Isologous diversification for robust development of cell society

A two-galectin network establishes mesenchymal condensation phenotype in limb development

Previous work showed that Gal-1A and Gal-8, two proteins belonging to the galactoside-binding galectin family, are the earliest determinants of the patterning of the skeletal elements of embryonic chicken limbs, and further, that their experimentally determined interactions in the embryonic limb bud can be interpreted via a reaction-diffusion-adhesion (2GL: two galectin plus ligands) model. Here, we use an ordinary differential equation-based approach to analyze the intrinsic switching modality of the 2GL network and characterize the network behavior independent of the diffusive and adhesive arms of the patterning mechanism. We identify two states: where the concentrations of both the galectins are respectively, negligible, and very high. This bistable switch-like system arises via a saddle-node bifurcation from a monostable state. For the case of mass-action production terms, we provide an explicit Lyapunov function for the system, which shows that it has no periodic solutions. Our model therefore predicts that the galectin network may exist in low expression and high expression states separated in space or time, without any intermediate states. We test these predictions in experiments performed with high density cultures of chick limb mesenchymal cells and observe that cells inside precartilage protocondensations express Gal-1A at a much higher rate than those outside, for which it was negligible. The Gal-1A and -8-based patterning network is therefore sufficient to partition the mesenchymal cell population into two discrete cell states with different developmental (chondrogenic vs. non-chondrogenic) fates. When incorporated into an adhesion and diffusion-enabled framework this system can generate a spatially patterned limb skeleton.

Multiscale modeling of vertebrate limb development

We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction-diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin-to-limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods.

Emergence of microbial diversity due to cross-feeding interactions in a spatial model of gut microbial metabolism

Background

The human gut contains approximately 10¹⁴ bacteria, belonging to hundreds of different species. Together, these microbial species form a complex food web that can break down nutrient sources that our own digestive enzymes cannot handle, including complex polysaccharides, producing short chain fatty acids and additional metabolites, e.g., vitamin K. Microbial diversity is important for colonic health: Changes in the composition of the microbiota have been associated with inflammatory bowel disease, diabetes, obesity and Crohn’s disease, and make the microbiota more vulnerable to infestation by harmful species, e.g., Clostridium difficile. To get a grip on the controlling factors of microbial diversity in the gut, we here propose a multi-scale, spatiotemporal dynamic flux-balance analysis model to study the emergence of metabolic diversity in a spatial gut-like, tubular environment. The model features genome-scale metabolic models (GEM) of microbial populations, resource sharing via extracellular metabolites, and spatial population dynamics and evolution.

Results

In this model, cross-feeding interactions emerge readily, despite the species’ ability to metabolize sugars autonomously. Interestingly, the community requires cross-feeding for producing a realistic set of short-chain fatty acids from an input of glucose, If we let the composition of the microbial subpopulations change during invasion of adjacent space, a complex and stratified microbiota evolves, with subspecies specializing on cross-feeding interactions via a mechanism of compensated trait loss. The microbial diversity and stratification collapse if the flux through the gut is enhanced to mimic diarrhea.

Conclusions

In conclusion, this in silico model is a helpful tool in systems biology to predict and explain the controlling factors of microbial diversity in the gut. It can be extended to include, e.g., complex nutrient sources, and host-microbiota interactions via the intestinal wall.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-017-0430-4) contains supplementary material, which is available to authorized users.

Symbiotic Cell Differentiation and Cooperative Growth in Multicellular Aggregates

As cells grow and divide under a given environment, they become crowded and resources are limited, as seen in bacterial biofilms and multicellular aggregates. These cells often show strong interactions through exchanging chemicals, as evident in quorum sensing, to achieve mutualism and division of labor. Here, to achieve stable division of labor, three characteristics are required. First, isogenous cells differentiate into several types. Second, this aggregate of distinct cell types shows better growth than that of isolated cells without interaction and differentiation, by achieving division of labor. Third, this cell aggregate is robust with respect to the number distribution of differentiated cell types. Indeed, theoretical studies have thus far considered how such cooperation is achieved when the ability of cell differentiation is presumed. Here, we address how cells acquire the ability of cell differentiation and division of labor simultaneously, which is also connected with the robustness of a cell society. For this purpose, we developed a dynamical-systems model of cells consisting of chemical components with intracellular catalytic reaction dynamics. The reactions convert external nutrients into internal components for cellular growth, and the divided cells interact through chemical diffusion. We found that cells sharing an identical catalytic network spontaneously differentiate via induction from cell-cell interactions, and then achieve division of labor, enabling a higher growth rate than that in the unicellular case. This symbiotic differentiation emerged for a class of reaction networks under the condition of nutrient limitation and strong cell-cell interactions. Then, robustness in the cell type distribution was achieved, while instability of collective growth could emerge even among the cooperative cells when the internal reserves of products were dominant. The present mechanism is simple and general as a natural consequence of interacting cells with limited resources, and is consistent with the observed behaviors and forms of several aggregates of unicellular organisms.

