A Mechanism for the Regulation of Ligament Frequency- Waves

8/2/2019 A Mechanism for the Regulation of Ligament Frequency- Waves

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J. theor. Biol. (1983) 102,477-486

A Mechanism for the Regulation of LigamentWidth Based on the Resonance Frequency of IonConcentration WavesJACK FERRIER

Medical Research Council Group in Periodontal Physiology, 4384Medical Sciences Build ing, University of Toronto, Toronto, Ontario,

M5S lA8, Canada(Received 26 August 1982, and in revised form 18 November 1982)A model is developed for a mechanism for the regulation of the width ofligament spaces and of other tissue spaces bounded by calcified surfaces.The proposed mechanism involves the transmission, detection, andretransmission of ion concentration waves by cells located on the calcifiedsurfaces. It is assumed that these cells can use the information regardingligament width contained in the resonance frequency of the cell-concentra-tion wave system. The assumptions of the proposed mechanism are suppor-ted by recent experimental evidence concerning the effect of electricalsignals on bone cells, the use of frequency-encoded information by cells,and the production of low frequency K pulses by osteoblast-like cells.The relation between resonance frequency and ligament width is derived,and the resonance frequencies corresponding to measured ligament widthsare shown to occur in the same frequency range as occur in the K pulsesemitted by bone cells. The model suggests definite experimental tests thatinvolve investigating the effect in vitro of ion concentrationwave frequencyon bone cell activity and hormone receptors.

The purpose of this paper is to develop the theory of a mechanism for theregulation of the width of ligament spaces. These spaces have widths ofthe order of hundreds of micrometers. A good example is the periodontalligament, which connects tooth to bone. The tooth to bone dimension ofthe periodontal ligament is known to be remarkably well regulated duringgrowth, during natural remodelling of the surrounding bone, and evenduring artificially induced (orthodontic) tooth movement (Melcher, 1976).The periodontal ligament is a fibrous connective tissue that is highly cellular,with the greatest concentration of cells occurring close to the mineralizedboundaries of the ligament (McCulloch, 1982). The typical dimensions ofthe tooth and ligament are such that, in a cross section perpendicular tothe longest axis of the tooth, the ligament has the appearance of a roughly

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0022-5193/83/120477+10$03.00/0 @ 1983Academic ressnc. (London) td.


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478 J. FERRIER

FIG. 1. A schematic diagram of the tooth-periodontal ligament-bone system, showingtypical relative dimensions of the tooth and ligament.

circular or elliptical annulus, with a width about 10 to 20% of the diameters(see Fig. 1).The proposed mechanism would provide information regarding the liga-ment width to the cells that reside on the two calcified surfaces that arethe boundaries of the ligament space. The model requires the assumptionthat these cells can transmit, detect, and retransmit ion concentration waves.It requires the further assumption that the cells can use the informationregarding ligament width contained in the resonance frequency of thecell-ion concentration wave system.The assumptions are based on the following experimental evidence. That

connective tissue cells can detect and act on information carried by changesin extracellular ion concentration is indicated by the stimulation of bonecells to increase bone formation rate by the application of electric currentsor fields (Brighton, Black & Pollack, 1979). The applied fields and theapplied currents that have been found to be effective would produce verysmall extracellular electric currents in the range l-10 kA/cm. Such cur-rents would be accompanied by very small electric potential gradients(5 1 mV/cm), and by very small changes in the extracellular ion concentra-tions at the cell surfaces. An upper bound on the change in extracellular


