Mathematical analysis of the oncornavirus maturation process (virion RNA conversion and morphological condensation)

The rate of the maturation process of avian myeloblastosis virus experimentally estimated on the basis of genomic viral RNA conversion and morphological transition of virions was mathematically analysed. Three mathematical models were suggested and fitted to experimental data. It was found that: (a) model of simple kinetics (Model 1) does not agree with experimental data. Therefore, two hypotheses were considered in further mathematical modelling: (b) virions are identical in time of budding: maturation is dependent on the presence of a virion component which is degraded with time (Model 2). This model agrees with experimental data in all stages of the maturation process. (c) Virions are released from cells at different stages of assembly (Model 3). This model differs from experimental data especially in early stages of maturation. The hypothesis used for the construction of Model 2 seems to be the most plausible to explain the maturation process and is in agreement with data of murine leukemia virus maturation which was found to be accomplished by cleavage of p70 precursor protein.     

Cleavage and gastrulation of the euphausiacean<Emphasis Type="Italic"> Meganyctiphanes norvegica</Emphasis> (Crustacea,Malacostraca)

According to the Articulata hypothesis the cleavage of arthropods must be derived from spiral cleavage. However, arthropods show a great variety of cleavage modes with a widespread occurrence of superficial cleavage. In the Malacostraca, holoblastic cleavage occurs in some taxa such as Amphipoda, Euphausiacea and Dendrobranchiata. In particular, the cleavage of euphausiaceans has been proposed to be a modified spiral cleavage. The cell lineage of early stages up to blastoderm formation of the euphausiacean Meganyctiphanes norvegica is reconstructed using recent methods of fluorescent staining. Only the oblique angle of the mitotic spindles during the transition from the 2- to the 4-cell stage resembles the spiral cleavage mode. At the 8-cell stage, four cells each form a pattern of two interlocking bands which is preserved until the 122-cell stage. One blastomere is delayed in division and shows an oblique division from the fourth cleavage on. It is the precursor cell of two enlarged and cleavage-arrested cells at the 32-cell stage. At the 62-cell stage, these two cells are surrounded by eight cells following a specific cell division pattern during the subsequent division cycles. The cleavage pattern of M. norvegica occurs in two mirror images. A comparative approach reveals distinct similarities between the early cleavage patterns of Euphausiacea and Dendrobranchiata which are suggested to be homologous. Furthermore, the relationships to non-malacostracan cleavage patterns are discussed. It is shown that the early cleavage pattern of M. norvegica does not offer an example of a spiral cleavage within arthropods.     

An immersed boundary method for simulating a single axisymmetric cell growth and division

For an animal cell, cytokinesis is the process by which a cell divides its cytoplasm to produce two daughter cells. We propose a new mathematical model for simulating cytokinesis. The proposed model is robust and realistic in deciding the position of the cleavage furrow and in defining the contractile force leading to cell division. We use an immersed boundary method to track the morphology of cell membrane during cytokinesis. For accurate calculation, we adaptively add and delete the immersed boundary points. We perform numerical simulations on the axisymmetric domain to have sufficient resolution and to incorporate three-dimensional effects such as anisotropic surface tension. Finally, we investigate the effects of each model parameter and compare a numerical result with the experimental data to demonstrate the efficiency and accuracy of our proposed method.     

Endocytosis resumes during late mitosis and is required for cytokinesis

Recent work has underscored the importance of membrane trafficking events during cytokinesis. For example, targeted membrane secretion occurs at the cleavage furrow in animal cells, and proteins that regulate endocytosis also influence the process of cytokinesis. Nonetheless, the prevailing dogma is that endosomal membrane trafficking ceases during mitosis and resumes after cell division is complete. In this study, we have characterized endocytic membrane trafficking events that occur during mammalian cell cytokinesis. We have found that, although endocytosis ceases during the early stages of mitosis, it resumes during late mitosis in a temporally and spatially regulated pattern as cells progress from anaphase to cytokinesis. Using fixed and live cell imaging, we have found that, during cleavage furrow ingression, vesicles are internalized from the polar region and subsequently trafficked to the midbody area during later stages of cytokinesis. In addition, we have demonstrated that cytokinesis is inhibited when clathrin-mediated endocytosis is blocked using a series of dominant negative mutants. In contrast to previous thought, we conclude that endocytosis resumes during the later stages of mitosis, before cytokinesis is completed. Furthermore, based on our findings, we propose that the proper regulation of endosomal membrane traffic is necessary for the successful completion of cytokinesis.     

