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Electrical signal transmission and gap junction regulation in a bone cell network: A cable model for an osteon

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Abstract

A cabel model is formulated to estimate the spatial distribution of intracellular electric potential and current, from the cement line to the lumen of an osteon, as the frequency of the loading and the conductance of the gap junction are altered. The model predicts that the characteristic diffusion time for the, spread of current along the membrane of the osteocytic processes, 0.03 sec, is nearly the same as the predicted pore pressure relaxation time in Zenget al. (Annals of Biomedical Engineering. 1994) for the draining of the bone fluid into the osteonal canal. This approximate equality of characteristic times causes the cable to behave as a high-pass, low-pass filter cascade with a maximum in the spectral response for the intracellular potential at approximately 30 Hz. This behavior could be related to the experimetns of Rubin and McLeod (Osteoporosis, Academic Press, 1996) which show that live bone appears to be selectively responsive to mechanical loading in a specific frequency range (15–30 Hz) for several species.

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Abbreviations

a :

radius of the osteocytic process, herea=0.075 μm

a gi :

radius of the open channel of theith connexon within a gap junction

A p :

intermediary parameter defined in Eq. 15; a function of ξ and θ

A :

intermediary parameter defined in Eq. 17; a function of ξ and θ

B p :

intermediary parameter defined in Eq. 15; a function of ξ and θ

c :

diffusion coefficient (m/sec2) of the bone fluid pressure within the canaliculi-lacunae system, evaluated through a poroelastic analysis

c m :

capacitance of the plasma membrane per unit surface areac m=0.01 Farad/m2

C m :

membrane capacitance per unit length of the cable (Farad/cm)

CmL,CmR:

membrane capacitance at the left and right end of the cable (Farad)

Ci,i=1,4:

coefficients used in Eq. 14 to expressV i *; expressions for them can be found in Eqs. 21 and 22

d :

diameter of the osteocytic process,d=2a=0.15 μm

e :

universal constant,e=2.718

I i :

intracellular current (Amp)

I m :

leakage current per unit length through the membrane (Amp/cm)

I ci :

nondimensionalized intracellular current,I ci =I i/(V c (1 Hz)L/R m)I ci =I *i (V c(ω)/V c(1 Hz))

I cir :

nondimensionalized intracellular current at the right (osteoblastic) end,I cir =I ci (x *=1)

I *i :

nondimensionalized intracellular current,I *i =I i/(V c(ω)L/R m)

I *ir :

nondimensionalized intracellular current at the right (osteoblastic) end,I cir =I ci (x *=1)

L :

length of the cable,L=105 μm

L g :

length of a connexon,L g=20 nm

L cp :

typical distance between two gap junctions in the bone cell network,L cp=35 μm

n :

number of open connexons in a gap junction; can be a fractional number to represent partially closed channels

p :

bone fluid pressure

r o :

annular thickness of an osteon,r o=105 μm

R i :

internal, longitudinal, resistance per unit length of the cable (Ω/cm)

R m :

resistance of the enclosing membrane of a unit length of the cylindrical cable (Ω/cm)

R o :

external, longitudinal, resistance per unit length of the cable (Ω/cm)

RmL,RmR:

membrane resistances at the left and right ends of the cable (Ω)

S,SL,SR:

leakage membrane areas along the whole cable, at the left, and right ends

t :

time (sec)

t e :

2π/ω, period of the external loading and its induced mechanical and electrical responses (sec)

t p :

r 2o /c, relaxation time of the bone fluid pressure (sec)

t * :

t/gt, nondimensional time

V c :

Vc(ω), amplitude of the stain generated streaming potential (used as the extracellular driving potential in the paper)

V i :

Vi(x, t), electrical potential inside the cable (volt)

V m :

Vm(x, t), transmembrane potential of the cable,Vm=Vi−Vc

V o :

Vo(x, t), electrical potential outside the cable (volt)

V ci :

