I will explain my manuscript under review for the moment. It is based on a field work of quadrats in Izu peninsula of Japan (sorry the link is only in Japanese). The draft is available as arXiv:1603.00959[q-bio.PE].
The fundamental idea of biology, species, is still vague in definition. A population is a group of organisms of the same species occupying a particular space at a particular time, while a species is defined as groups of actually or potentially interbreeding natural populations, which are reproductively isolated from other such groups. Although these terms are frequently used, there can be confusion due to their overlapping definitions, such as ‘ring species’ or sympatric speciation. Previously we showed that logarithmic distributions of the numbers of individuals among population ranks are well distinguished from non-logarithmic distributions among species ranks (Adachi, Evol. Biol. 42, 210).
In this paper, we utilized cellular slime molds –Dictyostelia– to discriminate dynamics between populations and species. Based on Price equation (describes the relationships among fitness, expectation & covariance of phenotypes), Riemann zeta function, R = T theory, Weil’s explicit formula and the idea analogous to supersymmetry depending on time (similar to transactional interpretation of quantum physics), we introduce a new complex metric: “small s”. It can measure the extent of difference from neutral and logarithmic distribution patterns of populations. In this idea, the fitness is analogical to fermion and s is analogical to boson. The introduction of entropy to the population/species distributions is achieved by Hubbell’s neutral theory.
The discrimination is clearly shown among our new metrics in Figure 1. The border of adaptive species and chaotic populations/species is Re(s) = 2, and the values of prime closed geodesics are |N(p)| = 2/3 for adaptive situation in species and |N(p)| = 1 for non-adaptive situation in species or in populations, respectively. Surprisingly, our new metrics show some quantizations of the values in adapted conditions, due to both adaptation and maximization of entropy. Furthermore, whether the species is adapted or in chaos is discriminated by 3 mod 4 or 1 mod 4 of corresponding prime. We expand this model to the patch with zeta-dominance model (PzDom). In the model, some quantization of a species “small s” metric is observed. We can calculate fitness of each individual simply by examining the distribution of the number of individuals in each population/species. We can also calculate time-dependent fitness function and a ‘Hubble parameter’ of a fitness space (e.g. Figure 2).
Furthermore, to quantify how these two concepts, the dynamics of populations and species, we applied the theory of statistical mechanics to the data. We can calculate the critical temperature/Weiss field and describe the potential phase transitions in terms of population and species dynamics; we can therefore estimate the phase and the extent of each of the various orders. In Figure 3, phase transitions were examined, and in Figure 4, discrimination was clearly shown among our new metrics. Our new metrics might be suitable parameters to examine phase transitions and differences of neutral populations/adaptive species. Finally, we applied a genetic model of tumor suppressor genes to explain the evolution of biological hierarchy observed among Dictyostelia. Overall, we aim to prove the existence of biological hierarchies, such as populations and species. Graphical summary is: Visual image of an upper half plane H of PzDom model. Non-trivial zeros for Riemann zeta function, which represent population burst/collapse (similar to Bose-Einstein condensation, prime of 2), is assumed to on the crossing points of Re(s) = 1/2 axis and horizontal broken lines according to Riemann hypothesis (note that scales of horizontal axis and vertical axis are different). Orange circles represent adaptive stages as species. Blue area represents future stage with [Vp = - ϕ, Re(s - 1) ~ -1/(3ϕ) and Im(s - 1) ~ e-1/(3ϕ) (Marino2014)], with converged states at blue circles. We also utilized Schwarz equation and ‘Hubble parameter’ to predict the situation of a unit time after the observation.
Finally, the required dimension for the model is 12 including fluctuations of 4 dimensional system, similar to superstring theory in material sciences. The evolution of fitness space resembles cyclic universe model. From these results, we discuss that biological system forms hierarchy to avoid deterioration due to being in neutral situation for a long time. The calculation requires only the data of population/species density in time course.
References:
On the Origin of Species: By Means of Natural Selection (Dover Thrift Editions)
Systematics and the Origin of Species from the Viewpoint of a Zoologist
The Unified Neutral Theory of Biodiversity and Biogeography (Monographs in Population Biology)
The Quantum Handshake: Entanglement, Nonlocality and Transactions
Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften)
