A Variational Approach to Morphogenesis : Recovering Spatial Phenotypic Features from Epigenetic Landscapes.

We model the process of cell fate determination of the flower Arabidopsis-thaliana employing a system of reaction-diffusion equations governed by a potential field. This potential field mimics the flower's epigenetic landscape as defined by Waddington. It is derived from the underlying genetic regulatory network (GRN), which is based on detailed experimental data obtained during cell fate determination in the early stages of development of the flower. The system of equations has a variational structure, and we use minimax techniques (in particular the Mountain Pass Lemma) to show that the minimal energy solution of our functional is, in fact, the one that traverses the epigenetic landscape (the potential field) in the spatial order that corresponds to the correct architecture of the flower, that is, following the observed geometrical features of the meristem. This approach can generally be applied to systems with similar structures to establish a genotype to phenotype correspondence. From a broader perspective, this problem is related to phase transition models with a multiwell vector potential, and the results and methods presented here can potentially be applied in this case.

Spatial dynamics of floral organ formation.

Understanding the emergence of biological structures and their changes is a complex problem. On a biochemical level, it is based on gene regulatory networks (GRN) consisting on interactions between the genes responsible for cell differentiation and coupled in a greater scale with external factors. In this work we provide a systematic methodological framework to construct Waddington's epigenetic landscape of the GRN involved in cellular determination during the early stages of development of angiosperms. As a specific example we consider the flower of the plant Arabidopsis thaliana. Our model, which is based on experimental data, recovers accurately the spatial configuration of the flower during cell fate determination, not only for the wild type, but for its homeotic mutants as well. The method developed in this project is general enough to be used in the study of the relationship between genotype-phenotype in other living organisms.

Objects and processes: Two notions for understanding biological information.

In spite of being ubiquitous in life sciences, the concept of information is harshly criticized. Uses of the concept other than those derived from Shannon׳s theory are denounced as metaphoric. We perform a computational experiment to explore whether Shannon׳s information is adequate to describe the uses of said concept in commonplace scientific practice. Our results show that semantic sequences do not have unique complexity values different from the value of meaningless sequences. This result suggests that quantitative theoretical frameworks do not account fully for the complex phenomenon that the term "information" refers to. We propose a restructuring of the concept into two related, but independent notions, and conclude that a complete theory of biological information must account completely not only for both notions, but also for the relationship between them.

Analysis of Shannon's entropy to contrast between the Embodied and Neurocentrist hypothesis of conscious experience.

We usually accept that consciousness is in the brain. This statement corresponds to a Neurocentrist view. However, with all the physical and physiological data currently available, a convincing explanation of how consciousness emerges has not been given this topic is aborded by Anil Seth (2021). Because of this, a natural question arises: Is consciousness really in the brain or not? If the answer is no, this corresponds to the Embodied perspective. We cannot discriminate between these two points of view because we cannot identify how the organism processes the information. If we try to measure information processing in the brain, then the Neurocentrist view is unavoidable. For example, the information integration theory of Tononi's research group and the global work area theory developed by Dehaene and Baars, focus solely on the brain without considering aspects of Embodied vision (See Tononi, 2021; Dehaene, 2021). In this article, we propose an index based on Shannon's entropy, capable of identifying the leading processing elements acting: Are they mainly inner or external? In order to validate it, we performed simulations with networks accounting for different amounts of internal and outer layers. Since Shannon's entropy is an abstract measure of the information content, this index is not dependent on the physical network nor the proportion of different layers. Therefore, we validate the index as free of bias. This index is a way to discriminate between Embodied from Neurocentrist hypotheses.

Uncoding the interdependency of tumor microenvironment and macrophage polarization: insights from a continuous network approach.