Unicellular organisms, when aggregated under limited resources, often exhibit behaviors akin to multicellular organisms, possibly without advanced regulation mechanisms, as observed in biofilms and bacterial colonies. Cells in an aggregate have to differentiate into several types that are specialized for different tasks, so that the growth rate should be enhanced by the division of labor among these cell types. To consider how a cell aggregate can acquire these properties, most theoretical studies have thus far assumed the fitness of an aggregate of cells and the ability of cell differentiation a priori. In contrast, we developed a dynamical-systems model consisting of cells without assuming predefined fitness. The model consists of catalytic-reaction networks for cellular growth. By extensive simulations and theoretical analysis of the model, we showed that cells growing under the condition of nutrient limitation and strong cell-cell interactions can differentiate with distinct chemical compositions. They achieve cooperative division of labor by exchanging the produced chemicals to attain a higher growth rate. The conditions for spontaneous cell differentiation and collective growth of cells are presented. The uncovered symbiotic differentiation and collective growth are akin to economic theory on division of labor and comparative advantage.

From globally coupled maps to complex-systems biology

Studies of globally coupled maps, introduced as a network of chaotic dynamics, are briefly reviewed with an emphasis on novel concepts therein, which are universal in high-dimensional dynamical systems. They include clustering of synchronized oscillations, hierarchical clustering, chimera of synchronization and desynchronization, partition complexity, prevalence of Milnor attractors, chaotic itinerancy, and collective chaos. The degrees of freedom necessary for high dimensionality are proposed to equal the number in which the combinatorial exceeds the exponential. Future analysis of high-dimensional dynamical systems with regard to complex-systems biology is briefly discussed.

Minimal model for stem-cell differentiation

To explain the differentiation of stem cells in terms of dynamical systems theory, models of interacting cells with intracellular protein expression dynamics are analyzed and simulated. Simulations were carried out for all possible protein expression networks consisting of two genes under cell-cell interactions mediated by the diffusion of a protein. Networks that show cell differentiation are extracted and two forms of symmetric differentiation based on Turing's mechanism and asymmetric differentiation are identified. In the latter network, the intracellular protein levels show oscillatory dynamics at a single-cell level, while cell-to-cell synchronicity of the oscillation is lost with an increase in the number of cells. Differentiation to a fixed-point-type behavior follows with a further increase in the number of cells. The cell type with oscillatory dynamics corresponds to a stem cell that can both proliferate and differentiate, while the latter fixed-point type only proliferates. This differentiation is analyzed as a saddle-node bifurcation on an invariant circle, while the number ratio of each cell type is shown to be robust against perturbations due to self-consistent determination of the effective bifurcation parameter as a result of the cell-cell interaction. Complex cell differentiation is designed by combing these simple two-gene networks. The generality of the present differentiation mechanism, as well as its biological relevance, is discussed.

Kinetic origin of heredity in a replicating system with a catalytic network

The origin of heredity is studied as a recursive state in a replicatingprotocell consisting of many molecule species in mutually catalyzingreaction networks. Protocells divide when the number of molecules, increasing due to replication, exceeds a certain threshold. We study how the chemicals in a catalytic network can form recursive production states in the presence of errors in the replication process. Depending on the balance between the total number of molecules in a cell and the number of molecule species, we have found three phases; a phase without a recursive production state, a phase with itinerancy over a few recursive states, and a phase with fixedrecursive production states. Heredity is realized in the latter two phaseswhere molecule species that are population-wise in the minority are preserved and control the phenotype of the cell. It is shown that evolvability is realized in the itinerancy phase, where a change in the number of minority molecules controls a change of the chemical state.

Structural Complexity of Early Embryos: A Study on the Nematode Caenorhabditis elegans

For analytical studies on the dynamics of gene expression, gene expressioncontrol and cellular interactions, the nematode Caenorhabditis elegans[C. elegans] is at present one of the best suited models [1-4].In this organism the genetic map and sequence is known [5], moreover theconstancy of its lineage tree allows a complete description of cellularclones giving rise to embryos. These characteristics have fostered detailedstudies on several aspects of development for this organism. Quantitativestudies of cellular movement, through time lapse cinematography of gastrulation, allows the description of cellular migrations giving rise to the final embryonic structure. In perspective, these studies coupledwith: genetic analysis, patterns of gene expression obtained throughmolecular techniques or other methods, open up the possibility of dynamicalstudies at the organismic scale. This possibility implies, first of all,a study of partitioning of space, and raise several problems in order todefine basic conceptual tools to be used in such studies. One of the mainproblems to handle in this respect is the definition of embryonic structurein a quantitative way. We will show that this aspect is a more generalcase of distance geometry approaches, as defined in protein folding studies.In this paper we discuss measures of the complexity for embryonal body plans,at the end of grastrulation. These can be applied to studies on the dynamicsof gene expression and phylogenetic studies with further experiments orsimulations.

