Evolutionary origins of Fibonacci phyllotaxis in land plants.

Fibonacci phyllotaxis is commonly seen in all major groups of land plants. While a precise correlation is found between the internal pattern of the primary vascular system and the external pattern of appendages on the stem surface, it remains a big question how this regularity of Fibonacci phyllotaxis came into being in the course of evolution. Here I address this problem with a model describing phylogenetic and ontogenetic changes in vascular phyllotaxis based on two hypotheses. The first is that the Fibonacci pattern of vascular connection is uniquely determined by the primary arrangement of incipient primordia, the sources of the primary signal system in vascular tissue differentiation. The second is that the surface-area-to-volume ratio of primary vascular tissues serves as a measure of fitness in evolution. The model explains the empirical rule on the manner in which vascular connection is reconfigured during ontogeny, especially during juvenile development. Fossil and phylogenetic evidence suggests that Fibonacci phyllotaxis appeared shortly after the innovation of indefinite lateral organ initiation in a regular sequence.

Optimal strategies and cost-benefit analysis of the [Formula: see text]-player weightlifting game.

The study of cooperation has been extensively studied in game theory. Especially, two-player two-strategy games have been categorized according to their equilibrium strategies and fully analysed. Recently, a grand unified game covering all types of two-player two-strategy games, i.e., the weightlifting game, was proposed. In the present study, we extend this two-player weightlifting game into an [Formula: see text]-player game. We investigate the conditions for pure strategy Nash equilibria and for Pareto optimal strategies, expressed in terms of the success probability and benefit-to-cost ratio of the weightlifting game. We also present a general characterization of [Formula: see text]-player games in terms of the proposed game. In terms of a concrete example, we present diagrams showing how the game category varies depending on the benefit-to-cost ratio. As a general rule, cooperation becomes difficult to achieve as group size increases because the success probability of weightlifting saturates towards unity. The present study provides insights into achieving behavioural cooperation in a large group by means of a cost-benefit analysis.

Practical Basis of the Geometric Mean Fitness and its Application to Risk-Spreading Behavior.

Temporal variations in population size under unpredictable environments are of primary concern in evolutionary ecology, where time scale enters as an important factor while setting up an optimization problem. Thus, short-term optimization with traditional (arithmetic) mean fitness may give a different result from long-term optimization. In the long-term optimization, the concept of geometric mean fitness has been received well by researchers and applied to various problems in ecology and evolution. However, the limit of applicability of geometric mean has not been addressed so far. Here we investigate this problem by analyzing numerically the probability distribution of a random variable obeying stochastic multiplicative growth. According to the law of large number, the expected value (i.e., arithmetic mean) manifests itself as a proper measure of optimization as the number of random processes increases to infinity. We show that the finiteness of this number plays a crucial role in arguing for the relevance of geometric mean. The geometric mean provides a satisfactory picture of the random variation in a long term above a crossover time scale that is determined by this number and the standard deviation of the randomly varying growth rates. We thus derive the applicability condition under which the geometric mean fitness is valid. We explore this condition in some examples of risk-spreading behavior.

A new long-term measure of sustainable growth under uncertainty.

The trade-off between short-term success and long-term sustainability is a common subject of great importance both in the biological evolution of organisms and in the economic activities of human beings. In evolutionary biology, bet-hedging theories have described it as the trade-off between the (within-generation) arithmetic mean fitness and the (between-generation) geometric mean fitness of a genotype. Accordingly, bet-hedging strategies observed in various organisms are regarded as optimizing the geometric mean fitness. To increase the geometric mean fitness signifies to suppress the between-generation variance in the mean fitness. Thus, this view is consistent with mean-variance portfolio analysis in which the standard deviation of a portfolio is regarded as a measure of risk. In the present study, we provide yet another measure of long-term sustainability, which is based on minimization of the probability of extinction/bankruptcy that randomly varying population/asset size after a long time becomes less than a certain small value. We present results for representative examples to show that the present criterion gives a qualitatively similar but quantitatively different prediction from the traditional ones. In particular, we emphasize that maximizing survival probability (i.e. minimizing extinction probability) is equivalent neither to maximizing geometric mean fitness nor to minimizing variance in mean fitness, while these three are consistently related to each other.

Improving environment drives dynamical change in social game structure.

