Evaluation of entropy and fractal dimension as biomarkers for tumor growth and treatment response using cellular automata.

Cell-based models provide a helpful approach for simulating complex systems that exhibit adaptive, resilient qualities, such as cancer. Their focus on individual cell interactions makes them a particularly appropriate strategy to study cancer therapies' effects, which are often designed to disrupt single-cell dynamics. In this work, we propose them as viable methods for studying the time evolution of cancer imaging biomarkers (IBM). We propose a cellular automata model for tumor growth and three different therapies: chemotherapy, radiotherapy, and immunotherapy, following well-established modeling procedures documented in the literature. The model generates a sequence of tumor images, from which a time series of two biomarkers: entropy and fractal dimension, is obtained. Our model shows that the fractal dimension increased faster at the onset of cancer cell dissemination. At the same time, entropy was more responsive to changes induced in the tumor by the different therapy modalities. These observations suggest that the prognostic value of the proposed biomarkers could vary considerably with time. Thus, it is essential to assess their use at different stages of cancer and for different imaging modalities. Another observation derived from the results was that both biomarkers varied slowly when the applied therapy attacked cancer cells scattered along the automatons' area, leaving multiple independent clusters of cells at the end of the treatment. Thus, patterns of change of simulated biomarkers time series could reflect on essential qualities of the spatial action of a given cancer intervention.

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.

Cell patterns emerge from coupled chemical and physical fields with cell proliferation dynamics: the Arabidopsis thaliana root as a study system.

A central issue in developmental biology is to uncover the mechanisms by which stem cells maintain their capacity to regenerate, yet at the same time produce daughter cells that differentiate and attain their ultimate fate as a functional part of a tissue or an organ. In this paper we propose that, during development, cells within growing organs obtain positional information from a macroscopic physical field that is produced in space while cells are proliferating. This dynamical interaction triggers and responds to chemical and genetic processes that are specific to each biological system. We chose the root apical meristem of Arabidopsis thaliana to develop our dynamical model because this system is well studied at the molecular, genetic and cellular levels and has the key traits of multicellular stem-cell niches. We built a dynamical model that couples fundamental molecular mechanisms of the cell cycle to a tension physical field and to auxin dynamics, both of which are known to play a role in root development. We perform extensive numerical calculations that allow for quantitative comparison with experimental measurements that consider the cellular patterns at the root tip. Our model recovers, as an emergent pattern, the transition from proliferative to transition and elongation domains, characteristic of stem-cell niches in multicellular organisms. In addition, we successfully predict altered cellular patterns that are expected under various applied auxin treatments or modified physical growth conditions. Our modeling platform may be extended to explicitly consider gene regulatory networks or to treat other developmental systems.

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.

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.