V ci (x*, t*)=Vi/Vc (1 Hz), nondimensional intracellular potential normalized with respect toVc (1 Hz)

V cir :

V ci (1,t*), value ofV ci at the right end of the cable,x*=1

V *i :

V *i (x*,t*)=Vi/Vc(ω), nondimensional intracellular potential normalized with respect toVc(ω)

V *m :

V *m (x*,t*)=Vm/Vc(ω), nondimensional transmenbrane potential normalized with respect toVc(ω)

V *o :

V *o (x*,t*)=Vo/Vc(ω), nondimensional extracellular potential normalized with respect toVc(ω)

V *ir :

V *i (1,t*), value ofV *i at the right end of the cable,x*=1

x :

axial coordinate (μm)

x * :

x/L, nondimensional axial coordinate

Δ:

intermediary variable defined in Eq. 16

ζL, ζR :

leakage area ratios at the left and right ends, respectively, relative to the total membrane leakage area along the cable, ζL,R=S L,R/S

ν:

ratio of the resistance of the gap junction to the resistance of a cell process of lengthL cp, defined in Eq. 2

θ:

ωτ=τ/(1/ω), ratio of membrane time constant τ to the time parameter associated with the external loading, 1/ω

λ:

λ=√R m /R i, electrical coupling length or decay length of the cellular cable (μm)

λmax :

coupling length of the cable when the gap junction resistance vanishes

ξ:

λ/L, ratio of the coupling length of the cable λ to the length of the cableL

ϱ:

resistivity of the cytoplasm (110 Ω−cm)

τ:

membrane time constant (sec), τ=R m C m

ω:

angular frequency of the external loading and its induced responses (rad/sec)

References

  1. Atkinson, P. J., and A. S. Hallsworth. The changing pore structure of aging human mandibular bone.Gerodontology 2:57–66, 1983.

    Article  Google Scholar 

  2. Bingmann, D., K. Schirrmacher, and D. Jones. Signalling in bone: Electrophysiological studies on cultured cells derived from calvarial fragments of rats.Cells Mater. 4:275–285, 1994

    Google Scholar 

  3. Brinley, F. J., Jr. Excitation and conduction in nerve fibers. In:Medical Physiology, (13th edition) vol. 1, edited by V. B. Mountcastle, Saint Louis: The C.V. Mosby Company, 1974, pp. 34–76.

    Google Scholar 

  4. Carola, R., J. P. Harley, and C. R. Noback.Human Anatomy and Physiology. New York: McGraw-Hill Publishing Company, 1990, p. 139.

    Google Scholar 

  5. Cole, S. Vibration and linear acceleration. In:The Body at Work: Biological Ergonomics, edited by W. T. Singleton. Cambridge: Cambridge University press, 1982, pp. 201–234.

    Google Scholar 

  6. Cooper, M. S., J. P. Miller, and S. E. Fraser. Electrophoretic repatterning of charged cytoplasmic molecules within tissues coupled by gap junctions by externally applied electric fields.Dev. Biol. 132:179–188, 1989.

    Article  PubMed  CAS  Google Scholar 

  7. Cowin, S. C., S. Weinbaum, and Y. Zeng. A case for the bone canaliculi as the anatomical site of strain generated potentials.J. Biomech. 28:1281–1297, 1995.

    Article  PubMed  CAS  Google Scholar 

  8. Doty, S. B. Morphological evidence of gap junctions between bone cells.Calcif. Tissue Int. 33:509–512, 1981.

    Article  PubMed  CAS  Google Scholar 

  9. Doty, S. B. Cell to cell communication in bone tissue. InThe Biological Mechanisms of Tooth Eruption and Root Resorption edited by Z. Davidovitch. Birmingham, AL, EBSCO Media, 1988, pp. 61–96.

    Google Scholar 

  10. Doty, S. B. Intercellular communication between blood vessels in bone and bone cells. InThe Biological Mechanisms of Tooth Movement and Craniofacial Adaptation, edited by Z. Davidovitch, Columbus, OH, The Ohio State University, 1992, pp. 73–83.

