A Mathematical Model for the Role of Cell Signal Transduction in the Initiation and Inhibition of Angiogenesis

Neovascular formation can be divided into three main stages (which may be overlapping): (1) changes within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel. In two previous papers, [Levine, H.A. and Sleeman, B.D. (1997) "A system of reaction diffusion equations arising in the theory of reinforced random walks" SIAM J. Appl. Math. 683-730; Levine, H.A., Sleeman, B.D. and Nilsen-Hamilton, M. (2001b) "Mathematical modelling of the onset of capillary formation initiating angiogenesis." J. Math. Biol. 195-238] the authors introduced a new approach to angiogenesis, based on the theory of reinforced random walks, coupled with a Michaelis-Menten type mechanism which views the endothelial vascular endothelial cell growth factor (VEGF) receptors as the catalyst for transforming into a proteolytic enzyme in order to model the first stage. It is the purpose of this paper to present a more descriptive yet not overly complicated mathematical model of the biochemical events that are initiated when VEGF interacts with endothelial cells and which result in the cell synthesis of proteolytic enzyme. We also delineate via chemical kinetics, three mechanisms by which one may inhibit angiogenesis (inhibition of growth factor, growth factor receptor and protease function).

[1]  N Paweletz,et al.  Tumor-related angiogenesis. , 1989, Critical reviews in oncology/hematology.

[2]  J. Bowersox,et al.  Chemotaxis of aortic endothelial cells in response to fibronectin. , 1982, Cancer research.

[3]  Andrew Costin,et al.  Biochemistry. Second Edition , 1983, The Yale Journal of Biology and Medicine.

[4]  M. Chaplain,et al.  A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching. , 1996, IMA journal of mathematics applied in medicine and biology.

[5]  J. Keiser,et al.  Vascular endothelial growth factor upregulates the expression of matrix metalloproteinases in vascular smooth muscle cells: role of flt-1. , 1998, Circulation research.

[6]  J. Avruch,et al.  Mammalian mitogen-activated protein kinase signal transduction pathways activated by stress and inflammation. , 2001, Physiological reviews.

[7]  B. Sleeman,et al.  A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis. , 2000, Mathematical biosciences.

[8]  E. Wouters,et al.  Factors contributing to alterations in skeletal muscle and plasma amino acid profiles in patients with chronic obstructive pulmonary disease. , 2000, The American journal of clinical nutrition.

[9]  S. Eckhardt.  Angiogenesis inhibitors as cancer therapy. , 1999, Hospital practice.

[10]  J. Folkman.  Angiogenesis--retrospect and outlook. , 1992, EXS.

[11]  Hans G. Othmer,et al.  Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[12]  H. Christensen,et al.  Simultaneous regulation of amino acid influx and efflux by system A in the hepatoma cell HTC. Ouabain simulates the starvation-induced derepression of system A amino acid transport. , 1983, The Journal of biological chemistry.

[13]  V. Terranova,et al.  Human endothelial cells are chemotactic to endothelial cell growth factor and heparin , 1985, The Journal of cell biology.

[14]  U Alon,et al.  Generation of oscillations by the p53-Mdm2 feedback loop: a theoretical and experimental study. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[15]  R. Hungate,et al.  Vascular Endothelial Growth Factor Receptor KDR Tyrosine Kinase Activity Is Increased by Autophosphorylation of Two Activation Loop Tyrosine Residues* , 1999, The Journal of Biological Chemistry.

[16]  H. Christensen,et al.  The two-way flux of cationic amino acids across the plasma membrane of mammalian cells is largely explained by a single transport system. , 1982, The Journal of biological chemistry.

[17]  H. Gutfreund,et al.  Enzyme kinetics , 1975, Nature.

[18]  T. Gelehrter,et al.  Derepression of amino acid transport by amino acid starvation in rat hepatoma cells. , 1977, The Journal of biological chemistry.

[19]  Jehoshua Bruck,et al.  Scaffold proteins may biphasically affect the levels of mitogen-activated protein kinase signaling and reduce its threshold properties. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Z. Han,et al.  Angiogenesis: state of the art. , 1999, International journal of hematology.

[21]  D. Dimitrov,et al.  Identification of the KDR tyrosine kinase as a receptor for vascular endothelial cell growth factor. , 1992, Biochemical and biophysical research communications.

[22]  H. Birkedal‐Hansen,et al.  Proteolytic activities of human fibroblast collagenase: hydrolysis of a broad range of substrates at a single active site. , 1990, Biochemistry.

[23]  B. Davis,et al.  Reinforced random walk , 1990 .

[24]  B. Sleeman,et al.  Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma , 2001, Bulletin of mathematical biology.

[25]  K. Kaibuchi,et al.  Small GTP-binding proteins. , 1992, International review of cytology.

[26]  Howard A. Levine,et al.  A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[27]  P. Rous,et al.  A METHOD FOR OBTAINING SUSPENSIONS OF LIVING CELLS FROM THE FIXED TISSUES, AND FOR THE PLATING OUT OF INDIVIDUAL CELLS , 1916, The Journal of experimental medicine.

[28]  K. Rakušan.  Coronary Angiogenesis , 1995, Annals of the New York Academy of Sciences.

[29]  M. Boffa,et al.  Plasma and Recombinant Thrombin-activable Fibrinolysis Inhibitor (TAFI) and Activated TAFI Compared with Respect to Glycosylation, Thrombin/Thrombomodulin-dependent Activation, Thermal Stability, and Enzymatic Properties* , 1998, The Journal of Biological Chemistry.

[30]  Kenneth M. Yamada,et al.  Fibronectins—adhesive glycoproteins of cell surface and blood , 1978, Nature.

[31]  Jonathan A. Sherratt,et al.  Models of epidermal wound healing , 1990, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[32]  M. Shibuya,et al.  Different signal transduction properties of KDR and Flt1, two receptors for vascular endothelial growth factor. , 1994, The Journal of biological chemistry.

[33]  E. Bauer,et al.  Vascular endothelial growth factor induces interstitial collagenase expression in human endothelial cells , 1992, Journal of cellular physiology.

[34]  B. Sleeman,et al.  Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma , 2001 .

[35]  A. Dalgleish,et al.  Angiogenesis inhibitors in cancer therapy. , 2003, Current opinion in investigational drugs.

[36]  B. Sleeman,et al.  Mathematical modeling of the onset of capillary formation initiating angiogenesis , 2001, Journal of mathematical biology.

[37]  Reinforced Random Walks.  Reinforced Random Walks , 2022 .

[38]  Donald Voet,et al.  Biochemistry, 2nd ed. , 1995 .