Differences in cell division rates drive the evolution of terminal differentiation in microbes

Multicellular differentiated organisms are composed of cells that begin by developing from a single pluripotent germ cell. In many organisms, a proportion of cells differentiate into specialized somatic cells. Whether these cells lose their pluripotency or are able to reverse their differentiated state has important consequences. Reversibly differentiated cells can potentially regenerate parts of an organism and allow reproduction through fragmentation. In many organisms, however, somatic differentiation is terminal, thereby restricting the developmental paths to reproduction. The reason why terminal differentiation is a common developmental strategy remains unexplored. To understand the conditions that affect the evolution of terminal versus reversible differentiation, we developed a computational model inspired by differentiating cyanobacteria. We simulated the evolution of a population of two cell types –nitrogen fixing or photosynthetic– that exchange resources. The traits that control differentiation rates between cell types are allowed to evolve in the model. Although the topology of cell interactions and differentiation costs play a role in the evolution of terminal and reversible differentiation, the most important factor is the difference in division rates between cell types. Faster dividing cells always evolve to become the germ line. Our results explain why most multicellular differentiated cyanobacteria have terminally differentiated cells, while some have reversibly differentiated cells. We further observed that symbioses involving two cooperating lineages can evolve under conditions where aggregate size, connectivity, and differentiation costs are high. This may explain why plants engage in symbiotic interactions with diazotrophic bacteria.

The evolution of multicellularity is one of the most fascinating topics of evolutionary biology. Without multicellularity the incredible diversity of extant life would not be possible. In many multicellular organisms with specialized cells, some cell types become terminally differentiated (somatic cells) and lose the ability to reproduce new organisms while other cells maintain this ability (germline). Little is known about the conditions that favor the evolution of terminal differentiation in multicellular organisms. To understand this problem we have developed a computational model, inspired by multicellular cyanobacteria, in which the cells in an organism composed of two cell types (photosynthetic and nitrogen fixing) are allowed to evolve from germline to soma cells. We find three striking results. First, faster dividing cell types always evolve to become the germline. Second, the conditions under which we find different outcomes from the model are in good agreement with the different forms of development observed in multicellular cyanobacteria. Third, some conditions lead to a symbiotic state in which the two cell types separate into different lineages evolving independently of one another. Remarkably, cyanobacteria are also known to engage in symbiotic relationships with plants, producing fixed nitrogen for the plant in exchange for carbohydrates.

The challenges facing systemic approaches in biology: an interview with kunihiko kaneko

Interviewed by Kumar Selvarajoo and Masa Tsuchiya at Institute for Advanced Biosciences, Keio University, Japan, I discuss my approach to biology, what I call complex systems biology. The approach is constructive in nature, and is based on dynamical systems theory and statistical physics. It is intended to understand universal characteristics of life systems; generic adaptation under noise, differentiation from stem cells in interacting cells, robustness and plasticity in evolution, and so forth. Current status and future directions in systems biology in Japan are also discussed.

Characterization of stem cells and cancer cells on the basis of gene expression profile stability, plasticity, and robustness: dynamical systems theory of gene expressions under cell-cell interaction explains mutational robustness of differentiated cells and suggests how cancer cells emerge

Here I present and discuss a model that, among other things, appears able to describe the dynamics of cancer cell origin from the perspective of stable and unstable gene expression profiles. In identifying such aberrant gene expression profiles as lying outside the normal stable states attracted through development and normal cell differentiation, the hypothesis explains why cancer cells accumulate mutations, to which they are not robust, and why these mutations create a new stable state far from the normal gene expression profile space. Such cells are in strong contrast with normal cell types that appeared as an attractor state in the gene expression dynamical system under cell-cell interaction and achieved robustness to noise through evolution, which in turn also conferred robustness to mutation. In complex gene regulation networks, other aberrant cellular states lacking such high robustness are expected to remain, which would correspond to cancer cells.

Predicting pancreas cell fate decisions and reprogramming with a hierarchical multi-attractor model

Cell fate reprogramming, such as the generation of insulin-producing β cells from other pancreas cells, can be achieved by external modulation of key transcription factors. However, the known gene regulatory interactions that form a complex network with multiple feedback loops make it increasingly difficult to design the cell reprogramming scheme because the linear regulatory pathways as schemes of causal influences upon cell lineages are inadequate for predicting the effect of transcriptional perturbation. However, sufficient information on regulatory networks is usually not available for detailed formal models. Here we demonstrate that by using the qualitatively described regulatory interactions as the basis for a coarse-grained dynamical ODE (ordinary differential equation) based model, it is possible to recapitulate the observed attractors of the exocrine and β, δ, α endocrine cells and to predict which gene perturbation can result in desired lineage reprogramming. Our model indicates that the constraints imposed by the incompletely elucidated regulatory network architecture suffice to build a predictive model for making informed decisions in choosing the set of transcription factors that need to be modulated for fate reprogramming.

Unified description of regeneration by coupled dynamical systems theory: intercalary/segmented regeneration in insect legs

Regeneration phenomena are ubiquitous in nature and are studied in a variety of experiments. Positional information and feedback-loop hierarchy are theories that have been proposed to explain ordering rules in regeneration; however, some regeneration phenomena violate the rules derived from them. In particular, grafted junction stumps with the same value/hierarchy sometimes lead to one extra segmented portion, termed segmented regeneration. To present a unified description of all insect leg regeneration phenomena, we propose a theoretical mechanism for regeneration without postulating positional information, by using a model that consists of intracellular reaction dynamics of chemical concentrations, cell-to-cell interactions, and an increase in cell number. As a normal developmental process, successive differentiation from pluripotent cells appears, as described by transition from cells with intracellular chaotic dynamics to those with oscillatory or fixed-point dynamics. By assigning chaotic and nonchaotic cell types to corresponding positions instead of positional information, intercalary, segmented, and tarsus regeneration are explained coherently. With this assignment of pluripotency to chaotic dynamics, a unified description of regeneration is obtained with some predictive value for new experiments.