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RESONANCE FREQUENCY AND LIGAMENT WIDTH 479ion concentration at the cell surface produced by an extracellular currentdensity of 1 uA/cm2 is lop4 mM (see Appendix A).There is some experimental evidence that many cell types can usefrequency-encoded information internally to regulate their function (Rapp,1979; Berridge & Rapp, 1979). In addition, a recent theoretical analysis(Rapp, Mees & Sparrow, 1981) shows that cellular regulation mechanismsusing frequency-encoded information internally would be inherently moreaccurate and much less susceptible to degradation by background noisethan mechanisms using amplitude-encoded information. Further, aspointed out by Goodwin & Cohen (1969), a system of information transmit-tal between cells based on waves can have a higher informational capacitythan a system based simply on molecular gradients. The model of Goodwin& Cohen is based on a wave-like propagation of signaling molecules whichare released by cells in a cascade. In their model, the phase differencebetween waves of different velocity would provide the cells with informationon their position relative to pacemaker cells that act as the source of thewaves.The assumption that connective tissue cells can transmit ion concentrationwaves is based on the recent finding of Ferrier ef al. (1982) that anosteoblast-like clone produced high magnitude low-frequency membranepotential fluctuations. Microelectrode data showed apparently randomfluctuations of several millivolts. A computer based analysis showed a steepslope in the power spectrum between 0.1 and 0.3 Hz, with considerablepower below 0.1 Hz. Analysis showed that the random fluctuations wereactually an incoherent sum of triangularly shaped pulses (which sometimescohered to form large spikes). The magnitude and polarity of the fluctu-ations strongly indicated that they resulted from an increase in the Kconductance of the cell membrane. Such conductance fluctuations wouldbe accompanied by pulses in the K membrane transport rate, which wouldhave the same frequency distribution as the membrane potential pulses.That is, starting from a steady state with zero net membrane flux foreach ion species, if there is a pulse change in conductance for one species(say for K), there will be a net flux of that species through the membranethat will persist during the time that the conductance is different from itszero flux value. A concurrent change in membrane potential will be pro-duced that will result in the fluxes of the other ion species changing in sucha way as to maintain the condition of zero net current through the cellmembrane (i.e. except for the extremely small currents required for changesin the charging of the membrane capacity, the net current integrated overthe cell boundary must be zero, to preserve electroneutrality). The resultof this will be pulsed extracellular fluxes for the ion species that have


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RESONANCE FREQUENCY AND LIGAMENT WIDTH 481can be obtained directly:

v = 2rf/k = (4rfD). (3)Combining equations (l), (2) and (3) results in the following equationfor the resonance or natural frequencies:fr = (rD/L)(n -A&edd2. (41

If the further assumption is made that A4cell


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482 J. FERRIERattenuated amplitude on the other side of the ligament of about 10e4 mM.Although this is a small amplitude, it is of the same magnitude as the ionconcentration changes produced at the cell surface by currents known toaffect bone cell activity (see Appendix A).The rate of bone formation by the cells must be an increasing functionof ligament width, while the rate of bone resorption by the cells must bea decreasing function of ligament width. The relation between resonancefrequency and ligament width given by equation (4) or (5) indicates thatthe rate of bone formation should be a decreasing function of resonancefrequency, while the rate of resorption should be an increasing function ofresonance frequency. While it is impossible at present to say anythingdefinite about how the cells might translate frequency into formation orresorption activity, the work of Rapp et al. (1981) provides a generaltheoretical picture of how a cell could use a change in frequency internallyto modulate cell activity. One of the cellular oscillators they discuss is afeedback loop involving internal [Ca*]. There is some evidence that asimilar oscillator is involved in potential fluctuations and oscillations inconnective tissue cells (Ferrier et al., 1982; Okada, Tsuchiya & Inouye,1979). In general, the work of Rapp ef al. (1981) supports the idea that achange in frequency of an internal oscillatory process can lead to a changein some other cell regulatory system. The work of Goodwin & Cohen(1969) is based on the idea that information regarding cell position canbe provided to cells by the phase difference between wave-like signalspropagating from cell to cell. They explore in some detail the possiblebiochemical connections between such phase-encoded information and cellregulatory processes.It is known that the formation/resorption activity of bone cells is stronglyaffected by various hormones which reinforce or interfere with each otherin complex ways (Heersche & Jez, 1981; Heersche ef al., 1980; Aubin eful., 1982). One possibility that seems well founded on the basis of theforegoing arguments is that the resonance frequency could modulate thenumber or functionality of hormone receptors. Since parathyroid hormone,PTH, is known to increase the net rate of bone resorption, at least in partby inhibiting the rate of bone formation by osteoblasts (e.g. Tam et uf.,1982), it can be hypothesized that an increase in resonance frequency(corresponding to a decrease in ligament width) would result in an increasein the number or functionality of PTH receptors in the osteoblast cellmembrane. This hypothesis could be tested in vitro, by subjectingosteobalst-like cell clones to concentration waves of various frequencies,and using established methods (Aubin et uZ., 1982) to measure the effectof this on PTH receptors.


