Mandri N. Obeyesekere

Numerical methods for the simulation of flow in root-soil systems

The numerical properties of approximation schemes for a model that simulates water transport in root-soil systems are considered. The model is derived in detail. It is based on a previously proposed model which is reformulated completely in terms of the water potential. The system of equations consists of a parabolic partial differential equation that contains a nonlinear capacity term coupled to two linear ordinary differential equations. A closed form solution is obtained for one of the latter equations. Finite element and finite difference schemes are defined to approximate the solution of the coupled system. Some new techniques which have wide applicability for analyzing the nonlinear capacity term are used, and optimal order error estimates are derived. A postprocessed water mass flux computation is also presented and shown to be superconvergent to the true flux. Computational results which verify the theoretical convergence rates are given.

A mathematical model of the regulation of the G1 phase of Rb +/+ and Rb —/— mouse embryonic fibroblasts and an osteosarcoma cell line

A mathematical model integrating the roles of cyclin D, cdk4, cyclin E, cdk2, E2F and RB in control of the G1 phase of the cell cycle is described. Experimental results described with murine embryo fibroblasts (MEFs), either Rb +/+ or Rb —/—, and with the RB‐deficient osteosarcoma cell line, Saos‐2, served as the basis for the formulation of this mathematical model. A model employing the known interactions of these six proteins does not reproduce the experimental observations described in the MEFs. The appropriate modelling of G1 requires the inclusion of a sensing mechanism which adjusts the activity of cyclin E/cdk2 in response to both RB concentration and growth factors. Incorporation of this sensing mechanism into the model allows it to reproduce most of the experimental results observed in Saos‐2 cells, Rb —/— MEFS, and Rb +/+ MEFs. The model also makes specific predictions which have not been tested experimentally.

Network topology determines dynamics of the mammalian MAPK1,2 signaling network: bifan motif regulation of C-Raf and B-Raf isoforms by FGFR and MC1R

Activation of the fibroblast growth factor (FGFR) and melanocyte stimulating hormone (MC1R) receptors stimulates B‐Raf and C‐Raf isoforms that regulate the dynamics of MAPK1,2 signaling. Network topology motifs in mammalian cells include feed‐for ward and feedback loops and bifans where signals from two upstream molecules integrate to modulate the activity of two downstream molecules. We computation ally modeled and experimentally tested signal processing in the FGFR/MC1R/B‐Raf/C‐Raf/MAPK1,2 net work in human melanoma cells; identifying 7 regulatory loops and a bifan motif. Signaling from FGFR leads to sustained activation of MAPK1,2, whereas signaling from MC1R results in transient activation of MAPK1,2. The dynamics of MAPK activation depends critically on the expression level and connectivity to C‐Raf, which is critical for a sustained MAPK1,2 response. A partially incoherent bifan motif with a feedback loop acts as a logic gate to integrate signals and regulate duration of activation of the MAPK signaling cascade. Further reducing a 106‐node ordinary differential equations network encompassing the complete network to a 6‐node network encompassing rate‐limiting processes sustains the feedback loops and the bifan, providing sufficient information to predict biological responses.—Muller, M., Obeyesekere, M., Mills, G. B., Ram, P. T. Network topology determines dynamics of the mammalian MAPK1,2 signaling network: bifan motif regulation of C‐Raf and B‐Raf isoforms by FGFR and MC1R. FASEB J. 22, 1393–1403 (2008)

Multiscale Angiogenesis Modeling Using Mixed Finite Element Methods

In this paper, we present a deterministic two-scale tissue-cellular approach for modeling growth factor-induced angiogenesis. The bioreaction-diffusion of chemotactic growth factors (CGF) is modeled at a tissue scale, whereas cell proliferation, capillary extension, branching, and anastomosis are modeled at a cellular scale. The capillary indicator function is used to bridge these two scales. The complete system of equations consists of parabolic PDEs coupled nonlinearly with a varying number of ODEs and algebraic equations. Our proposed schemes involve applying mixed finite element methods to approximate concentrations of CGF and a point-to-point tracking method to simulate sprout branching and anastomosis. Capillary extensions are computed by a system of ODEs. Here, both the continuous and discrete-in-time algorithms are analyzed using some new techniques for treating the nonlinear coupling terms. Error bounds for each of the processes---CGF reaction-diffusion, capillary extension, sprout branching, and...

