Linear Response Theory of Evolved Metabolic Systems.

Predicting cellular metabolic states is a central problem in biophysics. Conventional approaches, however, sensitively depend on the microscopic details of individual metabolic systems. In this Letter, we derived a universal linear relationship between the metabolic responses against nutrient conditions and metabolic inhibition, with the aid of a microeconomic theory. The relationship holds in arbitrary metabolic systems as long as the law of mass conservation stands, as supported by extensive numerical calculations. It offers quantitative predictions without prior knowledge of systems.

A linear reciprocal relationship between robustness and plasticity in homeostatic biological networks.

In physics of living systems, a search for relationships of a few macroscopic variables that emerge from many microscopic elements is a central issue. We evolved gene regulatory networks so that the expression of core genes (partial system) is insensitive to environmental changes. Then, we found the expression levels of the remaining genes autonomously increase to provide a plastic (sensitive) response. A feedforward structure from the non-core to core genes evolved autonomously. Negative proportionality was observed between the average changes in core and non-core genes, reflecting reciprocity between the macroscopic robustness of homeostatic genes and plasticity of regulator genes. The proportion coefficient between those genes is represented by their number ratio, as in the "lever principle", whereas the decrease in the ratio results in a transition from perfect to partial adaptation, in which only a portion of the core genes exhibits robustness against environmental changes. This reciprocity between robustness and plasticity was satisfied throughout the evolutionary course, imposing an evolutionary constraint. This result suggests a simple macroscopic law for the adaptation characteristic in evolved complex biological networks.

Autotoxin-mediated latecomer killing in yeast communities.

Cellular adaptation to stressful environments such as starvation is essential to the survival of microbial communities, but the uniform response of the cell community may lead to entire cell death or severe damage to their fitness. Here, we demonstrate an elaborate response of the yeast community against glucose depletion, in which the first adapted cells kill the latecomer cells. During glucose depletion, yeast cells release autotoxins, such as leucic acid and L-2keto-3methylvalerate, which can even kill the clonal cells of the ones producing them. Although these autotoxins were likely to induce mass suicide, some cells differentiated to adapt to the autotoxins without genetic changes. If nondifferentiated latecomers tried to invade the habitat, autotoxins damaged or killed the latecomers, but the differentiated cells could selectively survive. Phylogenetically distant fission and budding yeast shared this behavior using the same autotoxins, suggesting that latecomer killing may be the universal system of intercellular communication, which may be relevant to the evolutional transition from unicellular to multicellular organisms.

Microeconomics of Metabolism: The Warburg Effect as Giffen Behaviour.

Metabolic behaviours of proliferating cells are often explained as a consequence of rational optimization of cellular growth rate, whereas microeconomics formulates consumption behaviours as optimization problems. Here, we pushed beyond the analogy to precisely map metabolism onto the theory of consumer choice. We thereby revealed the correspondence between long-standing mysteries in both fields: the Warburg effect, a seemingly wasteful but ubiquitous strategy where cells favour aerobic glycolysis over more energetically efficient oxidative phosphorylation, and Giffen behaviour, the unexpected consumer behaviour where a good is demanded more as its price rises. We identified the minimal, universal requirements for the Warburg effect: a trade-off between oxidative phosphorylation and aerobic glycolysis and complementarity, i.e. impossibility of substitution for different metabolites. Thus, various hypotheses for the Warburg effect are integrated into an identical optimization problem with the same universal structure. Besides, the correspondence between the Warburg effect and Giffen behaviour implies that oxidative phosphorylation is counter-intuitively stimulated when its efficiency is decreased by metabolic perturbations such as drug administration or mitochondrial dysfunction; the concept of Giffen behaviour bridges the Warburg effect and the reverse Warburg effect. This highlights that the application of microeconomics to metabolism can offer new predictions and paradigms for both biology and economics.

Coordination of Polyploid Chromosome Replication with Cell Size and Growth in a Cyanobacterium.

