Evolutionary trajectories in the parameter space of the sectional model of spruce (Picea abies) crown: Emergence of the “protoplant”

Abstract

A sectional model of the dynamics of the system of regular spruce (Picea abies) branches, as well as submodels of the inhibition of initial growth and inter-verticillate branches, were extended to the range of (0, 3) of the fractаl parameter μ which reflects the association of green biomass with tree size (B ~ H ^μ). It is shown that the proto-spruce branches of the first three orders appear in the subrange of (0, 1) of the μ parameter. The branches of the first order appear at μ ≈ 0.25, while inter-verticillate branches appear at μ ≈ 1.4, which may be an element of spruce adaptation to unstable illumination conditions. The presence of green biomass at μ < 1.0 indicates that it can be represented as a set of spatial photosynthesizing points (hypothetical cyanobacteria). Therefore, the fractal properties of the set of these points located on a line segment are considered as the model. The condition of μ < 1.0 is shown to be true if the points are arranged in groups. In this case, μ is practically independent of groups allocation at the segment and depends only on the number and the type of distribution of points within the groups. For a fixed number of points in a group (for example, ng = 2) distributed randomly and uniformly, with increasing number of Ng groups, the parameter μ decreases from 1 to ≈0.25. Conversely, for a fixed large numberNg of groups with increasing from 2 number of points ng in the group, the parameter μ increases, tending to 1. Based on these fractal properties of placing the point groups, as well as on the hypothesis of the trophic nature of the organelle symbiogenesis in the eukaryotic cell, a two-stage mechanism of the emergence of protoplants was suggested. This mechanism is manifested in motion along the endosymbiosis trajectory, which is characterized by a constant number of points in a group and by increasing number of groups until, in the course of evolution, the host organism creates an infrastructure that allows the supplying of and interaction between cyanobacterial groups. At this stage, μ decreases from 1 to ≈0.25. When the infrastructure is created, and it becomes possible to increase the number of points in the group, the second stage begins. This stage is characterized by motion along the trajectory, which runs by doubling of cyanobacteria in the group (μ tends to 1). At the first stage, motion along such a composite trajectory results in a slow increase in the size of the photosynthetic system, even if the number of cyanobacterial groups grows exponentially. However, at the second stage, this size grows rapidly. The similar inhibition of initial growth is observed in today trees and manifests in the initial deceleration of the increase of branch orders.