The development of cooperation in human societies is a major unsolved problem in biological and social sciences. Extensive studies in game theory have shown that cooperative behaviour can evolve only under very limited conditions or with additional complexities, such as spatial structure. Non-trivial two-person games are categorized into three types of games, namely, the prisoner's dilemma game, the chicken game and the stag hunt game. Recently, the weight-lifting game has been shown to cover all five games depending on the success probability of weight lifting, which include the above three games and two trivial cases (all cooperation and all defection; conventionally not distinguished as separate classes). Here, we introduce the concept of the environmental value of a society. Cultural development and deterioration are represented by changes in this probability. We discuss cultural evolution in human societies and the biological communities of living systems.

Two-step mechanism of spiral phyllotaxis.

Fibonacci numbers such as 5, 8, and 13 occur in the spiral arrangement of lateral organs at shoot tips in plants. While the cone scales of conifers are normally arranged in 5 and 8 (or 8 and 13) curved rows in opposite directions, other numbers such as 4 and 7 (or 7 and 11) are found anomalously. The observed numbers still obey the Fibonacci rule, with the next number being the sum of the preceding two. Although these observations have been made for centuries, the underlying mechanisms of the numerical relationship have not been investigated. Here, we show that this phenomenon is caused by a two-step mechanism: (1) maintenance of a constant angle between consecutive lateral organs and (2) strong canalization of this angle to a specific value. The first step of the mechanism precedes the second step of the mechanism because the Fibonacci-rule pattern is due to the first step, while the second step distinguishes normal, anomalous and unobserved types. The current dominance of the normal type is a result of the evolutionary process of the second step.

Author Correction: Grazing enhances species diversity in grassland communities.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

A single 'weight-lifting' game covers all kinds of games.

Game theory has been studied extensively to answer why cooperation is promoted in human and animal societies. All games are classified into five games: the Prisoner's Dilemma, chicken game (including hawk-dove game), stag hunt game and two trivial games of either all cooperation or all defect, which are studied separately. Here, we propose a new game that covers all five game categories: the weight-lifting game. The player choose either to (1) carry a weight (cooperate: pay a cost) or (2) pretend to carry it (defect: pay no cost). The probability of success in carrying the weight depends on the number of cooperators, and the players either gain the success reward or pay the failure penalty. All five game categories appear in this game depending on the success probabilities for the number of cooperators. We prove that this game is exactly equivalent to the combination of all five games in terms of a pay-off matrix. This game thus provides a unified framework for studying all five types of games.

Bankruptcy is an inevitable fate of repeated investments with leverage.

Due to the globalization and computerization of financial and economic activities, numerous repetitive leveraged investments have become possible in stock markets and currency exchanges. In reality, repeated leveraged investments up to 100 times/day are possible via online access. With computer-aided programs, this repetition number may easily increase 1000 times/day. The possibility of bankruptcy in repeated leveraged investments has never been considered in actual practices because the probability of bankruptcy in a single investment trial is almost negligible. Here, we show that the extremely numerous repetitions have a considerable chance of bankruptcy overall, even if the probability of bankruptcy for a single investment is extremely close to zero. The exact relationship between the repetitions and the probability of bankruptcy is approximated well by n(0.63) = m, where 10n is the number of repetitions, 10-m is the bankruptcy probability of a single investment, and n(0.63) is the 63% chance of bankruptcy. Thus, extremely rare events can always lead to bankruptcy in continuously repeated investment, even if the possibility of such an event is almost null. We suggest that the avoidance measure of bankruptcy is necessary in numerous repeated investments even if a single trial is almost certain to win.

Grazing enhances species diversity in grassland communities.

In grassland studies, an intermediate level of grazing often results in the highest species diversity. Although a few hypotheses have been proposed to explain this unimodal response of species diversity to grazing intensity, no convincing explanation has been provided. Here, we build a lattice model of a grassland community comprising multiple species with various levels of grazing. We analyze the relationship between grazing and plant diversity in grasslands under variable intensities of grazing pressure. The highest species diversity is observed at an intermediate grazing intensity. Grazers suppress domination by the most superior species in birth rate, resulting in the coexistence of inferior species. This unimodal grazing effect disappears with the introduction of a small amount of nongrazing natural mortality. Unimodal patterns of species diversity may be limited to the case where grazers are the principal source of natural mortality.

The unified rule of phyllotaxis explaining both spiral and non-spiral arrangements.

