VirtualLeaf: an open-source framework for cell-based modeling of plant tissue growth and development

Plant organs, including leaves and roots, develop by means of a multilevel cross talk between gene regulation, patterned cell division and cell expansion, and tissue mechanics. The multilevel regulatory mechanisms complicate classic molecular genetics or functional genomics approaches to biological development, because these methodologies implicitly assume a direct relation between genes and traits at the level of the whole plant or organ. Instead, understanding gene function requires insight into the roles of gene products in regulatory networks, the conditions of gene expression, etc. This interplay is impossible to understand intuitively. Mathematical and computer modeling allows researchers to design new hypotheses and produce experimentally testable insights. However, the required mathematics and programming experience makes modeling poorly accessible to experimental biologists. Problem-solving environments provide biologically intuitive in silico objects ("cells", "regulation networks") required for setting up a simulation and present those to the user in terms of familiar, biological terminology. Here, we introduce the cell-based computer modeling framework VirtualLeaf for plant tissue morphogenesis. The current version defines a set of biologically intuitive C++ objects, including cells, cell walls, and diffusing and reacting chemicals, that provide useful abstractions for building biological simulations of developmental processes. We present a step-by-step introduction to building models with VirtualLeaf, providing basic example models of leaf venation and meristem development. VirtualLeaf-based models provide a means for plant researchers to analyze the function of developmental genes in the context of the biophysics of growth and patterning. VirtualLeaf is an ongoing open-source software project (http://virtualleaf.googlecode.com) that runs on Windows, Mac, and Linux.     

Mathematical modeling of collagen turnover in biological tissue

We present a theoretical and computational model for collagen turnover in soft biological tissues. Driven by alterations in the mechanical environment, collagen fiber bundles may undergo important chronic changes, characterized primarily by alterations in collagen synthesis and degradation rates. In particular, hypertension triggers an increase in tropocollagen synthesis and a decrease in collagen degradation, which lead to the well-documented overall increase in collagen content. These changes are the result of a cascade of events, initiated mainly by the endothelial and smooth muscle cells. Here, we represent these events collectively in terms of two internal variables, the concentration of growth factor TGF- $\beta $ and tissue inhibitors of metalloproteinases TIMP. The upregulation of TGF- $\beta $ increases the collagen density. The upregulation of TIMP also increases the collagen density through decreasing matrix metalloproteinase MMP. We establish a mathematical theory for mechanically-induced collagen turnover and introduce a computational algorithm for its robust and efficient solution. We demonstrate that our model can accurately predict the experimentally observed collagen increase in response to hypertension reported in literature. Ultimately, the model can serve as a valuable tool to predict the chronic adaptation of collagen content to restore the homeostatic equilibrium state in vessels with arbitrary micro-structure and geometry.     

Mathematical modeling of thrombus growth in mesenteric vessels

Richardson’s phenomenological mathematical model of the thrombi growth in microvessels is extended to describe the realistic features of the phenomenon. The main directions of the generalization of Richardson’s model are: (1) the dependence of platelet activation time on the distance from the injured vessel wall; (2) the non-homogeneity of the platelet distribution in blood flow in the vicinity of the vessel wall; (3) the adequate choice of the phenomenological function describing the dependence of blood velocity on the thrombus size. The generalization of the model corresponds to the main experimental results and theoretical considerations concerning thrombus formation obtained in recent years. The extended model permits to achieve qualitative agreement between model and experimental data.     

Mathematical modeling of solid cancer growth with angiogenesis

Background

Cancer arises when within a single cell multiple malfunctions of control systems occur, which are, broadly, the system that promote cell growth and the system that protect against erratic growth. Additional systems within the cell must be corrupted so that a cancer cell, to form a mass of any real size, produces substances that promote the growth of new blood vessels. Multiple mutations are required before a normal cell can become a cancer cell by corruption of multiple growth-promoting systems.

Methods

We develop a simple mathematical model to describe the solid cancer growth dynamics inducing angiogenesis in the absence of cancer controlling mechanisms.

Results

The initial conditions supplied to the dynamical system consist of a perturbation in form of pulse: The origin of cancer cells from normal cells of an organ of human body. Thresholds of interacting parameters were obtained from the steady states analysis. The existence of two equilibrium points determine the strong dependency of dynamical trajectories on the initial conditions. The thresholds can be used to control cancer.