The balance between pro- and anti-inflammatory immune system responses is crucial to preventing complex diseases like cancer. Macrophages are essential immune cells that contribute to this balance constrained by the local signaling profile of the tumor microenvironment. To understand how pro- and anti-inflammatory unbalance emerges in cancer, we developed a theoretical analysis of macrophage differentiation that is derived from activated monocytes circulating in the blood. Once recruited to the site of inflammation, monocytes can be polarized based on the specific interleukins and chemokines in the microenvironment. To quantify this process, we used a previous regulatory network reconstructed by our group and transformed Boolean Network attractors of macrophage polarization to an ODE scheme, it enables us to quantify the activation of their genes in a continuous fashion. The transformation was developed using the interaction rules with a fuzzy logic approach. By implementing this approach, we analyzed different aspects that cannot be visualized in the Boolean setting. For example, this approach allows us to explore the dynamic behavior at different concentrations of cytokines and transcription factors in the microenvironment. One important aspect to assess is the evaluation of the transitions between phenotypes, some of them characterized by an abrupt or a gradual transition depending on specific concentrations of exogenous cytokines in the tumor microenvironment. For instance, IL-10 can induce a hybrid state that transits between an M2c and an M2b macrophage. Interferon- γ can induce a hybrid between M1 and M1a macrophage. We further demonstrated the plasticity of macrophages based on a combination of cytokines and the existence of hybrid phenotypes or partial polarization. This mathematical model allows us to unravel the patterns of macrophage differentiation based on the competition of expression of transcriptional factors. Finally, we survey how macrophages may respond to a continuously changing immunological response in a tumor microenvironment.

From ABC genes to regulatory networks, epigenetic landscapes and flower morphogenesis: making biological sense of theoretical approaches.

The ABC model postulates that expression combinations of three classes of genes (A, B and C) specify the four floral organs at early stages of flower development. This classic model provides a solid framework to study flower development and has been the foundation for multiple studies in different plant species, as well as for new evolutionary hypotheses. Nevertheless, it has been shown that in spite of being necessary, these three gene classes are not sufficient for flower organ specification. Rather, flower organ specification depends on complex interactions of several genes, and probably other non-genetic factors. Being useful to study systems of complex interactions, mathematical and computational models have enlightened the origin of the A, B and C stereotyped and robust expression patterns and the process of early flower morphogenesis. Here, we present a brief introduction to basic modeling concepts and techniques and review the results that these models have rendered for the particular case of the Arabidopsis thaliana flower organ specification. One of the main results is the uncovering of a robust functional module that is sufficient to recover the gene configurations characterizing flower organ primordia. Another key result is that the temporal sequence with which such gene configurations are attained may be recovered only by modeling the aforementioned functional module as a noisy or stochastic system. Finally, modeling approaches enable testable predictions regarding the role of non-genetic factors (noise, mechano-elastic forces, etc.) in development. These predictions, along with some perspectives for future work, are also reviewed and discussed.

General theory of genotype to phenotype mapping: derivation of epigenetic landscapes from N-node complex gene regulatory networks.

We propose a systematic methodology to construct a probabilistic epigenetic landscape of cell-fate attainment associated with N-node Boolean genetic regulatory networks. The general derivation proposed here is exemplified with an Arabidopsis thaliana network underlying floral organ determination grounded on qualitative experimental data.

Epigenetic forest and flower morphogenesis.

This paper studies the epigenetic process that leads to Angiosperms' flower architecture (flowering plants). As a case study, we analyze the flower Arabidopsis thaliana's GRN obtained during cell fate determination in the early stages of the flower's development, which was constructed in a previous work using experimental data. We start by constructing and analyzing the Epigenetic Forest, a discrete representation of Waddington's Epigenetic Landscape, obtained as the transition graph of the discrete dynamical system associated with the GRN. Next, we propose an optimization problem to model morphogenesis by defining a biologically meaningful function that accounts for the work involved in cell specialization. Finally, the problem is solved using a genetic algorithm. The optimal solution found by the algorithm correctly recovers the flower's architecture, as observed in wild type flowers and recovered in other theoretical works. Even though the case study addresses this specific problem, the method is directly applicable to other GRN's with attractors consisting of equilibrium points only and could be extended to the situation where there are periodic attractors.

Gene regulatory network models for plant development.

Accumulated genetic data are stimulating the use of mathematical and computational tools for studying the concerted action of genes during cell differentiation and morphogenetic processes. At the same time, network theory has flourished, enabling analyses of complex systems that have multiple elements and interactions. Reverse engineering methods that use genomic data or detailed experiments on gene interactions have been used to propose gene network architectures. Experiments on gene interactions incorporate enough detail for relatively small developmental modules and thus allow dynamical analyses that have direct functional interpretations. Generalities are beginning to emerge. For example, biological genetic networks are robust to environmental and genetic perturbations. Such dynamical studies also enable novel predictions that can lead to further experimental tests, which might then feedback to the theoretical analyses. This interplay is proving productive for understanding plant development. Finally, both experiments on gene interactions and theoretical analyses allow the identification of frequent or fixed evolutionary solutions to developmental problems, and thus are contributing to an understanding of the genetic basis of the evolution of development and body plan.