Leaf-like appendages of different plant groups are arranged in common phyllotaxis patterns categorized into two types: spiral and non-spiral arrangements. The adaptive reason for this morphological convergence is unknown. In the non-spiral arrangement, the divergence angle between successive leaves is a simple fraction of 360°, e.g. distichy, decussate and whorled phyllotaxis. In the spiral arrangement, the divergence angle of nascent leaves at the shoot apex is fixed at the golden angle 137.5°, whereas those of the developed leaves varies within a sequence of Fibonacci fractions, such as 1/3, 2/5, 3/8, 5/13, etc. The optimality of the golden angle has been shown recently by assuming that the pattern of developed leaves varies during growth in a manner depending on the divergence angle of nascent leaves. Here we propose a unified rule of phyllotaxis to explain both types of arrangement: the developed leaves form vertical rows along the stem. In the non-spiral arrangement, nascent to developed leaves always follow this rule, so that the number of leaf rows is kept constant irrespective of stem growth. In the spiral arrangement, developed leaves attain this rule by adjusting the divergence angle from the golden angle. The spiral arrangement is adaptive in that the number of leaf rows varies during growth depending on shoot thickness.

Asymptotic stability of a modified Lotka-Volterra model with small immigrations.

Predator-prey systems have been studied intensively for over a hundred years. These studies have demonstrated that the dynamics of Lotka-Volterra (LV) systems are not stable, that is, exhibiting either cyclic oscillation or divergent extinction of one species. Stochastic versions of the deterministic cyclic oscillations also exhibit divergent extinction. Thus, we have no solution for asymptotic stability in predator-prey systems, unlike most natural predator-prey interactions that sometimes exhibit stable and persistent coexistence. Here, we demonstrate that adding a small immigration into the prey or predator population can stabilize the LV system. Although LV systems have been studied intensively, there is no study on the non-linear modifications that we have tested. We also checked the effect of the inclusion of non-linear interaction term to the stability of the LV system. Our results show that small immigrations invoke stable convergence in the LV system with three types of functional responses. This means that natural predator-prey populations can be stabilized by a small number of sporadic immigrants.

Multi-species coexistence in Lotka-Volterra competitive systems with crowding effects.

Classical Lotka-Volterra (LV) competition equation has shown that coexistence of competitive species is only possible when intraspecific competition is stronger than interspecific competition, i.e., the species inhibit their own growth more than the growth of the other species. Note that density effect is assumed to be linear in a classical LV equation. In contrast, in wild populations we can observed that mortality rate often increases when population density is very high, known as crowding effects. Under this perspective, the aggregation models of competitive species have been developed, adding the additional reduction in growth rates at high population densities. This study shows that the coexistence of a few species is promoted. However, an unsolved question is the coexistence of many competitive species often observed in natural communities. Here, we build an LV competition equation with a nonlinear crowding effect. Our results show that under a weak crowding effect, stable coexistence of many species becomes plausible, unlike the previous aggregation model. An analysis indicates that increased mortality rate under high density works as elevated intraspecific competition leading to the coexistence. This may be another mechanism for the coexistence of many competitive species leading high species diversity in nature.

Synthesis, Characterization and Protonation Behavior of Quinoxaline-Fused Porphycenes.

9,10-Quinoxaline-fused porphycenes 1a-H2 and 1b-H2 were synthesized by intramolecular McMurry coupling. As a result of the annulation of the quinoxaline moiety on the porphycene skeleton, 1a-H2 and 1b-H2 display absorption and fluorescence in the near infra-red (NIR) region. Additionally, the quinoxaline moieties of 1a-H2 and 1b-H2 act as electron-withdrawing groups, introducing lower reduction potentials than for pristine porphycene. The protonation occurred at the nitrogen atoms in the cavity of freebase porphycenes and at the quinoxaline moieties for their nickel complexes to give diprotonic species.

Optimal designs of mollusk shells from bivalves to snails.

Bivalve, ammonite and snail shells are described by a small number of geometrical parameters. Raup noted that the vast majority of theoretically possible shell forms do not occur in nature. The constraint factors that regulate the biased distribution of natural form have long since been an open problem in evolution. The problem of whether natural shell form is a result of optimization remains unsolved despite previous attempts. Here we solve this problem by considering the scaling exponent of shell thickness as a morphological parameter. The scaling exponent has a drastic effect on the optimal design of shell shapes. The observed characteristic shapes of natural shells are explained in a unified manner as a result of optimal utilization of shell material resources, while isometric growth in thickness leads to impossibly tight coiling.

Optimal hash arrangement of tentacles in jellyfish.

At first glance, the trailing tentacles of a jellyfish appear to be randomly arranged. However, close examination of medusae has revealed that the arrangement and developmental order of the tentacles obey a mathematical rule. Here, we show that medusa jellyfish adopt the best strategy to achieve the most uniform distribution of a variable number of tentacles. The observed order of tentacles is a real-world example of an optimal hashing algorithm known as Fibonacci hashing in computer science.