    Google Scholar 

  11. Harrigan, T. P. and J. J. Hamilton. Bone strain sensation via transmembrane potential changes in surface osteoblasts: loading rate and microstructural implications.J. Biomech. 26:183–200, 1993.

    Article  PubMed  CAS  Google Scholar 

  12. Hille, B.Ionic channels of Excitable Membranes (second edition) Sunderland, MA. Sinauer Associates Inc., 1992, p. 9.

    Google Scholar 

  13. Hobbie, R. K.Intermediate Physics for Medicine and Biology (second edition). New York: John Wiley & Sons, 1988, p. 174.

    Google Scholar 

  14. Jeansonne, B. G., F. F. Feagin, R. W. Mcinn, R. L. Shoemaker, and W. S. Rehm. Cell-to-cell communication of osteoblasts.J. Dent. Res. 58:1415–1423, 1979.

    PubMed  CAS  Google Scholar 

  15. Klein-Nulend, J., A. van der Plas, C. M. Semeins, N. E. Ajubi, J. A. Frangos, P. J. Nijweide, and E. H. Burger. Sensitivity of osteocytes to biomechanical stressin vitro.FASEB J. 9:441–445, 1995.

    PubMed  CAS  Google Scholar 

  16. Loewenstein, W. R. Junctional intercellular communication: The cell-to-cell membrane channel.Physiol. Rev. 61:829–913, 1981.

    PubMed  CAS  Google Scholar 

  17. Marotti, G. Three-dimensional study of osteocytic lacunae.Metab. Bone Dis., Relat. Res. 2:S223-S229 1980.

    Article  Google Scholar 

  18. Matthews, J. L. Bone structure and ultrastructure. In:Fundamental and Clinical Bone Physiology, edited by M. R. Urist, Philadelphia, PA: J. B. Lippincott Company, 1980, pp. 4–44.

    Google Scholar 

  19. Otter, M. W., V. R. Palmieri, D. D. Wu, K. G. Seiz, L. A. MacGinitie, and G. V. B. Cochran. A comparative analysis of streaming potentialsin vivo andin vitro.J. Orthop. Res. 10:710–719, 1992.

    Article  PubMed  CAS  Google Scholar 

  20. Pilla, A. A., P. R. Nasser, and J. J. Kaufman. Gap junction impedance, tissue dielectrics and thermal noiser limits for EMF bioeffects.Bioelectrochem. Bioenerg. 35:63–69, 1994.

    Article  CAS  Google Scholar 

  21. Pollack, S. R., N. Petrov, R. Salzstein, G. Brankov, and R. Blagoeva. An anatomical model for streaming potentials in osteons.J. Biomech. 17:627–636, 1984.

    Article  PubMed  CAS  Google Scholar 

  22. Rasmussen, H., and P. Bordier.The Physiological and Cellular Basis of Metabolic Bone Disease. Baltimore: The Williams & Wilkins Company, 1974, pp. 8–69.

    Google Scholar 

  23. Rubin, C. T., and K. J. McLeod. Inhibition of osteopenia by biophysical intervention.Osteoporosis chap. 14, edited by R. Marcus, D. Feldman, and J. Kelsey, New York: Academic Press, 1996, pp. 351–371.

    Google Scholar 

  24. Salzstein, R. A., S. R. Pollack, A. F. T. Mak, and N. Petrov. Electromechanical potentials in cortical bone-I. A continuum approach.J. Biomech 20:261–270, 1987.

    Article  PubMed  CAS  Google Scholar 

  25. Salzstein, R. A. and S. R. Pollack. Electromechanical potentials in cortical bone-II. Experimental analysis.J. Biomech. 20:271–280, 1987.

    Article  PubMed  CAS  Google Scholar 

  26. Schirrmacher, K., I. Schmitz, E. Winterhager, O. Traub, F. Brümmer, D. Jones, and D. Bingmann. Characterization of gap junctions between osteoblast-like cells in culture.Calcif. Tissue Int. 51:285–290, 1992.