Equivalent genetic regulatory networks in different contexts recover contrasting spatial cell patterns that resemble those in Arabidopsis root and leaf epidermis: a dynamic model.

In Arabidopsis thaliana, leaf and root epidermis hairs exhibit contrasting spatial arrangements even though the genetic networks regulating their respective cell-fate determination have very similar structures and components. We integrated available experimental data for leaf and root hair patterning in dynamic network models which may be reduced to activator-inhibitor models. This integration yielded expected results for these kinds of dynamic models, including striped and dotted cell patterns which are characteristic of root and leaf epidermis, respectively. However, these formal tools have led us to novel insights on current data and to put forward precise hypotheses which can be addressed experimentally. In particular, despite subtle differences in the root and leaf networks, these have equivalent dynamical behaviors. Our simulations also suggest that only when a biasing signal positively affects an activator in the network, the system recovers striped cellular patterns similar to those of root epidermis. We also postulate that cell shape may affect pattern stability in the root. Our results thus support the idea that in this and other cases, contrasting spatial cell patterns and other evolutionary morphogenetic novelties originate from conserved genetic network modules subject to divergent contextual traits.

Modeling some properties of circadian rhythms.

Mathematical models have been very useful in biological research. From the interaction of biology and mathematics, new problems have emerged that have generated advances in the theory, suggested further experimental work and motivated plausible conjectures. From our perspective, it is absolutely necessary to incorporate modeling tools in the study of circadian rhythms and that without a solid mathematical framework a real understanding of them will not be possible. Our interest is to study the main process underlying the synchronization in the pacemaker of a circadian system: these mechanisms should be conserved in all living beings. Indeed, from an evolutionary perspective, it seems reasonable to assume that either they have a common origin or that they emerge from similar selection circumstances. We propose a general framework to understand the emergence of synchronization as a robust characteristic of some cooperative systems of non-linear coupled oscillators. In a first approximation to the problem we vary the topology of the network and the strength of the interactions among oscillators. In order to study the emergent dynamics, we carried out some numerical computations. The results are consistent with experiments reported in the literature. Finally, we proposed a theoretical framework to study the phenomenon of synchronization in the context of circadian rhythms: the dissipative synchronization of nonautonomous dynamical systems.

Single-cell profiling of epigenetic modifiers identifies PRDM14 as an inducer of cell fate in the mammalian embryo.

Cell plasticity or potency is necessary for the formation of multiple cell types. The mechanisms underlying this plasticity are largely unknown. Preimplantation mouse embryos undergo drastic changes in cellular potency, starting with the totipotent zygote through to the formation of the pluripotent inner cell mass (ICM) and differentiated trophectoderm in the blastocyst. Here, we set out to identify and functionally characterize chromatin modifiers that define the transitions of potency and cell fate in the mouse embryo. Using a quantitative microfluidics approach in single cells, we show that developmental transitions are marked by distinctive combinatorial profiles of epigenetic modifiers. Pluripotent cells of the ICM are distinct from their differentiated trophectoderm counterparts. We show that PRDM14 is heterogeneously expressed in 4-cell-stage embryos. Forced expression of PRDM14 at the 2-cell stage leads to increased H3R26me2 and can induce a pluripotent ICM fate. Our results shed light on the epigenetic networks that govern cellular potency and identity in vivo.

Dynamic network-based epistasis analysis: boolean examples.

In this article we focus on how the hierarchical and single-path assumptions of epistasis analysis can bias the inference of gene regulatory networks. Here we emphasize the critical importance of dynamic analyses, and specifically illustrate the use of Boolean network models. Epistasis in a broad sense refers to gene interactions, however, as originally proposed by Bateson, epistasis is defined as the blocking of a particular allelic effect due to the effect of another allele at a different locus (herein, classical epistasis). Classical epistasis analysis has proven powerful and useful, allowing researchers to infer and assign directionality to gene interactions. As larger data sets are becoming available, the analysis of classical epistasis is being complemented with computer science tools and system biology approaches. We show that when the hierarchical and single-path assumptions are not met in classical epistasis analysis, the access to relevant information and the correct inference of gene interaction topologies is hindered, and it becomes necessary to consider the temporal dynamics of gene interactions. The use of dynamical networks can overcome these limitations. We particularly focus on the use of Boolean networks that, like classical epistasis analysis, relies on logical formalisms, and hence can complement classical epistasis analysis and relax its assumptions. We develop a couple of theoretical examples and analyze them from a dynamic Boolean network model perspective. Boolean networks could help to guide additional experiments and discern among alternative regulatory schemes that would be impossible or difficult to infer without the elimination of these assumption from the classical epistasis analysis. We also use examples from the literature to show how a Boolean network-based approach has resolved ambiguities and guided epistasis analysis. Our article complements previous accounts, not only by focusing on the implications of the hierarchical and single-path assumption, but also by demonstrating the importance of considering temporal dynamics, and specifically introducing the usefulness of Boolean network models and also reviewing some key properties of network approaches.