Biophysical optimality of the golden angle in phyllotaxis.

Plant leaves are arranged around a stem axis in a regular pattern characterized by common fractions, a phenomenon known as phyllotaxis or phyllotaxy. As plants grow, these fractions often transition according to simple rules related to Fibonacci sequences. This mathematical regularity originates from leaf primordia at the shoot tip (shoot apical meristem), which successively arise at fixed intervals of a divergence angle, typically the golden angle of 137.5°. Algebraic and numerical interpretations have been proposed to explain the golden angle observed in phyllotaxis. However, it remains unknown whether phyllotaxis has adaptive value, even though two centuries have passed since the phenomenon was discovered. Here, I propose a new adaptive mechanism explaining the presence of the golden angle. This angle is the optimal solution to minimize the energy cost of phyllotaxis transition. This model accounts for not only the high precision of the golden angle but also the occurrences of other angles observed in nature. The model also effectively explains the observed diversity of rational and irrational numbers in phyllotaxis.

Synthesis and electrochemical properties of porphycene-diketopyrrolopyrrole conjugates.

The selective iodination of 2,7,12,17-tetrahexylporphycene 1 was successfully accomplished by using N-iodosuccinimide in the presence of activators to give 3-iodoporphycene 2 and 3,13-diiodoporphycene 3a. These iodinated porphycenes can be used as the substrates for palladium-catalyzed coupling to prepare porphycene-diketopyrrolopyrrole conjugates in two steps. The connection of the diketopyrrolopyrrole units to porphycenes broadened their absorption spectra and increased the intensity of the Q-bands due to the electronic interactions between the porphycene and diketopyrrolopyrrole moieties.

FISH identification of Helicoverpa armigera and Mamestra brassicae chromosomes by BAC and fosmid probes.

Since the Bombyx mori genome sequence was published, conserved synteny between B. mori and some other lepidopteran species has been revealed by either FISH (fluorescence in situ hybridization) with BAC (bacterial artificial chromosome) probes or linkage analysis. However, no species belonging to the Noctuidae, the largest lepidopteran family which includes serious polyphagous pests, has been analyzed so far with respect to genome-wide conserved synteny and gene order. For that purpose, we selected the noctuid species Helicoverpa armigera and Mamestra brassicae, both with n = 31 chromosomes. Gene-defined fosmid clones from M. brassicae and BAC clones from a closely related species of H. armigera, Heliothis virescens, were used for a FISH analysis on pachytene chromosomes. We recognized all H. armigera chromosomes from specific cross-hybridization signals of 146 BAC probes. With 100 fosmid clones we identified and characterized all 31 bivalents of M. brassicae. Synteny and gene order were well conserved between the two noctuid species. The comparison with the model species B. mori (n = 28) showed the same phenomenon for 25 of the 28 chromosomes. Three chromosomes (#11, #23 and #24) had two counterparts each in H. armigera and M. brassicae. Since n = 31 is the modal chromosome number in Lepidoptera, the noctuid chromosomes probably represent an ancestral genome organization of Lepidoptera. This is the first identification of a full karyotype in Lepidoptera by means of BAC cross-hybridization between species. The technique shows the potential to expand the range of analyzed species efficiently.

EST sequencing and fosmid library construction in a non-model moth, Mamestra brassicae, for comparative mapping.

Genome data are useful for both basic and applied research; however, it is difficult to carry out large-scale genome analyses using species with limited genetic or genomic resources. Here, we describe a cost-effective method to analyze the genome of a non-model species, using the cabbage moth, Mamestra brassicae (Lepidoptera: Noctuidae). First, we conducted expression sequence tag (EST) analysis. In this analysis, we performed PCR-based prescreening of a non-normalized embryonic cDNA library to eliminate already sequenced cDNAs from further sequencing, which significantly increased the percentage of unique genes. Next, we constructed a fosmid library of M. brassicae and isolated 120 clones containing 119 putative single copy genes by PCR-based screening with primer sets designed from the ESTs. Finally, we showed that the isolated fosmid clones could be used as probes for multicolor fluorescence in situ hybridization (FISH) analysis against an M. brassicae chromosome and confirmed conserved gene order between M. brassicae and the silkworm, Bombyx mori. Thus, we developed new genomic resources for comparative genome analysis in M. brassicae using robust and relatively low cost methods that can be applied to any non-model organism.