On the biomechanics of cytokinesis in animal cells

The material properties of the cell membrane are discussed. Various theories concerning the mechanism of cytokinesis in animal cells are presented. The currently accepted mechanism is that of active muscle-like contraction of the furrow base itself. A mathematical model is developed based on this theory. The cell membrane is modelled as a spherical membrane of nonlinear, elastic material. The membrane undergoes large deformations under the action of a contractile ring force in its equatorial plane. The numerical procedure employed in the solution of the governing equations is explained. The numerical results are compared with the experimental observations available in the literature. It is concluded that the cell membrane stiffness increases during the early stages of cleavage and it, later, decreases. The cell membrane division is a biomechanical instability problem. The factors that may facilitate or block cleavage are discussed. The experimental evidences that support the conjectures of the model are pointed out.     

Reconstitution of endoplasmic reticulum in rapidly dividing cells of early Xenopus embryos

The cytology of early blastomeres of Xenopus laevis embryos was examined. Particular attention was given to the organization of the nuclear envelope of karyomeres (chromosome vesicles) and the endoplasmic reticulum (ER) at different stages in early cleavage cycles of frog development. Nuclear envelope formation was observed to occur rapidly around individual chromosomes during early anaphase, and karyomeres fused subsequently to yield the final nucleus during telophase. Endoplasmic reticulum in the perinuclear cytoplasm was observed to be vesicular during metaphase and cisternal in form during telophase. Following microinjection of rat liver rough microsomes into early blastomeres, heterologous ER components were identified by electron microscope immunocytochemistry. The foreign ER was observed as large, reconstituted cisternae at stages in the cell cycle when the nuclear envelope was intact. Therefore, transplanted ER maintained the capacity to reconstitute in the cytoplasm of a rapidly dividing cell. In an attempt to better assess ER structure at the metaphase stage of the cell cycle, we next slowed down the division process by treating Xenopus embryos with anti-microtubule agents. Treatment with critical concentrations of colchicine, nocodazole, or vinblastine led to cleavage arrest but not to inhibition of the nuclear cycle. Following such treatment, homologous ER was observed in a vesicular form at all stages of the nuclear cycle. Heterologous ER, however, identified by immunocytochemistry in microinjected cells treated with nocodazole, displayed both vesicular and cisternal forms. We conclude that microinjected ER membranes exhibit cell-cycle-specific behavior, which is different from that of the host cell ER.     

Cell lineage,axis formation,and the origin of germ layers in the amphipod crustacean Orchestia cavimana

Embryos of the amphipod crustacean Orchestia cavimana are examined during cleavage, gastrulation, and segmentation by using in vivo labelling. Single blastomeres of the 8- and 16-cell stages were labelled with DiI to trace cell lineages. Early cleavage follows a distinct pattern and the a/p and d/v body axes are already determined at the 4- and 8-cell stages, respectively. In these stages, the germinal rudiment and the naupliar mesoderm can be traced back to a single blastomere each. In addition, the ectoderm and the postnaupliar mesoderm are separated into right and left components. At the16-cell stage, naupliar ectoderm is divided from the postnaupliar ectoderm, and extraembryonic lineages are separated from postnaupliar mesoderm and endoderm. From our investigation, it is evident that the cleavage pattern and cell lineage of Orchestia cavimana are not of the spiral type. Furthermore, the results of the labelling show many differences to cleavage patterns and cell lineages in other crustaceans, in particular, other Malacostraca. The cleavage and cell lineage patterns of the amphipod Orchestia are certainly derived within Malacostraca, whose ancestral cleavage mode was most likely of the superficial type. On the other hand, Orchestia exhibits a stereotyped cell division pattern during formation and differentiation of the germ band that is typical for malacostracans. Hence, a derived (apomorphic) early cleavage pattern is the ontogenetic basis for an evolutionarily older cell division pattern of advanced developmental stages. O. cavimana offers the possibility to trace the lineages and the fates of cells from early developmental stages up to the formation of segmental structures, including neurogenesis at a level of resolution that is not matched by any other arthropod system.     