Conclusions

Cancer can be settled in an organ if the following combination matches: better fitness of cancer cells, decrease in the efficiency of the repairing systems, increase in the capacity of sprouting from existing vascularization, and higher capacity of mounting up new vascularization. However, we show that cancer is rarely induced in organs (or tissues) displaying an efficient (numerically and functionally) reparative or regenerative mechanism.     

A general framework for modeling growth and division of mammalian cells

Background  

Modeling the cell-division cycle has been practiced for many years. As time has progressed, this work has gone from understanding the basic principles to addressing distinct biological problems, e.g., the nature of the restriction point, how checkpoints operate, the nonlinear dynamics of the cell cycle, the effect of localization, etc. Most models consist of coupled ordinary differential equations developed by the researchers, restricted to deal with the interactions of a limited number of molecules. In the future, cell-cycle modeling--and indeed all modeling of complex biologic processes--will increase in scope and detail.     

A framework for whole-cell mathematical modeling

The default framework for modeling biochemical processes is that of a constant-volume reactor operating under steady-state conditions. This is satisfactory for many applications, but not for modeling growth and division of cells. In this study, a whole-cell modeling framework is developed that assumes expanding volumes and a cell-division cycle. A spherical newborn cell is designed to grow in volume during the growth phase of the cycle. After 80% of the cycle period, the cell begins to divide by constricting about its equator, ultimately affording two spherical cells with total volume equal to twice that of the original. The cell is partitioned into two regions or volumes, namely the cytoplasm (Vcyt) and membrane (Vmem), with molecular components present in each. Both volumes change during the cell cycle; Vcyt changes in response to osmotic pressure changes as nutrients enter the cell from the environment, while Vmem changes in response to this osmotic pressure effect such that membrane thickness remains invariant. The two volumes change at different rates; in most cases, this imposes periodic or oscillatory behavior on all components within the cell. Since the framework itself rather than a particular set of reactions and components is responsible for this behavior, it should be possible to model various biochemical processes within it, affording stable periodic solutions without requiring that the biochemical process itself generates oscillations as an inherent feature. Given that these processes naturally occur in growing and dividing cells, it is reasonable to conclude that the dynamics of component concentrations will be more realistic than when modeled within constant-volume and/or steady-state frameworks. This approach is illustrated using a symbolic whole cell model.     

Mathematical foundations for modeling poikilotherm mortality

A mathematical basis for incorporating mortality into models of insect development is presented. There has been much attention given to modeling insect development, but the quantitative treatment of mortality, especially of immature stages, is often overlooked. This paper investigates mortality as it affects immature-organism development, proceeding from the simple to the complex. The methods presented are illustrated with examples taken from boll-weevil immature mortality within cotton floral buds.     

Mathematical modeling for mixed culture growth of two bacterial populations with opposite substrate preferences

An unstructured mathematical model is proposed for mixed culture growth of two different bacterial species that exhibit "opposite" substrate preferences in response to the "same" environmental conditions. The model incorporates enzymatic control mechanisms such as induction, repression, and inhibition in the microorganisms as manifested in their preferential utilization of substrates and microbial interactions such as amensalism and competition. The model predicts cell mass, substrate concentrations, dissolved oxygen tension, as well as key enzyme levels. The predictions of the model are compared with experimental data for pure culture growth and for mixed culture growth on two substrates, glucose and citrate, in a batch reactor.     

Mathematical modeling of bioassays

The high affinity and specificity of biological receptors determine the demand for and the intensive development of analytical systems based on use of these receptors. Therefore, theoretical concepts of the mechanisms of these systems, quantitative parameters of their reactions, and relationships between their characteristics and ligand–receptor interactions have become extremely important. Many mathematical models describing different bioassay formats have been proposed. However, there is almost no information on the comparative characteristics of these models, their assumptions, and predic- tive insights. In this review we suggested a set of criteria to classify various bioassays and reviewed classical and contempo- rary publications on these bioassays with special emphasis on immunochemical analysis systems as the most common and in-demand techniques. The possibilities of analytical and numerical modeling are discussed, as well as estimations of the minimum concentrations that may be detected in bioassays and recommendations for the choice of assay conditions.     

Mathematical modeling of growth and alkaloid production in Claviceps purpurea batch fermentation

An new systematic approach for describing Claviceps purpurea growth and ergot alkaloid production during batch fermentation is presented. The model is based on microbial life, as the main characteristic for microbial development during fermentation process. The aging process of the microorganism is represented by life function, defined in microbial life space. The life space is defined as a measure in which the observer follows the development of a biosystem through physiological and morphological changes of a microorganism. As a consequence of such approach the relativistic theory is recognized. To validate the model developed, a test on growth and alkaloid synthesis data from an industrial batch fermentation was performed. (c) 1993 John Wiley & Sons, Inc.     