Mathematical models for the cellular concentrations of cyclin and MPF

Several mathematical models have been proposed for regulation of the cell cycle in early embryos by cyclin and maturation-promoting factor (MPF). In this paper the previously proposed models for cyclin and MPF activity are analyzed, and the validity of those models based on the mathematical behavior of their solutions and on physical considerations are discussed. In addition, three further models are proposed that exhibit the periodic behavior necessary for modeling the mitotic clock but that do not have certain of the limitations of the other models.

A model for regulation of the cell cycle incorporating cyclin A, cyclin B and their complexes

Abstract. t. A mathematical model for the cell cycle is proposed that incorporates the known biochemical reactions involving both cyclin A and cyclin B, the interactions of these cyclins with cdc2 and cdk2, and the controlling effects of cdc25 and weel. The model also postulates the existence of an as yet unknown phosphatase involved in the formation of maturation promoting factor. The model produces solutions that agree qualitatively with a wide variety of experimentally observed cell‐cycle behavior. Conditions under which the model could explain the initial rapid divisions of embryonic cells and the transition to the slower somatic cell cycle are also discussed.

A mathematical model of haemopoiesis as exemplified by CD34 + cell mobilization into the peripheral blood

Abstract.  A mathematical model for the kinetics of haemopoietic cells, including CD34+ cells, is proposed. This minimal model reflects the known kinetics of haemopoietic progenitor cells, including peripheral blood CD34+ cells, white blood cells and platelets, in the presence of granulocyte colony‐stimulating factor. Reproducing known perturbations within this system, subjected to granulocyte colony‐stimulating factor treatment and apheresis of peripheral blood progenitor cells (CD34+ cells) in healthy individuals allows validation of the model. Predictions are made with this model for reducing the length of time with neutropenia after high‐dose chemotherapy. Results based on this model indicate that myelosuppressive treatment together with infusion of CD34+ peripheral blood progenitor cells favours a faster recovery of the haemopoietic system than with granulocyte colony‐stimulating factor alone. Additionally, it predicts that infusion of white blood cells and platelets can relieve the symptoms of neutropenia and thrombocytopenia, respectively, without drastically hindering the haemopoietic recovery period after high dose chemotherapy.

Model Predictions of MDM2 Mediated Cell Regulation

In this work we present a mathematical approach to elucidate possible mechanisms involving mdm2 in the regulation of the cell cycle. It has been experimentally shown that the over-expression of MDM2 leads to decoupling of DNA synthesis with mitosis resulting in polyploidy cells with multiple copies of their genomes. The function of MDM2 that decouples the DNA synthesis phase (S) and the Mitosis phase (M) is unclear. To answer this question, we first formulate a mathematical model of the dynamics of the cell cycle regulatory proteins during the DNA synthesis phase and mitosis. This model is then tested for bifurcation that produces period doubling cascades that we relate to the biological event of polyploidy. The model formulation, the underlying biology, and the bifurcation results to delineate the unknown function of MDM2 are presented. Based on reproducing known experimental result of polyploidy in MDM2 overexpressed cells, we propose several possible functions of mdm2, i.e., possible interactions with the other cell cycle regulating proteins that will result in decoupling the S and M phases. We conclude that the most likely unknown function of MDM2 leading to the decoupling of the S and M phases is an obstruction of the activation of cdc25C by MDM2.

Mathematical analysis of a 3-variable cell cycle model

Mathematical analysis is performed on a 3-variable nonlinear ordinary differential equation system which had been previously introduced to model the regulation of the G1 phase of the cell cycle. The nature and stability of the models steady states and periodic solutions are described. These results are obtained via linear stability analysis, bifurcation theory and computational techniques using AUTO. The model exhibits different types of bifurcations. The bifurcation results are further confirmed by numerical simulations. This original model (three variables) and a model for a special biological case (the original reduced to two variables) are compared. Some mathematical properties of the 3-variable model are preserved, while others are lost, when the model is reduced to a 2-variable system. The possible biological relevance of the mathematical results is discussed.

A model for water uptake in plants

A model is developed to describe water flow through soil containing roots. The governing equation is Richards equation with a sink term representing extraction of water by the root system. Radial and axial flow in the root system is modeled as a resistance network, with radial resistances obtained approximately from a single root radial model and axial resistances obtained using the Hagen-Poisuille law. The analysis leads to a nonlinear parabolic partial differential equation coupled with a second order two-point boundary value problem. A maximum principle is proved for the system, giving uniqueness and continuous dependence on the data. Results of simulations are presented.