Pattern formation and evolution on plants

Abstract

Phyllotaxis, namely the arrangement of phylla (leaves, florets, etc.) has intrigued natural scientists for over four hundred years. Statistics show that about 90\% of the spiral patterns has their numbers of spirals belonging to two consecutive members of the regular Fibonacci sequence. (Fibonacci(-like) sequences refer to any sequences constructed with the addition rule $a_{j+2}=a_{j}+a_{j+1}$, while the regular Fibonacci sequence refers to the particular sequences 1,1,2,3,5,8,13,...) Historical research on pattern formation on plants, tracing back to as early as four hundred years ago, was mostly geometry based. Current studies focus on the activities on the cellular level and study initiation of primordia (the initial undifferentiated form of phylla) as a morphogenesis process cued by some signal. The nature of the signal and the mechanisms governing the distribution of the signal are still under investigation. The two top candidates are the biochemical hormone auxin distribution and the mechanical stresses in the plant surface (tunica). We built a model which takes into consideration the interactions between these mechanisms. In addition, this dissertation explores both analytically and numerically the conditions for the Fibonacci-like patterns to continuously evolve (i.e. as the mean radius of the generative annulus changes over time, the numbers of spirals in the pattern increase or decreases along the same Fibonacci-like sequence), as well as for different types of pattern transitions to occur. The essential condition for the Fibonacci patterns to continuously evolve is that the patterns are formed annulus by annulus on a circular domain and the pattern-forming mechanism is dominated by a quadratic nonlinearity. The predominance of the regular Fibonacci pattern is determined by the pattern transitions at early stages of meristem growth. Furthermore, Fibonacci patterns have self-similar structures across different radii, and there exists a one-to-one mapping between any two Fibonacci-like patterns. The possibility of unifying the previous theory of optimal packing on phyllotaxis and the solutions of current mechanistic partial differential equations is discussed.