A framework for modeling of juxtacrine signaling systems

Juxtacrine signaling is intercellular communication, in which the receptor of the signal (typically a protein) as well as the ligand (also typically a protein, responsible for the activation of the receptor) are anchored in the plasma membranes, so that in this type of signaling the activation of the receptor depends on direct contact between the membranes of the cells involved. Juxtacrine signaling is present in many important cellular events of several organisms, especially in the development process. We propose a generic formal model (a modeling framework) for juxtacrine signaling systems that is a class of discrete dynamic systems. It possesses desirable characteristics in a good modeling framework, such as: a) structural similarity with biological models, b) capacity of operating in different scales of time, and c) capacity of explicitly treating both the events and molecular elements that occur in the membrane, and those that occur in the intracellular environment and that are involved in the juxtacrine signaling process. We have implemented this framework and used it to develop a new three-level discrete model for the neurogenic network and its participation in neuroblast segregation. This paper presents the details of this framework and its current status.     

A log-linear modeling framework for selective mixing.

Nonrandom mixing can significantly alter the diffusion path of an infectious disease such as AIDS that requires intimate contact. Recent attempts to model this effect have sought a general framework capable of representing both simple and arbitrarily complicated mixing structures, and of solving the balancing problem in a nonequilibrium multigroup population. Log-linear models are proposed here as a general framework for solving the first problem. This approach offers several additional benefits: The parameters used to govern the mixing have a simple, intuitive interpretation, the framework provides a statistically sound basis for the estimation of these parameters from mixing-matrix data, and the resulting estimates are easily integrated into compartmental models for diffusion. A modified selection model is proposed to solve the second problem of generalizing the selection process to nonequilibrium populations. The distribution of contacts under this model is derived and is found to satisfy the assumptions of statistical inference for log-linear models. Together these techniques provide an integrated and flexible framework for modeling the role of selective mixing in the spread of disease.     

A framework for structured modeling of skeletal muscle

The aim of this study is to present a detailed continuum mechanics formulation, and the corresponding algorithms, to predict the deformation of skeletal muscle at different structural levels, starting from the muscle fiber level. The model is used to investigate force production and structural changes during isometric and dynamic contractions of the cat medial gastrocnemius. From a comparison with experimental data obtained in our own laboratories, we conclude that the model faithfully predicts all of the observations pertaining to force production, fascicle length and angle of pennation under various test conditions.     

Mathematical modeling for the pathogenesis of Alzheimer's disease

Despite extensive research, the pathogenesis of neurodegenerative Alzheimer's disease (AD) still eludes our comprehension. This is largely due to complex and dynamic cross-talks that occur among multiple cell types throughout the aging process. We present a mathematical model that helps define critical components of AD pathogenesis based on differential rate equations that represent the known cross-talks involving microglia, astroglia, neurons, and amyloid-β (Aβ). We demonstrate that the inflammatory activation of microglia serves as a key node for progressive neurodegeneration. Our analysis reveals that targeting microglia may hold potential promise in the prevention and treatment of AD.     

Mathematical modeling of planar polarity

Although it is well established that the Frizzled receptor is involved in the transmission of polarity information from cell to cell in the Drosophila cuticle, its precise role is still unclear. A recent paper by presents a mathematical model of a feedback loop-based mechanism for propagation of polarity between cells that can account for the known functions of Frizzled.     

Mathematical modeling of tumor-induced angiogenesis

Journal of Mathematical Biology -     

Mathematical modeling of cancer radiovirotherapy

Cancer virotherapy represents a dynamical system that requires mathematical modeling for complete understanding of the outcomes. The combination of virotherapy with radiation (radiovirotherapy) has been recently shown to successfully eliminate tumors when virotherapy alone failed. However, it introduces a new level of complexity. We have developed a mathematical model, based on population dynamics, that captures the essential elements of radiovirotherapy. The existence of corresponding equilibrium points related to complete cure, partial cure, and therapy failure is proved and discussed. The parameters of the model were estimated by fitting to experimental data. By using simulations we analyzed the influence of parameters that describe the interaction between virus and tumor cell on the outcome of the therapy. Furthermore, we evaluated relevant therapeutic scenarios for radiovirotherapy, and offered elements for optimization.