Protease-dependent uncoating of a complex retrovirus

Although retrovirus egress and budding have been partly unraveled, little is known about early stages of the replication cycle. In particular, retroviral uncoating, a process during which incoming retroviral cores are altered to allow the integration of the viral genome into host chromosomes, is poorly understood. To get insights into these early events of the retroviral cycle, we have used foamy complex retroviruses as a model. In this report, we show that a protease-defective foamy retrovirus is noninfectious, although it is still able to bud and enter target cells efficiently. Similarly, a retrovirus mutated in an essential viral protease-dependent cleavage site in the central part of Gag is noninfectious. Following entry, wild-type and mutant retroviruses are able to traffic along microtubules towards the microtubule-organizing center (MTOC). However, whereas nuclear import of Gag and of the viral genome was observed for the wild-type virus as early as 8 hours postinfection, incoming capsids and genome from mutant viruses remained at the MTOC. Interestingly, a specific viral protease-dependent Gag cleavage product was detected only for the wild-type retrovirus early after infection, demonstrating that cleavage of Gag by the viral protease at this stage of the virus life cycle is absolutely required for productive infection, an unprecedented observation among retroviruses.     

A mathematical model for embryonic cell division based on a surface “cleavage field”

The process whereby a fertilized egg divides to give rise to an embryo, i.e. the process of cleavage—which can be considered, in some sense, as the early phase of embryonic differentiation—exhibit in many species a precise geometry. Such a geometry may be altered within certain limits, as was done in various classical experiments, and yet normal differentiation may occur. However, since the pattern of cleavage is clearly under genetic control, any model of cleavage should incorporate some device apt to produce a specific geometry. In this paper, a model for embryonic cell division based on a surface “cleavage field” is described. This surface field may be interpreted, for instance, as the surface density of sources of active transport of ions which diffuse into the cell, although other interpretations —such as the surface density of specific binding sites or functional membrane receptors, etc.—are possible. Assumptions relating the geometry of cleavage to the geometry of the level surfaces of the ionic concentration are given together with a discussion of the change in the surface field due to cleavage. Finally, a simplified two-dimensional version of the model is presented which develops interesting patterns of “cleavagerd”, calculated by computer, similar in many ways to those of real threedimensional embryos.     

Mathematical model for early development of the sea urchin embryo

In Xenopus and Drosophila, the nucleocytoplasmic ratio controls many aspects of cell-cycle remodeling during the transitory period that leads from fast and synchronous cell divisions of early development to the slow, carefully regulated growth and divisions of somatic cells. After the fifth cleavage in sea urchin embryos, there are four populations of differently sized blastomeres, whose interdivision times are inversely related to size. The inverse relation suggests nucleocytoplasmic control of cell division during sea urchin development as well. To investigate this possibility, we developed a mathematical model based on molecular interactions underlying early embryonic cell-cycle control. Introducing the nucleocytoplasmic ratio explicitly into the molecular mechanism, we are able to reproduce many physiological features of sea urchin development.     