Homologous chromosome number (ploidy) has diversified among bacteria, archaea, and eukaryotes over evolution. In bacteria, model organisms such as Escherichia coli possess a single chromosome encoding the entire genome during slow growth. In contrast, other bacteria, including cyanobacteria, maintain multiple copies of individual chromosomes (polyploid). Although a correlation between ploidy level and cell size has been observed in bacteria and eukaryotes, it is poorly understood how replication of multicopy chromosomes is regulated and how ploidy level is adjusted to cell size. In addition, the advantages conferred by polyploidy are largely unknown. Here we show that only one or a few multicopy chromosomes are replicated at once in the cyanobacterium Synechococcus elongatus and that this restriction depends on regulation of DnaA activity. Inhibiting the DnaA intrinsic ATPase activity in S. elongatus increased the number of replicating chromosomes and chromosome number per cell but did not affect cell growth. In contrast, when cell growth rate was increased or decreased, DnaA level, DnaA activity, and the number of replicating chromosomes also increased or decreased in parallel, resulting in nearly constant chromosome copy number per unit of cell volume at constant temperature. When chromosome copy number was increased by inhibition of DnaA ATPase activity or reduced culture temperature, cells exhibited greater resistance to UV light. Thus, it is suggested that the stepwise replication of the genome enables cyanobacteria to maintain nearly constant gene copy number per unit of cell volume and that multicopy chromosomes function as backup genetic information to compensate for genomic damage.IMPORTANCE Polyploidy has evolved many times across the kingdom of life. The relationship between cell growth and chromosome replication in bacteria has been studied extensively in monoploid model organisms such as Escherichiacoli but not in polyploid organisms. Our study of the polyploid cyanobacterium Synechococcus elongatus demonstrates that replicating chromosome number is restricted and regulated by DnaA to maintain a relatively stable gene copy number/cell volume ratio during cell growth. In addition, our results suggest that polyploidy confers resistance to UV, which damages DNA. This compensatory polyploidy is likely necessitated by photosynthesis, which requires sunlight and generates damaging reactive oxygen species, and may also explain how polyploid bacteria can adapt to extreme environments with high risk of DNA damage.

Metabolic dynamics restricted by conserved carriers: Jamming and feedback.

To uncover the processes and mechanisms of cellular physiology, it first necessary to gain an understanding of the underlying metabolic dynamics. Recent studies using a constraint-based approach succeeded in predicting the steady states of cellular metabolic systems by utilizing conserved quantities in the metabolic networks such as carriers such as ATP/ADP as an energy carrier or NADH/NAD+ as a hydrogen carrier. Although such conservation quantities restrict not only the steady state but also the dynamics themselves, the latter aspect has not yet been completely understood. Here, to study the dynamics of metabolic systems, we propose adopting a carrier cycling cascade (CCC), which includes the dynamics of both substrates and carriers, a commonly observed motif in metabolic systems such as the glycolytic and fermentation pathways. We demonstrate that the conservation laws lead to the jamming of the flux and feedback. The CCC can show slow relaxation, with a longer timescale than that of elementary reactions, and is accompanied by both robustness against small environmental fluctuations and responsiveness against large environmental changes. Moreover, the CCC demonstrates robustness against internal fluctuations due to the feedback based on the moiety conservation. We identified the key parameters underlying the robustness of this model against external and internal fluctuations and estimated it in several metabolic systems.

Robustness of spatial patterns in buffered reaction-diffusion systems and its reciprocity with phase plasticity.

The robustness of spatial patterns against perturbations is an indispensable property of developmental processes for organisms, which need to adapt to changing environments. Although specific mechanisms for this robustness have been extensively investigated, little is known about a general mechanism for achieving robustness in reaction-diffusion systems. Here, we propose a buffered reaction-diffusion system, in which active states of chemicals mediated by buffer molecules contribute to reactions, and demonstrate that robustness of the pattern wavelength is achieved by the dynamics of the buffer molecule. This robustness is analytically explained as a result of the scaling properties of the buffered system, which also lead to a reciprocal relationship between the wavelength's robustness and the plasticity of the spatial phase upon external perturbations. Finally, we explore the relevance of this reciprocity to biological systems.

