like elements of a group, 2) object-like places of a category, 3) square poles (rather than pentagon poles) that enable unique identifications of single-strand DNA sequences from the shape of its polygonal line, 4) creation/annihilation morphisms that generate/erase tokens, 5) Cartesian products 'Z5×Z2×…' that enable conversions between DNA and RNA sequences, 6) somatic recombinations (VDJ recombinations) for antibodies displayed concretely in category-theoretic form, 7) 'identity protein Δ' translated from a triplet of identity bases 'EEE' as an advanced concept from our previous display of the canonical central dogma, 8) illustrations of an incidence-matrix-like matrix A that includes operators as coordinates, and 9) basic topics concerning the canonical central dogma being displayed concretely using concepts of conventional category theory such as 'adjoint', 'adjoint functor', 'natural transformation', 'Yoneda's lemma' and 'Kan extension'. These ideas provide more advanced tools that expand our previous model concerning nucleic-acid-base sequences. Despite the nascent nature of our methodology, our hybrid model has potential in a variety of applications, illustrated using molecular/genetic sequences, in particular providing a simple dynamic/visual representation. With further improvements, this approach may prove effective in reducing the need for large data-storing systems.

Symmetrical treatment of "Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition", for major depressive disorders

BACKGROUND

We previously presented a group theoretical model that describes psychiatric patient states or clinical data in a graded vector-like format based on modulo groups. Meanwhile, the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5, the current version), is frequently used for diagnosis in daily psychiatric treatments and biological research. The diagnostic criteria of DSM-5 contain simple binominal items relating to the presence or absence of specific symptoms. In spite of its simple form, the practical structure of the DSM-5 system is not sufficiently systemized for data to be treated in a more rationally sophisticated way. To view the disease states in terms of symmetry in the manner of abstract algebra is considered important for the future systematization of clinical medicine.

RESULTS

We provide a simple idea for the practical treatment of the psychiatric diagnosis/score of DSM-5 using depressive symptoms in line with our previously proposed method. An expression is given employing modulo-2 and -7 arithmetic (in particular, additive group theory) for Criterion A of a 'major depressive episode' that must be met for the diagnosis of 'major depressive disorder' in DSM-5. For this purpose, the novel concept of an imaginary value 0 that can be recognized as an explicit 0 or implicit 0 was introduced to compose the model. The zeros allow the incorporation or deletion of an item between any other symptoms if they are ordered appropriately. Optionally, a vector-like expression can be used to rate/select only specific items when modifying the criterion/scale. Simple examples are illustrated concretely.

CONCLUSIONS

Further development of the proposed method for the criteria/scale of a disease is expected to raise the level of formalism of clinical medicine to that of other fields of natural science.