Nuclear cysteine-protease involved in male chromatin remodeling after fertilization is ubiquitously distributed during sea urchin development

Previously we have identified a cysteine-protease involved in male chromatin remodeling which segregates into the nuclei of the two blastomeres at the first cleavage division. Here we have investigated the fate of this protease during early embryogenesis by immunodetecting this protein with antibodies elicited against its N-terminal sequence. As shown in this report, the major 60 kDa active form of this protease was found to be present in the extracts of chromosomal proteins obtained from all developmental stages analyzed. In morula and gastrula the 70 kDa inactive precursor, which corresponds to the major form of the zymogen found in unfertilized eggs, was detected. In plutei larvas, the major 60 kDa form of this enzyme was found together with a higher molecular weight precursor (90 kDa) which is consistent with the less abundant zymogen primarily detected in unfertilized eggs. As reported here, either the active protease or its zymogens were visualized in most of the embryonic territories indicating that this enzyme lacks a specific pattern of spatial-temporal developmental segregation. Taken together our results indicate that this protease persists in the embryo and is ubiquitously distributed up to larval stages of development, either as an active enzyme and/or as an inactive precursor. These results suggest that this enzyme may display yet unknown functions during embryonic development that complement its role in male chromatin remodeling after fertilization.     

Immunological detection of changes in genomic DNA methylation during early zebrafish development.

DNA methylation reprogramming, the erasure of DNA methylation patterns shortly after fertilization and their reestablishment during subsequent early development, is essential for proper mammalian embryogenesis. In contrast, the importance of this process in the development of non-mammalian vertebrates such as fish is less clear. Indeed, whether or not any widespread changes in DNA methylation occur at all during cleavage and blastula stages of fish in a fashion similar to that shown in mammals has remained controversial. Here we have addressed this issue by applying the techniques of Southwestern immunoblotting and immunohistochemistry with an anti-5-methylcytosine antibody to the examination of DNA methylation in early zebrafish embryos. These techniques have recently been utilized to demonstrate that development-specific changes in genomic DNA methylation also occur in Drosophila melanogaster and Dictyostelium discoideum, both organisms for which DNA methylation was previously not thought to occur. Our data demonstrate that genome-wide changes in DNA methylation occur during early zebrafish development. Although zebrafish sperm DNA is strongly methylated, the zebrafish genome is not detectably methylated through cleavage and early blastula stages but is heavily remethylated in blastula and early gastrula stages.     

Syntaxin Is Required for Cell Division

We recently identified a single family member homologue of syntaxin in the sea urchin. Syntaxin is present throughout development, and in rapidly dividing cleavage stage embryos it is present on numerous vesicles at the cell cortex. We hypothesized that syntaxin mediates essential membrane fusion events during early embryogenesis, reasoning that the vesicles and/or their contents are important for development. Here we show that functional inactivation of syntaxin with either Botulinum neurotoxin C1, which specifically proteolyzes syntaxin, or antibodies against syntaxin results in an inhibition of cell division. These observations suggest that syntaxin is essential for membrane fusion events critical for cell division.     

Determination of cell division axes in the early embryogenesis of Caenorhabditis elegans

The establishment of cell division axes was examined in the early embryonic divisions of Caenorhabditis elegans. It has been shown previously that there are two different patterns of cleavage during early embryogenesis. In one set of cells, which undergo predominantly determinative divisions, the division axes are established successively in the same orientation, while division axes in the other set, which divide mainly proliferatively, have an orthogonal pattern of division. We have investigated the establishment of these axes by following the movement of the centrosomes. Centrosome separation follows a reproducible pattern in all cells, and this pattern by itself results in an orthogonal pattern of cleavage. In those cells that divide on the same axis, there is an additional directed rotation of pairs of centrosomes together with the nucleus through well-defined angles. Intact microtubules are required for rotation; rotation is prevented by inhibitors of polymerization and depolymerization of microtubules. We have examined the distribution of microtubules in fixed embryos during rotation. From these and other data we infer that microtubules running from the centrosome to the cortex have a central role in aligning the centrosome-nuclear complex.     