Dynamics robustness of cascading systems.

A most important property of biochemical systems is robustness. Static robustness, e.g., homeostasis, is the insensitivity of a state against perturbations, whereas dynamics robustness, e.g., homeorhesis, is the insensitivity of a dynamic process. In contrast to the extensively studied static robustness, dynamics robustness, i.e., how a system creates an invariant temporal profile against perturbations, is little explored despite transient dynamics being crucial for cellular fates and are reported to be robust experimentally. For example, the duration of a stimulus elicits different phenotypic responses, and signaling networks process and encode temporal information. Hence, robustness in time courses will be necessary for functional biochemical networks. Based on dynamical systems theory, we uncovered a general mechanism to achieve dynamics robustness. Using a three-stage linear signaling cascade as an example, we found that the temporal profiles and response duration post-stimulus is robust to perturbations against certain parameters. Then analyzing the linearized model, we elucidated the criteria of when signaling cascades will display dynamics robustness. We found that changes in the upstream modules are masked in the cascade, and that the response duration is mainly controlled by the rate-limiting module and organization of the cascade's kinetics. Specifically, we found two necessary conditions for dynamics robustness in signaling cascades: 1) Constraint on the rate-limiting process: The phosphatase activity in the perturbed module is not the slowest. 2) Constraints on the initial conditions: The kinase activity needs to be fast enough such that each module is saturated even with fast phosphatase activity and upstream changes are attenuated. We discussed the relevance of such robustness to several biological examples and the validity of the above conditions therein. Given the applicability of dynamics robustness to a variety of systems, it will provide a general basis for how biological systems function dynamically.

Reciprocity Between Robustness of Period and Plasticity of Phase in Biological Clocks.

Circadian clocks exhibit the robustness of period and plasticity of phase against environmental changes such as temperature and nutrient conditions. Thus far, however, it is unclear how both are simultaneously achieved. By investigating distinct models of circadian clocks, we demonstrate reciprocity between robustness and plasticity: higher robustness in the period implies higher plasticity in the phase, where changes in period and in phase follow a linear relationship with a negative coefficient. The robustness of period is achieved by the adaptation on the limit cycle via a concentration change of a buffer molecule, whose temporal change leads to a phase shift following a shift of the limit-cycle orbit in phase space. Generality of reciprocity in clocks with the adaptation mechanism is confirmed with theoretical analysis of simple models, while biological significance is discussed.

Kinetic memory based on the enzyme-limited competition.

Cellular memory, which allows cells to retain information from their environment, is important for a variety of cellular functions, such as adaptation to external stimuli, cell differentiation, and synaptic plasticity. Although posttranslational modifications have received much attention as a source of cellular memory, the mechanisms directing such alterations have not been fully uncovered. It may be possible to embed memory in multiple stable states in dynamical systems governing modifications. However, several experiments on modifications of proteins suggest long-term relaxation depending on experienced external conditions, without explicit switches over multi-stable states. As an alternative to a multistability memory scheme, we propose "kinetic memory" for epigenetic cellular memory, in which memory is stored as a slow-relaxation process far from a stable fixed state. Information from previous environmental exposure is retained as the long-term maintenance of a cellular state, rather than switches over fixed states. To demonstrate this kinetic memory, we study several models in which multimeric proteins undergo catalytic modifications (e.g., phosphorylation and methylation), and find that a slow relaxation process of the modification state, logarithmic in time, appears when the concentration of a catalyst (enzyme) involved in the modification reactions is lower than that of the substrates. Sharp transitions from a normal fast-relaxation phase into this slow-relaxation phase are revealed, and explained by enzyme-limited competition among modification reactions. The slow-relaxation process is confirmed by simulations of several models of catalytic reactions of protein modifications, and it enables the memorization of external stimuli, as its time course depends crucially on the history of the stimuli. This kinetic memory provides novel insight into a broad class of cellular memory and functions. In particular, applications for long-term potentiation are discussed, including dynamic modifications of calcium-calmodulin kinase II and cAMP-response element-binding protein essential for synaptic plasticity.