Noisy cell growth rate leads to fluctuating protein concentration in bacteria

The present study discusses a prime cause of fluctuating protein concentrations, which play a significant role in generating phenotypic diversity in bacteria. A genetic circuit integrated in a bacterial genome was used to evaluate the cell-to-cell variation in protein concentration. A simple dynamic model, comprising terms for synthesis and dilution, was used to elucidate the contributions of distinct noises to the fluctuation in cell protein concentration. Experimental and theoretical results demonstrated that noise in the rate of increase in cell volume (cell growth rate) serves as a source of extrinsic noise that accounts for dozens of percent of the total noise, whereas intrinsic noise in protein synthesis makes only a moderate contribution to the fluctuation in protein concentration. This suggests that such external noise in the cell growth rate has a global effect on cellular components, resulting in a large fluctuation in protein concentration in bacterial cells.

Chaotic expression dynamics implies pluripotency: when theory and experiment meet

Background

During normal development, cells undergo a unidirectional course of differentiation that progressively decreases the number of cell types they can potentially become. Pluripotent stem cells can differentiate into several types of cells, but terminally differentiated cells cannot differentiate any further. A fundamental problem in stem cell biology is the characterization of the difference in cellular states, e.g., gene expression profiles, between pluripotent stem cells and terminally differentiated cells.

Presentation of the hypothesis

To address the problem, we developed a dynamical systems model of cells with intracellular protein expression dynamics and interactions with each other. According to extensive simulations, cells with irregular (chaotic) oscillations in gene expression dynamics have the potential to differentiate into other cell types. During development, such complex oscillations are lost successively, leading to a loss of pluripotency. These simulation results, together with recent single-cell-level measurements in stem cells, led us to the following hypothesis regarding pluripotency: Chaotic oscillation in the expression of some genes leads to cell pluripotency and affords cellular state heterogeneity, which is supported by itinerancy over quasi-stable states. Differentiation stabilizes these states, leading to a loss of pluripotency.

Testing the hypothesis

To test the hypothesis, it is crucial to measure the time course of gene expression levels at the single-cell level by fluorescence microscopy and fluorescence-activated cell sorting (FACS) analysis. By analyzing the time series of single-cell-level expression data, one can distinguish whether the variation in protein expression level over time is due only to stochasticity in expression dynamics or originates from the chaotic dynamics inherent to cells, as our hypothesis predicts. By further analyzing the expression in differentiated cell types, one can examine whether the loss of pluripotency is accompanied by a loss of oscillation.

Implications of the hypothesis

Recovery of pluripotency from determined cells is a long-standing aspiration, from both scientific and clinical perspectives. Our hypothesis suggests a feasible route to recover the potential to differentiate, i.e., by increasing the variety of expressed genes to restore chaotic expression dynamics, as is consistent with the recent generation of induced pluripotent stem (iPS) cells.

Reviewers

This article was reviewed by David Krakauer, Jeroen van Zon (nominated by Rob de Boer), and Williams S. Hlavacek.

How selection affects phenotypic fluctuation

The large degree of phenotypic fluctuation among isogenic cells highlighted by recent studies on stochastic gene expression confers fitness on some individuals through a ‘bet-hedging' strategy, when faced with different selective environments. Under a single selective environment, the fluctuation may be suppressed through evolution, as it prevents maintenance of individuals around the fittest state and/or function. However, as fluctuation can increase phenotypic diversity, similar to mutation, it may contribute to the survival of individuals even under a single selective environment. To discuss whether the fluctuation increases over the course of evolution, cycles of mutation and selection for higher GFP fluorescence were carried out in Escherichia coli. Mutant genotypes possessing broad GFP fluorescence distributions with low average values emerged under strong selection pressure. These ‘broad mutants' appeared independently on the phylogenetic tree and increased fluctuations in GFP fluorescence were attributable to the variance in mRNA abundance. In addition to the average phenotypic change by genetic mutation, the observed increase in phenotypic fluctuation acts as an evolutionary strategy to produce an extreme phenotype under severe selective environments.

Shaping robust system through evolution

Biological functions are generated as a result of developmental dynamics that form phenotypes governed by genotypes. The dynamical system for development is shaped through genetic evolution following natural selection based on the fitness of the phenotype. Here we study how this dynamical system is robust to noise during development and to genetic change by mutation. We adopt a simplified transcription regulation network model to govern gene expression, which gives a fitness function. Through simulations of the network that undergoes mutation and selection, we show that a certain level of noise in gene expression is required for the network to acquire both types of robustness. The results reveal how the noise that cells encounter during development shapes any network's robustness, not only to noise but also to mutations. We also establish a relationship between developmental and mutational robustness through phenotypic variances caused by genetic variation and epigenetic noise. A universal relationship between the two variances is derived, akin to the fluctuation-dissipation relationship known in physics.