The segregation of inner and outer cells in porcine embryos follows a different pattern compared to the segregation in mouse embryos

The mammalian blastocyst consists of an inner cell mass (ICM) enclosed by the trophectoderm. The origin of these two cell populations lies in the segregation of inner and outer cells in the early morula. In the present study, the segregation of inner and outer cells has been studied in porcine embryos and is compared with segregation in mouse embryos. For this, nuclei of inner and outer cells were differentially labelled with two fluorochromes after partial complement-mediated lysis of the outer cells. In porcine and mouse embryos compaction and the first appearance of inner cells occur at different stages of development. In porcine embryos compaction was observed as early as the 4-cell stage, while in mouse embryos compaction occurred in the 8-cell stage. The first inner cells segregated in porcine embryos which were in the transition from four to eight cells and inner cells were added during two subsequent cell cycles. In mouse embryos inner cells segregated predominantly during the fourth cleavage division. From the results obtained we conclude that the segregation of inner and outer cells follows a different pattern in mouse and in porcine embryos.     

Calpain Translocation during Muscle Fiber Necrosis and Regeneration in Dystrophin-Deficient Mice

Previous studies have shown that calpains are autolytically cleaved during the disease process of mdx dystrophy, a mouse model for Duchenne muscular dystrophy, indicating that calpains may be activated and play a role in proteolysis that occurs in muscular dystrophy (J. Biol. Chem.270(18), 10909–10914, 1995). In the present study, we investigated the location of calpain in dystrophic muscle fibers over the course of mdx dystrophy, to relate the protease distribution to its state of activation, and to determine whether calpain translocation was an early event in mdx dystrophy. Immunolabeling of healthy, fully differentiated muscle fibers showed calpain present throughout the cytosol, but more concentrated near the plasma membrane. However, degenerating mdx fibers did not contain higher concentrations of calpain at the plasma membrane and showed only a homogeneous, cytosolic distribution. Calpain distribution was similarly diffuse in young myotubes and regenerating fibers with increased cytosolic concentration in early myotubes. Calpain distribution in adult mdx tissue was similar to that occurring in healthy, fully differentiated fibers, although adult mdx fibers displayed higher concentrations of membrane-associated calpain than those observed in C57 controls. The association of calpain with the plasma membrane was verified by immunoblots of isolated sarcolemmal membrane from adult mdx and control muscle which showed calpain present predominantly in the cytosol along with some membrane association. Thus, changes in calpain distribution coincide with changes in enzymatic cleavage over the course of mdx dystrophy shown previously. Furthermore, the stages of pathology at which calpain cleavage is least coincides with those stages when calpain is most concentrated at the cell membrane, suggesting that calpain is retained in an inactive form at the plasma membrane.     

The nuclear division and parasexual cycle inPenicillium

The vegetative nuclear division inPenicillium differs from classical mitosis, and a model for the division process is presented. In early divisional stages the interconnected chromosomes are arranged in a ring which breaks, giving rise to a linear configuration which divides by longitudinal splitting. The break may occur with equal probability between each of the chromosomes. At the end of the division process the daughter nuclei regain ring structure. One of the chromosomes is believed to represent a separation center for all the chromosomes. In diploid strains the two haploid genomes show a close somatic association and the nuclear configurations occurring during the divisional cycle are identical to those in the haploids. The double break of the ring structures in diploids will give recombinant nuclei in certain cases. The model explains the available data of the parasexual cycle, and both diploidization and haploidization are believed to represent singlestep processes.     

Mathematical Model of Stem Cell Differentiation and Tissue Regeneration with Stochastic Noise

Differentiation and self-renewal of stem cells is an essential process for the maintenance of tissue composition. The promise of novel medical therapies combined with the complexity of this process encourage us to employ numerical and mathematical methods. This will allow us to understand better the mechanisms which regulate stem cell behaviour. Perturbations to the cellular environment may have an influence on the death rate, proliferation rate and on the fraction of self-renewal at every stage of differentiation. In this paper, we present mathematical study of the effect of stochastic noise on the process of tissue regeneration. Here, a system of Itô stochastic differential equations with linear diffusion coefficients that is based on a deterministic model of multistage cell lineages is investigated. Numerical simulations of the stochastic model are shown for a different number of stages of differentiation. Interactions between the noise, added to the different stages, are characterised using numerical simulations. The long-time behaviour of the two-dimensional version of the model is fully characterised; asymptotic stability of the related Markov semigroup is proved using the theory of the Markov semigroups and the method of the Khasminskií function.