The unique, hierarchical patterns of leaf veins have attracted extensive attention in recent years. However, it remains unclear how biological and mechanical factors influence the topology of leaf veins. In this paper, we investigate the optimization mechanisms of leaf veins through a combination of experimental measurements and numerical simulations. The topological details of three types of representative plant leaves are measured. The experimental results show that the vein patterns are insensitive to leaf shapes and curvature. The numbers of secondary veins are independent of the length of the main vein, and the total length of veins increases linearly with the leaf perimeter. By integrating biomechanical mechanisms into the topology optimization process, a transdisciplinary computational method is developed to optimize leaf structures. The numerical results show that improving the efficiency of nutrient transport plays a critical role in the morphogenesis of leaf veins. Contrary to the popular belief in the literature, this study shows that the structural performance is not a key factor in determining the venation patterns. The findings provide a deep understanding of the optimization mechanism of leaf veins, which is useful for the design of high-performance shell structures.

How cells utilize instructions provided by genes and integrate mechanical forces generated by tissue growth to produce morphology is a fundamental question of biology. Dermal bones of the vertebrate cranial vault are formed through the direct differentiation of mesenchymal cells on the neural surface into osteoblasts through intramembranous ossification. Here we join a self-organizing Turing mechanism, computational biomechanics, and experimental data to produce a 3D representative model of the growing cerebral surface, cranial vault bones, and sutures. We show how changes in single parameters regulating signaling during osteoblast differentiation and bone formation may explain cranial vault shape variation in craniofacial disorders. A key result is that toggling a parameter in our model results in closure of a cranial vault suture, an event that occurred during evolution of the cranial vault and that occurs in craniofacial disorders. Our approach provides an initial and important step toward integrating biomechanics into the genotype phenotype map to explain the production of variation in head morphology by developmental mechanisms.

In our previous paper we introduced morphogenesis and post-embryonic life as arising from cells interacting via coupled chemical, electrical and mechanical processes occurring across multiple organization levels. We reviewed these processes from the perspectives of developmental biology and how they relate to physics-based constitutive equations that are well suited to model intercellular interactions\' fields. In this paper we will describe a knowledge representation and architectural design strategy that can organize and encode the biochemical, biological and biophysical data necessary to represent and model the highly specialized and diversified cells that constitute living tissues. Since there are about 200 different types of cells in mammalian tissues, a huge amount of molecular, cellular and tissue data must be accounted for. This data cannot be incorporated in an ad hoc manner but, on the contrary, must be organized according to some sound principles. We give an overview of these principles and describe how they can be incorporated as proper features of a Knowledge Base System (KBS) dedicated to computational morphogenesis (CM).

In order to understand living organisms, considerable experimental efforts and resources have been devoted to correlate genes and their expressions with cell, tissue, organ and whole organisms\' phenotypes. This data driven approach to knowledge discovery has led to many breakthrough in our understanding of healthy and diseased states, and is paving the way to improve the diagnosis and treatment of diseases. Complementary to this data-driven approach, computational models of biological systems based on first principles have been developed in order to deepen our understanding of the multi-scale dynamics that drives normal and pathological biological functions. In this paper we describe the biological, physical and mathematical concepts that led to the design of a Computational Morphogenesis (CM) platform baptized Generic Modeling and Simulating Platform (GMSP). Its role is to generate realistic 3D multi-scale biological tissues from virtual stem cells and the intended target applications include in virtuo studies of normal and abnormal tissue (re)generation as well as the development of complex diseases such as carcinogenesis. At all space-scales of interest, biological agents interact with each other via biochemical, bioelectrical, and mechanical fields that operate in concert during embryogenesis, growth and adult life. The spatio-temporal dependencies of these fields can be modeled by physics-based constitutive equations that we propose to examine in relation to the landmark biological events that occur during embryogenesis.

Bones of the murine cranial vault are formed by differentiation of mesenchymal cells into osteoblasts, a process that is primarily understood to be controlled by a cascade of reactions between extracellular molecules and cells. We assume that the process can be modeled using Turing\'s reaction-diffusion equations, a mathematical model describing the pattern formation controlled by two interacting molecules (activator and inhibitor). In addition to the processes modeled by reaction-diffusion equations, we hypothesize that mechanical stimuli of the cells due to growth of the underlying brain contribute significantly to the process of cell differentiation in cranial vault development. Structural analysis of the surface of the brain was conducted to explore the effects of the mechanical strain on bone formation. We propose a mechanobiological model for the formation of cranial vault bones by coupling the reaction-diffusion model with structural mechanics. The mathematical formulation was solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data that provide precise three dimensional (3D) measures of murine cranial geometry and cranial vault bone formation for specific embryonic time points. The results of this study suggest that mechanical strain contributes information to specific aspects of bone formation. Our mechanobiological model predicts some key features of cranial vault bone formation that were verified by experimental observations including the relative location of ossification centers of individual vault bones, the pattern of cranial vault bone growth over time, and the position of cranial vault sutures.

Bones of the cranial vault are formed by the differentiation of mesenchymal cells into osteoblasts on a surface that surrounds the brain, eventually forming mineralized bone. Signaling pathways causative for cell differentiation include the actions of extracellular proteins driven by information from genes. We assume that the interaction of cells and extracellular molecules, which are associated with cell differentiation, can be modeled using Turing\'s reaction-diffusion model, a mathematical model for pattern formation controlled by two interacting molecules (activator and inhibitor). In this study, we hypothesize that regions of high concentration of an activator develop into primary centers of ossification, the earliest sites of cranial vault bone. In addition to the Turing model, we use another diffusion equation to model a morphogen (potentially the same as the morphogen associated with formation of ossification centers) associated with bone growth. These mathematical models were solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data showing skull bone formation in mouse at different embryonic days in mice carrying disease causing mutations and their unaffected littermates. The results show that the relative locations of the five ossification centers that form in our model occur at the same position as those identified in experimental data. As bone grows from these ossification centers, sutures form between the bones.