Dynamical patterning modules: physico-genetic determinants of morphological development and evolution

The shapes and forms of multicellular organisms arise by the generation of new cell states and types and changes in the numbers and rearrangements of the various kinds of cells. While morphogenesis and pattern formation in all animal species are widely recognized to be mediated by the gene products of an evolutionarily conserved 'developmental-genetic toolkit', the link between these molecular players and the physics underlying these processes has been generally ignored. This paper introduces the concept of 'dynamical patterning modules' (DPMs), units consisting of one or more products of the 'toolkit' genes that mobilize physical processes characteristic of chemically and mechanically excitable meso- to macroscopic systems such as cell aggregates: cohesion, viscoelasticity, diffusion, spatiotemporal heterogeneity based on lateral inhibition and multistable and oscillatory dynamics. We suggest that ancient toolkit gene products, most predating the emergence of multicellularity, assumed novel morphogenetic functions due to change in the scale and context inherent to multicellularity. We show that DPMs, acting individually and in concert with each other, constitute a 'pattern language' capable of generating all metazoan body plans and organ forms. The physical dimension of developmental causation implies that multicellular forms during the explosive radiation of animal body plans in the middle Cambrian, approximately 530 million years ago, could have explored an extensive morphospace without concomitant genotypic change or selection for adaptation. The morphologically plastic body plans and organ forms generated by DPMs, and their ontogenetic trajectories, would subsequently have been stabilized and consolidated by natural selection and genetic drift. This perspective also solves the apparent 'molecular homology-analogy paradox', whereby widely divergent modern animal types utilize the same molecular toolkit during development by proposing, in contrast to the Neo-Darwinian principle, that phenotypic disparity early in evolution occurred in advance of, rather than closely tracked, genotypic change.

When does division of labor lead to increased system output?

This paper develops a set of simplified dynamical models with which to explore the conditions under which division of labor leads to optimized system output, as measured by the rate of production of a given product. We consider two models: in the first model, we consider the flow of some resource into a compartment, and the conversion of this resource into some product. In the second model, we consider the growth of autoreplicating systems. In this case, we divide the replication and metabolic tasks among different agents. The general features that emerge from our models is that division of labor is favored when the resource to agent ratio is at intermediate values, and when the time cost associated with transporting intermediate products is small compared to characteristic process times. The results of our model are consistent with the behavior of the cellular slime mold Dictyostelium discodeum, which switches from a single-celled to a multi-celled state when resources become limited. We also argue that division of labor in the context of our replication model suggests an evolutionary basis for the emergence of the stem-cell-based tissue architecture in complex organisms. Finally, the results of this paper may be useful for understanding how, in an economic context, firm productivity is maximized at intermediate firm sizes.

Evolution of robustness to noise and mutation in gene expression dynamics

Phenotype of biological systems needs to be robust against mutation in order to sustain themselves between generations. On the other hand, phenotype of an individual also needs to be robust against fluctuations of both internal and external origins that are encountered during growth and development. Is there a relationship between these two types of robustness, one during a single generation and the other during evolution? Could stochasticity in gene expression have any relevance to the evolution of these types of robustness? Robustness can be defined by the sharpness of the distribution of phenotype; the variance of phenotype distribution due to genetic variation gives a measure of 'genetic robustness', while that of isogenic individuals gives a measure of 'developmental robustness'. Through simulations of a simple stochastic gene expression network that undergoes mutation and selection, we show that in order for the network to acquire both types of robustness, the phenotypic variance induced by mutations must be smaller than that observed in an isogenic population. As the latter originates from noise in gene expression, this signifies that the genetic robustness evolves only when the noise strength in gene expression is larger than some threshold. In such a case, the two variances decrease throughout the evolutionary time course, indicating increase in robustness. The results reveal how noise that cells encounter during growth and development shapes networks' robustness to stochasticity in gene expression, which in turn shapes networks' robustness to mutation. The necessary condition for evolution of robustness, as well as the relationship between genetic and developmental robustness, is derived quantitatively through the variance of phenotypic fluctuations, which are directly measurable experimentally.

Derivation of rigorous conditions for high cell-type diversity by algebraic approach

The development of a multicellular organism is a dynamic process. Starting with one or a few cells, the organism develops into different types of cells with distinct functions. We have constructed a simple model by considering the cell number increase and the cell-type order conservation, and have assessed conditions for cell-type diversity. This model is based on a stochastic Lindenmayer system with cell-to-cell interactions for three types of cells. In the present model, we have successfully derived complex but rigorous algebraic relations between the proliferation and transition rates for cell-type diversity by using a symbolic method: quantifier elimination (QE). Surprisingly, three modes for the proliferation and transition rates have emerged for large ratios of the initial cells to the developed cells. The three modes have revealed that the equality between the development rates for the highest cell-type diversity is reduced during the development process of multicellular organisms. Furthermore, we have found that the highest cell-type diversity originates from order conservation.