Homeostasis of the period of post-translational biochemical oscillators.

Generally, circadian clocks or biological oscillations are resistant to external conditions such as temperature and nutrient concentration. We propose that enzyme-limited competition provides a general mechanism of homeostasis of the period of post-translational oscillators based on protein modifications, and demonstrate it by nutrient compensation in a theoretical model of cyanobacterial circadian clock. The rate change by nutrient concentration is counterbalanced by the amount of available free enzyme, which occurs because of the competition among the various substrates for the limited enzyme. The temperature and nutrient compensation are determined by the postulate that the catalytic modification reactions are rate limiting.

Generic temperature compensation of biological clocks by autonomous regulation of catalyst concentration.

Circadian clocks--ubiquitous in life forms ranging from bacteria to multicellular organisms--often exhibit intrinsic temperature compensation; the period of circadian oscillators is maintained constant over a range of physiological temperatures, despite the expected Arrhenius form for the reaction coefficient. Observations have shown that the amplitude of the oscillation depends on the temperature but the period does not; this suggests that although not every reaction step is temperature independent, the total system comprising several reactions still exhibits compensation. Here we present a general mechanism for such temperature compensation. Consider a system with multiple activation energy barriers for reactions, with a common enzyme shared across several reaction steps. The steps with the highest activation energy rate-limit the cycle when the temperature is not high. If the total abundance of the enzyme is limited, the amount of free enzyme available to catalyze a specific reaction decreases as more substrates bind to the common enzyme. We show that this change in free enzyme abundance compensates for the Arrhenius-type temperature dependence of the reaction coefficient. Taking the example of circadian clocks with cyanobacterial proteins KaiABC, consisting of several phosphorylation sites, we show that this temperature compensation mechanism is indeed valid. Specifically, if the activation energy for phosphorylation is larger than that for dephosphorylation, competition for KaiA shared among the phosphorylation reactions leads to temperature compensation. Moreover, taking a simpler model, we demonstrate the generality of the proposed compensation mechanism, suggesting relevance not only to circadian clocks but to other (bio)chemical oscillators as well.

Circadian transcriptional regulation by the posttranslational oscillator without de novo clock gene expression in Synechococcus.

Circadian rhythms are a fundamental property of most organisms, from cyanobacteria to humans. In the unicellular obligately photoautotrophic cyanobacterium Synechococcus elongatus PCC 7942, essentially all promoter activities are controlled by the KaiABC-based clock under continuous light conditions. When Synechococcus cells are transferred from the light to continuous dark (DD) conditions, the expression of most genes, including the clock genes kaiA and kaiBC, is rapidly down-regulated, whereas the KaiC phosphorylation cycle persists. Therefore, we speculated that the posttranslational oscillator might not drive the transcriptional circadian output without de novo expression of the kai genes. Here we show that the cyanobacterial clock regulates the transcriptional output even in the dark. The expression of a subset of genes in the genomes of cells grown in the dark was dramatically affected by kaiABC nullification, and the magnitude of dark induction was dependent on the time at which the cells were transferred from the light to the dark. Moreover, under DD conditions, the expression of some dark-induced gene transcripts exhibited temperature-compensated damped oscillations, which were nullified in kaiABC-null strains and were affected by a kaiC period mutation. These results indicate that the Kai protein-based posttranslational oscillator can drive the circadian transcriptional output even without the de novo expression of the clock genes.