Modeling of the kinetics of vitamin D_3 in osteoblastic cells

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2013, 10(2): 319-344. doi: 10.3934/mbe.2013.10.319

Modeling of the kinetics of vitamin D$_3$ in osteoblastic cells

Robert P. Gilbert ^1, , Philippe Guyenne ^1, and Ying Liu ^1,

1.	Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, United States, United States, United States

Received May 2012 Revised November 2012 Published January 2013

Abstract
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A one-dimensional model for the transport of vitamin D$_3$ in an osteoblast cell is proposed, from its entry through the membrane to its activation of RANKL synthesis in the nucleus. In the membrane and cytoplasm, the transport of D$_3$ and RANKL is described by a diffusion process, while their interaction in the nucleus is modeled by a reaction-diffusion process. For the latter, an integral equation involving the boundary conditions, as well as an asymptotic solution in the regime of small concentrations, are derived. Numerical simulations are also performed to investigate the kinetics of D$_3$ and RANKL through the entire cell. Comparison between the asymptotics and numerics in the nucleus shows an excellent agreement. To our knowledge, this is the first time, albeit using a simple model, a description of the complete passage of D$_3$ through the cell membrane, the cytoplasm, into the cell nucleus, and finally the production of RANKL with its passage to the exterior of the cell, has been modeled.

Keywords: numerical simulations., receptors, reaction-diffusion, osteoblasts, RANKL.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Robert P. Gilbert, Philippe Guyenne, Ying Liu. Modeling of the kinetics of vitamin D$_3$ in osteoblastic cells. Mathematical Biosciences & Engineering, 2013, 10 (2) : 319-344. doi: 10.3934/mbe.2013.10.319

References:

[1]	A. Atri, J. Amundson, D. Clapham and J. Sneyd, A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis Oocyte, Biophys. J., 65 (1993), 1727-1739.
[2]	F. Bronner, Cytoplasmic transport of calcium and other inorganic ions, Comp. Biochem. Physiol., 115B (1996), 313-317.
[3]	J. Buchanan, R. P. Gilbert and M. J. Ou, The kinetics of vitamin $D_3$ in the osteoblastic cell, Submitted to J. Theor. Biol., (2012).
[4]	E. M. Costa and D. Feldman, Measurement of 1,25-Dihydroxyvitamin D3 receptor turnover by dense amino acid labeling: Changes during receptor up-regulation by vitamin D metabolites, Endocrinology, 120 (1987), 1173-1178.
[5]	M. C. Farach-Carson and P. J. Davis, Steroid hormone interactions with target cells: Cross talk between membrane and nuclear pathways, J. Pharm. Exp. Therap., 307 (2003), 839-845.
[6]	R. Gilbert, A. Panasenko and A. Vasilic, Acoustic Propagation in a Random Saturated Medium: The Monophasic Case, Math. Mehods Appl. Sciences, 33 (2010), 2206-2214.
[7]	K. Hackl and S. Ilic, Application of the multiscale FEM to the modeling of cancellous bone, Biomechan. Model. Mechanobiol., 9 (2010), 87-102.
[8]	V. V. Jikov, S. M. Kozlov and O.A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer, Berlin 1994.
[9]	J. Keener and J. Sneyd, "Mathematical Physiology," Springer, Berlin, 1998.
[10]	S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims and L. M. Wahl, Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling, Bone, 33 (2003), 206-215.
[11]	D. L. Lacey, E. Timms, h.-L. Tan, M. J. Kelley, C. R. Dunstan, T. Burgess, R. Elliott, A. Columbero, G. Elliott, S. Scully, H. Hsu, J. Sullivan, N. Hawkins, E. Davy, C. Capparelli, A. Eli, Y. X. Qian, S. Kaufman, I. Sarosi, V. Shalhoub, G. Senaldi, J. Guo, J. Delaney and W. J. Boyle, Osteoprotegerin ligand is a cytokine that regulates osteoclast differentiation and activation, Cell, 93 (1998), 165-176.
[12]	D. A. Lauffenburger and J. Linderman, "Receptors: Models for Binding, Trafficking and Signaling," Oxford University Press, New York, 1996.
[13]	V. Lemaire, F. L. Tobin, L. D. Greller, C. R. Cho and L. J. and Suva, Modeling the interactions between osteoblast and osteoclast activities in bone remodeling, J. Theor. Biol., 229 (2004), 293-309. doi: 10.1016/j.jtbi.2004.03.023.
[14]	I. Nemere, 24,25-Dihydroxyvitamin D3 suppresses the rapid actions of 1,25 Dihydroxyvitamin D3 and parathyroid hormone on calcium transport in chick intestine, Bone Miner. Res., 14 (1999), 1543-1549.
[15]	I. Nemere, N. Garbi, G. J. Hammerling and R. C. Khanal, Intestinal cell calcium uptake and the targeted knockout of the 1,25D3-MARRS (Membrane-associated, rapid response steroid-binding) receptor/PDIA3/Erp57, J. Biol. Chem., 285 (2010), 31859-31866.
[16]	I. Nemere, R. J. Pietras and P. F. Blackmore, Membrane receptors for membrane hormones: Signal transduction and physiological significance, J. Cell Biochem., 88 (2003), 438-445.
[17]	A. W. Norman, "Rapid Biological Responses Mediated by $1\alpha,25$-Dihydroxyvitamin D$_3$," In Vitamin D (D. Feldman and F. H. Glorieux and J. W. Pike), Academic Press, New York, 1997.
[18]	I. Novak, P. Kraikivski and B. Slepchenko, Diffusion in cytoplasm: Effects of excluded volume due to internal membranes and cytoskeletal structures, Biophys. J., 97 (2009), 758-767.
[19]	I. L. Novak, F. Gao, P. Kraikivski and B. Slepchenko, B. M. Diffusion amid random overlapping obstacles: Similarities, invariants, aproximations, J. Chem. Phys., 134 (2011), 154104.
[20]	J. W. Pike, "The Vitamin D Receptor and Its Gene," In Vitamin D (D. Feldman, F. H. Glorieux and J. W. Pike), Academic Press, New York, 1997, 105-125.
[21]	W. S. Simonet, D. L. Lacey, C. R. Dunstan, M. Kelley, M.-S. Chang, R. Lothy, H. Q. Nguyen, S. Wooden, L. Bennett, T. Boone, G. Shimamoto, M. DeRose, R. Elliott, A. Columbero, H.-L. Tan, G. Trail, J. Sullivan, E. Davy, N. Bucay, L. Renshaw-Gregg, T. M. Hughes, D. Hill, W. Pattison, P. Campbell, S. Sander, G. Van, J. Tarpley, P. Derby, R. Lee and W. J. Boyle, Osteoprotegerin: A novel secreted protein involved in the regulation of bone density, Cell, 89 (1997), 309-319.
[22]	P. Slepchenko, I. Semenova, I. Zaliopin. and V. Rodianaov, Switching of membrane organelles between cytoskeletal transport systems is determined by regulation of the microtubule-based transport, J. Cell Biol., 179 (2007), 635-641.
[23]	G. K. Witfield, P. W. Jurutka, C. A. Hausler, J.-C. Hsieh, T. K. Barthel, E. T. Jacobs, C. E. Dominguez, M. L. Thatcher and M. R. Hausler, "Nuclear Vitamin D Receptor: Control of Gene Transcription, and Novel Bioactions," In Vitamin D (D. Feldman, F. H. Glorieux and J. W. Pike), Academic Press, New York, 1997, 219-261.

show all references

References:

[1]	A. Atri, J. Amundson, D. Clapham and J. Sneyd, A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis Oocyte, Biophys. J., 65 (1993), 1727-1739.
[2]	F. Bronner, Cytoplasmic transport of calcium and other inorganic ions, Comp. Biochem. Physiol., 115B (1996), 313-317.
[3]	J. Buchanan, R. P. Gilbert and M. J. Ou, The kinetics of vitamin $D_3$ in the osteoblastic cell, Submitted to J. Theor. Biol., (2012).
[4]	E. M. Costa and D. Feldman, Measurement of 1,25-Dihydroxyvitamin D3 receptor turnover by dense amino acid labeling: Changes during receptor up-regulation by vitamin D metabolites, Endocrinology, 120 (1987), 1173-1178.
[5]	M. C. Farach-Carson and P. J. Davis, Steroid hormone interactions with target cells: Cross talk between membrane and nuclear pathways, J. Pharm. Exp. Therap., 307 (2003), 839-845.
[6]	R. Gilbert, A. Panasenko and A. Vasilic, Acoustic Propagation in a Random Saturated Medium: The Monophasic Case, Math. Mehods Appl. Sciences, 33 (2010), 2206-2214.
[7]	K. Hackl and S. Ilic, Application of the multiscale FEM to the modeling of cancellous bone, Biomechan. Model. Mechanobiol., 9 (2010), 87-102.
[8]	V. V. Jikov, S. M. Kozlov and O.A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer, Berlin 1994.
[9]	J. Keener and J. Sneyd, "Mathematical Physiology," Springer, Berlin, 1998.
[10]	S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims and L. M. Wahl, Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling, Bone, 33 (2003), 206-215.
[11]	D. L. Lacey, E. Timms, h.-L. Tan, M. J. Kelley, C. R. Dunstan, T. Burgess, R. Elliott, A. Columbero, G. Elliott, S. Scully, H. Hsu, J. Sullivan, N. Hawkins, E. Davy, C. Capparelli, A. Eli, Y. X. Qian, S. Kaufman, I. Sarosi, V. Shalhoub, G. Senaldi, J. Guo, J. Delaney and W. J. Boyle, Osteoprotegerin ligand is a cytokine that regulates osteoclast differentiation and activation, Cell, 93 (1998), 165-176.
[12]	D. A. Lauffenburger and J. Linderman, "Receptors: Models for Binding, Trafficking and Signaling," Oxford University Press, New York, 1996.
[13]	V. Lemaire, F. L. Tobin, L. D. Greller, C. R. Cho and L. J. and Suva, Modeling the interactions between osteoblast and osteoclast activities in bone remodeling, J. Theor. Biol., 229 (2004), 293-309. doi: 10.1016/j.jtbi.2004.03.023.
[14]	I. Nemere, 24,25-Dihydroxyvitamin D3 suppresses the rapid actions of 1,25 Dihydroxyvitamin D3 and parathyroid hormone on calcium transport in chick intestine, Bone Miner. Res., 14 (1999), 1543-1549.
[15]	I. Nemere, N. Garbi, G. J. Hammerling and R. C. Khanal, Intestinal cell calcium uptake and the targeted knockout of the 1,25D3-MARRS (Membrane-associated, rapid response steroid-binding) receptor/PDIA3/Erp57, J. Biol. Chem., 285 (2010), 31859-31866.
[16]	I. Nemere, R. J. Pietras and P. F. Blackmore, Membrane receptors for membrane hormones: Signal transduction and physiological significance, J. Cell Biochem., 88 (2003), 438-445.
[17]	A. W. Norman, "Rapid Biological Responses Mediated by $1\alpha,25$-Dihydroxyvitamin D$_3$," In Vitamin D (D. Feldman and F. H. Glorieux and J. W. Pike), Academic Press, New York, 1997.
[18]	I. Novak, P. Kraikivski and B. Slepchenko, Diffusion in cytoplasm: Effects of excluded volume due to internal membranes and cytoskeletal structures, Biophys. J., 97 (2009), 758-767.
[19]	I. L. Novak, F. Gao, P. Kraikivski and B. Slepchenko, B. M. Diffusion amid random overlapping obstacles: Similarities, invariants, aproximations, J. Chem. Phys., 134 (2011), 154104.
[20]	J. W. Pike, "The Vitamin D Receptor and Its Gene," In Vitamin D (D. Feldman, F. H. Glorieux and J. W. Pike), Academic Press, New York, 1997, 105-125.
[21]	W. S. Simonet, D. L. Lacey, C. R. Dunstan, M. Kelley, M.-S. Chang, R. Lothy, H. Q. Nguyen, S. Wooden, L. Bennett, T. Boone, G. Shimamoto, M. DeRose, R. Elliott, A. Columbero, H.-L. Tan, G. Trail, J. Sullivan, E. Davy, N. Bucay, L. Renshaw-Gregg, T. M. Hughes, D. Hill, W. Pattison, P. Campbell, S. Sander, G. Van, J. Tarpley, P. Derby, R. Lee and W. J. Boyle, Osteoprotegerin: A novel secreted protein involved in the regulation of bone density, Cell, 89 (1997), 309-319.
[22]	P. Slepchenko, I. Semenova, I. Zaliopin. and V. Rodianaov, Switching of membrane organelles between cytoskeletal transport systems is determined by regulation of the microtubule-based transport, J. Cell Biol., 179 (2007), 635-641.
[23]	G. K. Witfield, P. W. Jurutka, C. A. Hausler, J.-C. Hsieh, T. K. Barthel, E. T. Jacobs, C. E. Dominguez, M. L. Thatcher and M. R. Hausler, "Nuclear Vitamin D Receptor: Control of Gene Transcription, and Novel Bioactions," In Vitamin D (D. Feldman, F. H. Glorieux and J. W. Pike), Academic Press, New York, 1997, 219-261.

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