Turing pattern with proportion preservation

Although Turing pattern is one of the most universal mechanisms for pattern formation, in its standard model the number of stripes changes with the system size, since the wavelength of the pattern is invariant. It fails to preserve the proportionality of the pattern, i.e. the ratio of the wavelength to the size, that is often required in biological morphogenesis. To get over this problem, we show that the Turing pattern can preserve proportionality by introducing a catalytic chemical whose concentration depends on the system size. Several plausible mechanisms for such size dependence of the concentration are discussed. Following this general discussion, two models are studied in which arising Turing patterns indeed preserve the proportionality. Relevance of the present mechanism to biological morphogenesis is discussed from the viewpoint of its generality, robustness, and evolutionary accessibility.

Dynamical advantages of scale-free networks

A dynamical analysis of common network topologies is given and it is reported that a scale-free structure has two vital and distinctive features. First, complex but nevertheless reproducible states exist and, second, single-site induced state switching reminiscent of gene-expression control exists also. This indicates that scale-free networks have key dynamical advantages over other network topologies that could have contributed to their evolutionary success and thus may provide another reason for their prevalence in nature.

Evolutionary origin of power-laws in a biochemical reaction network: embedding the distribution of abundance into topology

The evolutionary origin of general statistics in a biochemical reaction network is studied here to explain the power-law distribution of reaction links and the power-law distribution of chemical abundance. Using cell models with catalytic reaction networks, we have confirmed that the power-law distribution for the abundance of chemicals emerges by the selection of cells with higher growth rates, as suggested in our previous study [Phys. Rev. Lett. 90, 088102 (2003)]. Through further evolution, this inhomogeneity in chemical abundance is shown to be embedded in the distribution of links, leading to the power-law distribution. We analyze the mechanism of this embedding and discuss the generality of the results.

Periodic forcing of a mathematical model of the eukaryotic cell cycle

In a differential equation model of the molecular network governing cell growth and division, cell cycle phases and transitions through checkpoints are associated with certain bifurcations of the underlying vector field. If the cell cycle is driven by another rhythmic process, interactions between forcing and bifurcations lead to emergent orbits and oscillations. In this paper, by varying the amplitude and frequency of forcing of the synthesis rates of regulatory proteins and the mass growth rate in a minimal model of the eukaryotic cell cycle, we study changes of the probability distributions of interdivision time and mass at division. By computing numerically the Lyapunov exponent of the model, we show that the splitting of probability distributions is associated with mode-locked solutions. We also introduce a simple, integrate-and-fire model to analyze mode locking in the cell cycle.

Externally controlled attractor selection in a high-dimensional system

A high-dimensional dynamical system with global couplings that can serve as a prototype for systems with very large numbers of attractors like memory is investigated and shown to be controllable by external inputs. By changing the duration that noise is added, final attractors are selected as the number of degrees of freedom of the nonsynchronized elements decreases one by one over time. Furthermore, it is found that this selection of attractors is also possible by controlling the sweeping speed of a parameter. The mechanism for this controlled selection is explained and shown to be rather general. Applications to attractor switching are given.

Selection of initial conditions for recursive production of multicellular organisms

The development of a multicellular organism is a dynamic process. Starting from one or a few cells, the organism becomes a set of cells with different types that form well-determined patterns. It is rather surprising that differentiation in cell types and formation of controlled patterns are compatible, because the former gives morphogenetic diversification whereas the latter implies recursive production of a cell ensemble, reducing individual differences. We studied this problem by taking a simple cell model with intracellular reaction dynamics of chemical concentrations, cell-cell interactions, and increase in cell numbers. We observed successive differentiation from a cell type with diverse chemicals and chaotic concentration dynamics to cell types with oscillatory or fixed-point dynamics, leading to morphogenetic diversity in a spatial pattern. We further show that, by starting from an initial object consisting of both the former cell type with diverse chemicals and the latter differentiated cell type, the recursive production of a multicellular organism with morphogenetic diversity is possible. By relating the former type to a cell in the vegetal pole and the latter to one in the animal pole, classic experimental results with separation of blastomeres in sea urchin eggs are coherently explained, while some predictions are made for in vitro morphogenesis from embryonic stem cells.

Conceptual models to understand tissue stem cell organization

PURPOSE OF REVIEW

Theoretic and, in particular, mathematic models can help biologists to select and design experiments, to highlight general principles, to discriminate similar and to link different phenomena, and to predict novel features. Specifically, they contribute to an understanding of latent mechanisms and crucial parameters of biologic processes. The following review gives an overview of recent developments in the field of hematopoietic tissue stem cell modeling.

RECENT FINDINGS

A number of experimental findings on heterogeneity, flexibility, and plasticity of hematopoietic and other tissue stem cells are challenging the classic stem cell concept of a predefined intrinsic stem cell program. Self-organizing systems provide a more elegant and comprehensive alternative to explain experimental data.

SUMMARY

Within the last few decades, modeling approaches in stem cell biology have evolved and now encompass a broad spectrum of phenomena, ranging from the cellular level to the tissue level. The application of theoretic models is currently suggesting that we abandon the classic assumption of a strict developmental hierarchy and understand stem cell organization as a dynamic, functional process. Such a perspective has implications for a prospective characterization of tissue stem cells (eg, regarding gene expression profiles and genetic regulation patterns).