Interlinked nonlinear subnetworks underlie the formation of robust cellular patterns in Arabidopsis epidermis: a dynamic spatial model.

BACKGROUND: Dynamical models are instrumental for exploring the way information required to generate robust developmental patterns arises from complex interactions among genetic and non-genetic factors. We address this fundamental issue of developmental biology studying the leaf and root epidermis of Arabidopsis. We propose an experimentally-grounded model of gene regulatory networks (GRNs) that are coupled by protein diffusion and comprise a meta-GRN implemented on cellularised domains. RESULTS: Steady states of the meta-GRN model correspond to gene expression profiles typical of hair and non-hair epidermal cells. The simulations also render spatial patterns that match the cellular arrangements observed in root and leaf epidermis. As in actual plants, such patterns are robust in the face of diverse perturbations. We validated the model by checking that it also reproduced the patterns of reported mutants. The meta-GRN model shows that interlinked sub-networks contribute redundantly to the formation of robust hair patterns and permits to advance novel and testable predictions regarding the effect of cell shape, signalling pathways and additional gene interactions affecting spatial cell-patterning. CONCLUSION: The spatial meta-GRN model integrates available experimental data and contributes to further understanding of the Arabidopsis epidermal system. It also provides a systems biology framework to explore the interplay among sub-networks of a GRN, cell-to-cell communication, cell shape and domain traits, which could help understanding of general aspects of patterning processes. For instance, our model suggests that the information needed for cell fate determination emerges from dynamic processes that depend upon molecular components inside and outside differentiating cells, suggesting that the classical distinction of lineage versus positional cell differentiation may be instrumental but rather artificial. It also suggests that interlinkage of nonlinear and redundant sub-networks in larger networks is important for pattern robustness. Pursuing dynamic analyses of larger (genomic) coupled networks is still not possible. A repertoire of well-characterised regulatory modules, like the one presented here, will, however, help to uncover general principles of the patterning-associated networks, as well as the peculiarities that originate diversity.

Floral morphogenesis: stochastic explorations of a gene network epigenetic landscape.

In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5-10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.

A gene regulatory network model for cell-fate determination during Arabidopsis thaliana flower development that is robust and recovers experimental gene expression profiles.

Flowers are icons in developmental studies of complex structures. The vast majority of 250,000 angiosperm plant species have flowers with a conserved organ plan bearing sepals, petals, stamens, and carpels in the center. The combinatorial model for the activity of the so-called ABC homeotic floral genes has guided extensive experimental studies in Arabidopsis thaliana and many other plant species. However, a mechanistic and dynamical explanation for the ABC model and prevalence among flowering plants is lacking. Here, we put forward a simple discrete model that postulates logical rules that formally summarize published ABC and non-ABC gene interaction data for Arabidopsis floral organ cell fate determination and integrates this data into a dynamic network model. This model shows that all possible initial conditions converge to few steady gene activity states that match gene expression profiles observed experimentally in primordial floral organ cells of wild-type and mutant plants. Therefore, the network proposed here provides a dynamical explanation for the ABC model and shows that precise signaling pathways are not required to restrain cell types to those found in Arabidopsis, but these are rather determined by the overall gene network dynamics. Furthermore, we performed robustness analyses that clearly show that the cell types recovered depend on the network architecture rather than on specific values of the model's gene interaction parameters. These results support the hypothesis that such a network constitutes a developmental module, and hence provide a possible explanation for the overall conservation of the ABC model and overall floral plan among angiosperms. In addition, we have been able to predict the effects of differences in network architecture between Arabidopsis and Petunia hybrida.