    Article  PubMed  CAS  Google Scholar 

  27. Schirrmacher, K., D. Nonhoff, D. Bingmann, and P. R. Brink. Calcium effects of gap junctions between rat osteoblast-like cellsin vitro.Pflügers Arch. Eur. J. Physiol. 431(Suppl.):R92, 1996.

    Google Scholar 

  28. Schirrmacher, K., R. Grümmer, S. V. Ramanan, and P. R. Brink. Voltage sensitivity of gap junctions in osteoblast-like cellsin vitro.Pflügers Arch. Eur. J. Physiol. 431(Suppl.):R93, 1996.

    Google Scholar 

  29. Scott, G. C., and E. Korostoff. Oscillatory and step response to electromechanical phenomena in human and bovine bone.J. Biomech. 23:127–143, 1990.

    Article  PubMed  CAS  Google Scholar 

  30. Socolar, S. J., and W. R. Loewenstein, Methods for studying transmission through permeable cell-to-cell junctions. In:Methods in Membrane Biology, vol. 10, edited by E. D. Korn, New York: Plenum Press, 1979, pp. 123–179.

    Google Scholar 

  31. Spray, D. C. Physiological and pharmacological regulation of gap junction channels. In:Molecular Mechanisms of Epithelial Cell Junctions: From Development to Disease, edited by S. Citi, Austin, TX: R. G. Landes Company, 1994, pp. 195–215.

    Google Scholar 

  32. Starkebaum, W., W. R. Pollack, and E. Korostoff. Microlectrode studies of stress-generated potentials in four-point bending of bone.J. Biomed. Mater. Res. 13:729–751, 1979.

    Article  PubMed  CAS  Google Scholar 

  33. Wallace, R. A.Biology, the World of Life, (fourth edition). Glenview, IL: Scott. Foresman & Company, 1987, p. 111.

    Google Scholar 

  34. Weinbaum, S., S. C. Cowin, and Y. Zeng. Excitation of osteocytes by mechanical loading-induced bone fluid shear stresses.J. Biomech. 27:339–360, 1994.

    Article  PubMed  CAS  Google Scholar 

  35. Xia, S. L., and J. Ferrier. Propagation of a calcium pulse between osteoblastic cells.Biochem. Biophys. Res. Commun. 186:1212–1219, 1992.

    Article  PubMed  CAS  Google Scholar 

  36. Yamaguchi, D. T., J. T. Huang, D. Ma. Regulation of gap junction intercellular communication by pH in MC3T3-E1 osteoblastic cells.J. Bone Miner. Res. 10:1891–1899, 1995.

    Article  PubMed  CAS  Google Scholar 

  37. Zeng, Y., S. C. Cowin, and S. Weinbaum. A fiber matrix model for fluid flow and streaming potentials in the canaliculi of an osteon.Ann. Biomed. Eng. 22:280–292, 1994.

    Article  PubMed  CAS  Google Scholar 

  38. Zhang, D., and S. C. Cowin. Oscillatory bending of a poroelastic beam.J. Mech. Phys. Solids 42:1575–1599, 1994.

    Article  Google Scholar 

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Authors and Affiliations

  1. CUNY Graduate School, The City College of City University of New York, New York, NY

    Dajun Zhang, Stephen C. Cowin & Sheldon Weinbaum

  2. Department of Mechanical Engineering, The City College of City University of New York, 140th St. and Convent Ave., 10031, New York, NY, U.S.A.

    Dajun Zhang, Stephen C. Cowin & Sheldon Weinbaum

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Zhang, D., Cowin, S.C. & Weinbaum, S. Electrical signal transmission and gap junction regulation in a bone cell network: A cable model for an osteon. Ann Biomed Eng 25, 357–374 (1997). https://doi.org/10.1007/BF02648049

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