Dynamical systems basis of metamorphosis: diversity and plasticity of cellular states in reaction diffusion network

Dynamics maintaining diversity of cell types in a multi-cellular system are studied in relation to the plasticity of cellular states. By adopting a simple theoretical framework for intra-cellular chemical reaction dynamics and considering the division and death of cells, the development of cells is studied. Cell differentiation process is found to occur through instability in transient dynamics and cell-cell interaction. In long-term behavior, extinction of multiple cells is repeated, which leads to itinerancy over successive quasi-stable multi-cellular states consisting of different types of cells. By defining the plasticity of a cellular state, it is shown that the plasticity of cells decreases before the extinction of most cell types, from which diversity and plasticity are later recovered. In the following, a decrease of plasticity occurs again, leading to the next extinction. This cycle of diversification and extinction is repeated. Relevance of our results to development and evolution is briefly discussed.

Robust development as a consequence of generated positional information

The origin and robustness of morphogenesis are studied by dynamical system modeling of a cell society, in which cells possessing internal chemical reaction dynamics interact with each other through their mutual interaction with diffusive chemicals in a two-dimensional medium. It is found that stem-type cells differentiate into various cell types (where a cell 'type' is defined by a type of intra-cellular dynamics) due to a dynamic instability caused by cell-cell interactions in a manner described by the isologous diversification theory. The differentiations are spatially regulated by the concentration of chemicals in the medium, while the chemical concentrations are locally influenced by the intra-cell dynamics. Through this reciprocal relationship, chemical concentrations come to exhibit spatial variation as differentiated cell types begin to emerge, and as a result the regulation exercised by the chemical concentrations become spatially inhomogeneous. This reinforces the process of differentiation, through which spatial patterns of differentiated cells appear. Within this reciprocal relationship, the concentration gradients are read and interpreted by the cell as positional information. A spatial order of cells realized in this process represents a stable state of the system governed by this reciprocal relationship, and that the developmental process through which this state is realized is robust with respect to perturbations. The dependence of the morphogenesis on history and the community effect in cell differentiation are also discussed.

Collective specification of cellular development

Studies of chimeras and in vivo development demonstrate that cell lineages are often quite variable, apparently in response to chance perturbations. This points to an apparent contradiction: although individual cells are the units of genetic information and differentiation, not all cellular events need be precise for the development of functional organisms. The social organization of ants can serve as a metaphor that helps understand the mechanisms that underlie such development. Ants suggest that continued cellular interactions and environmental conditions could specify the proportion and general location of specialized units. Leaf venation is used as a concrete example of this general principle. A signal produced continuously by all cells specifies a requirement for vein differentiation. The cells that respond by differentiation then transport the signal away from the leaf; this removal acting as a feedback indicating that the requirement is being met. Because transport increases during vein differentiation, early initiation occurs in excess and vein 'competition' for the signal assures an acceptable outcome. Such specification would be robust since it does not depend on events in any single cell, and chance events, rather than being corrected or reversed, may be built upon in reaching an expected, collective phenotype. The absence of detailed information preceding development distinguishes this hypothesis from the common alternatives of a program or blueprint. Collective specification would have important implications for developmental plasticity and evolution.

Origin of multicellular organisms as an inevitable consequence of dynamical systems

The origin of multicellular organisms is studied by considering a cell system that satisfies minimal conditions, that is, a system of interacting cells with intracellular biochemical dynamics, and potentiality in reproduction. Three basic features in multicellular organisms-cellular diversification, robust developmental process, and emergence of germ-line cells-are found to be general properties of such a system. Irrespective of the details of the model, such features appear when there are complex oscillatory dynamics of intracellular chemical concentrations. Cells differentiate from totipotent stem cells into other cell types due to instability in the intracellular dynamics with cell-cell interactions, as explained by our isologous diversification theory (Furusawa and Kaneko, 1998a; Kaneko and Yomo, 1997). This developmental process is shown to be stable with respect to perturbations, such as molecular fluctuations and removal of some cells. By further imposing an adequate cell-type-dependent adhesion force, some cells are released, from which the next generation cell colony is formed, and a multicellular organism life-cycle emerges without any finely tuned mechanisms. This recursive production of multicellular units is stabilized if released cells are few in number, implying the separation of germ cell lines. Furthermore, such an organism with a variety of cellular states and robust development is found to maintain a larger growth speed as an ensemble by achieving a cooperative use of resources, compared to simple cells without differentiation. Our results suggest that the emergence of multicellular organisms is not a "difficult problem" in evolution, but rather is a natural consequence of a cell colony that can grow continuously.

On a kinetic origin of heredity: minority control in a replicating system with mutually catalytic molecules

As the first step in an investigation of the origin of genetic information, we study how some species of molecules are preserved over cell generations and play an important role in controlling the growth of a cell. We consider a model consisting of protocells. Each protocell contains two mutually catalysing molecule species (X and Y), each of which has catalytically active and inactive types. One of the species Y is assumed to have a slower synthesis speed. Through divisions of the protocells, the system reaches and remains in a state in which there are only a few active Y and almost no inactive Y molecules in most protocells, through the selection of very rare fluctuations. In this state, the active Y molecules are shown to control the behavior of the protocell. The minority molecule species act as the carrier of heredity, due to the relatively discrete nature of its population, in comparison with the majority species which behaves statistically in accordance with the law of large numbers. The minority controlled state may give rise to a selection pressure for mechanisms that ensure the transmission of the minority molecule. Once those mechanisms are in place, the minority molecule becomes the ideal storage device for information to be transmitted across generations, thus giving rise to "genetic information". The relevance of this minority controlled state to evolvability is also discussed.

Theory of robustness of irreversible differentiation in a stem cell system: chaos hypothesis

Based on an extensive study of a dynamical systems model of the development of a cell society, a novel theory for stem cell differentiation and its regulation is proposed as the "chaos hypothesis". Two fundamental features of stem cell systems-stochastic differentiation of stem cells and the robustness of a system due to regulation of this differentiation-are found to be general properties of a system of interacting cells exhibiting chaotic intra-cellular reaction dynamics and cell division, whose presence does not depend on the detail of the model. It is found that stem cells differentiate into other cell types stochastically due to a dynamical instability caused by cell-cell interactions, in a manner described by the Isologous Diversification theory. This developmental process is shown to be stable not only with respect to molecular fluctuations but also with respect to the removal of cells. With this developmental process, the irreversible loss of multipotency accompanying the change from a stem cell to a differentiated cell is shown to be characterized by a decrease in the chemical diversity in the cell and of the complexity of the cellular dynamics. The relationship between the division speed and the loss of multipotency is also discussed. Using our model, some predictions that can be tested experimentally are made for a stem cell system.

Transitions induced by the discreteness of molecules in a small autocatalytic system

The autocatalytic reaction system with a small number of molecules is studied numerically by stochastic particle simulations. A novel state due to fluctuation and discreteness in molecular numbers is found, characterized as an extinction of molecule species alternately in the autocatalytic reaction loop. Phase transition to this state with changes of the system size and flow is studied, while a single-molecule switch of the molecule distributions is reported. The relevance of the results to intracellular processes is briefly discussed.

Sympatric speciation: compliance with phenotype diversification from a single genotype

A novel mechanism for sympatric speciation that takes into account complex bioprocesses within each individual organism is proposed. According to dynamical systems theory, organisms with identical genotypes can possess differentiated physiological states and may coexist 'symbiotically' through appropriate mutual interaction. With mutations, the phenotypically differentiated organisms gradually come to possess distinct genotypes while maintaining their symbiotic relationship. This symbiotic speciation is robust against sexual recombination, because offspring of mixed parentage with intermediate genotypes are less fit than their parents. This leads to sterility of the hybrid. Accordingly, a basis for mating preference also arises.

Origin of complexity in multicellular organisms

Through extensive studies of dynamical system modeling cellular growth and reproduction, we find evidence that complexity arises in multicellular organisms naturally through evolution. Without any elaborate control mechanism, these systems can exhibit complex pattern formation with spontaneous cell differentiation. Such systems employ a "cooperative" use of resources and maintain a larger growth speed than simple cell systems, which exist in a homogeneous state and behave "selfishly." The relevance of the diversity of chemicals and reaction dynamics to the growth of a multicellular organism is demonstrated. Chaotic biochemical dynamics are found to provide the multipotency of stem cells.

Complex organization in multicellularity as a necessity in evolution

By introducing a dynamical system model of a multicellular system, it is shown that an organism with a variety of differentiated cell types and a complex pattern emerges through cell-cell interactions even without postulating any elaborate control mechanism. Such an organism is found to maintain a larger growth speed as an ensemble, by achieving a cooperative use of resources, than do simple homogeneous cells, which behave "selfishly." This suggests that the emergence of multicellular organisms with complex organization is a necessity in evolution. According to our theoretical model, there initially appear multipotent stem cells, which undergo stochastic differentiation to other cell types. With development and differentiation, both the chemical diversity and the complexity of intra-cellular dynamics are decreased, as a general consequence of our system. Robustness of the developmental process is also confirmed.

Evolution of genetic codes through isologous diversification of cellular states

Evolution of genetic codes is studied as change in the choice of enzymes that are used to synthesize amino acids from the genetic information of nucleic acids. We propose the following theory: the differentiation of physiological states of a cell allows for a choice of enzymes, and this choice is later fixed genetically through evolution. To demonstrate this theory, a dynamical systems model consisting of the concentrations of metabolites, enzymes, amino acyl tRNA synthetase, and tRNA - amino acid complexes in a cell is introduced and studied numerically. It is shown that the biochemical states of cells are differentiated by cell-cell interactions, and each differentiated type starts to use a different synthetase. Through the mutation of genes, this difference in the genetic code is amplified and stabilized. The relevance of this theory to the evolution of non-universal genetic code in mitochondria is suggested. The present theory is based on our recent theory of isologous symbiotic speciation, which is briefly reviewed. According to the theory, phenotypes of organisms are first differentiated into distinct types through the interaction and developmental dynamics, even though they have identical genotypes; later, with mutation in the genotype, the genotype also differentiates into discrete types, while maintaining the "symbiotic" relationship between the types. Relevance of the theory to natural as well as artificial evolution is discussed.