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RESONANCE FREQUENCY AND LIGAMENT WIDTH 483Another experimental test of the proposed mechanism would be to

subject organ cultures containing bone and active bone cells (Tenenbaum& Heersche, 1982) to concentration waves. The rate of bone formation orresorption as a function of wave frequency could then be directly measured.It should be possible to produce K concentration waves in such systemssimply by applying pressure pulses to a KCl-filled pipette, or by injectingalternating current into the extracellular medium through a K-selectiveelectrode placed near the cell layer.The model developed in this paper has some limited similarities to amodel proposed by Newman & Frisch (1979), in which standing waves ofmorphogen concentration would regulate the spatial dimensions in embry-onic skeletal development. The model of Goodwin & Cohen (1969) inwhich the phase difference between wave-like signals provides informationon cell position also has some limited similarities to the model proposedhere. More generally, there are of course many examples of biologicalsystems using wave phenomena to convey information. The unique aspectof the present model is that it predicts specific effects of the frequency ofK concentration waves on the bone formation/resorption activity of bonecells. The model could also apply to the regulation of other connectivetissue spaces delineated by calcified surfaces, such as the Haversian canalsystem within bone (Pritchard, 1972), the canals of which have anapproximately cylindrical geometry with a diameter of the same magnitudeas the width of periodontal ligament.To summarize, the mechanism proposed here for the regulation ofligament width involves the transmission, detection, and retransmission ofion concentration waves by cells on the calcified surfaces. The central ideaof the hypothesis is that these cells can use the fundamental resonancefrequency of the cell-ion concentration wave system as a measure of theligament width. There is some experimental support for each of the assump-tions involved in the model. The model does suggest definite experimentaltests that can be carried out with existing in vitro techniques.

REFERENCESAUBIN, J. E., HEERS CHE, J. N. M., MERRILEES, M. J. & SODEK. J. (1982). J. cdl Biol .

92,452.BERRIDGE, M. J. & RAP P, P. E. (1979). J. exp. Biol . 81,217.BRIGHTON, C. T., BLAC K, J. & POLLACK, S. R. (eds) (1979). Elecrr ical Properties of Bone

and Cartilage. Experimental Effects and Clin ical Ap plicatio ns. New York: Grune & Stratton.CARS LAW, H. S. & JAEGE R, J. C. (1959). Conductivity of Heat in Solid s, p, 65. New York:

Oxford University Pre ss.FERRIER, J. M. (1980). 1. theor. Biol. 85, 793.FERRIER, J. M. (1981a). J. theor. Bio l. 92, 363.


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484 J. FERRIERFERRIER, J. M. (1981b). J. &or. Biol . 93,495.FERRIER, J., DIXON, J., ILLEMAN, A., DILLON, E. & SMITH, I. (1982). J. cell . Physiol.

113,267.GOODWIN, B. C. & COHEN, M. H. (1969). J. theor. Biol . 25,49.HEERS CHE, J. N. M. & JEZ, D. H. (1981). Prostuglu ndins. 21,401.HEERS CHE, J. N. M., JEZ, D. H., AUBIN, J. & SODEK, J. (1980). Hormonal Control of

Calciu m Metabolism (Cohn, D. V., Talm age, R. V., Mathews, J. L., eds). pp. 157-162.Amsterdam: Excerpta Medica.

MELCHER, A. H. (1976). In: Orbans Oral Histology (Bhaskar, S. N., ed.) pp. 206-233.Saint L ouis: C. V. Mosby.

MCCULLOCH, C. (1982). Progenitor Cel ls in the PeriodontalLigamen t. Ph.D. The sis, Univer-sity of Toronto.

NEWMAN, S. A. & FRISCH, H. L. (1979). Scien ce. 205,662.OKADA, Y., TSUCHIYA, W. & INOUYE, A. (1979). J. membr. Bio l. 47,357.PRITCH ARD, J. J. (1972). In: The Biochem istry and Physiology ofBone, (Bourne, G. H., ed.)pp. 11-14. New York: Acad emic Press.RAP P, P. E. (1979). J. exp. Bio l. 81,281.RAPP, P. E., MEES, A. I. & SPARROW, C. T. (1981). J. theor. B iol. 90,531.TAM, C. S., HEERS CHE, J. N. M., MURRAY, T. M. & PARSON S, J. A. (1982). Endocrinology.

110, 506.TENENBAUM,H.C.& HEERSCHE,J.N.M. (1982). C&if. Tissu e Znt. 34.76.WE I, J. & Russ, M. B. (1977). J. theor. Bio l. 66,775.

APPENDIX AAn upper bound for the change in ion concentration at the cell surfaceresulting from an externally applied extracellular current can be calculatedas follows. The extracellular electrical potential gradient, VJI, resulting froma current density, 1, is given by V@ = -I/g, where (T is conductivity of theextracellular medium. In going across a cell in the direction of the current

there must be a change in potential of about AIL, = BV+k, where B is thecell dimension in the direction of the current. Almost all of this AI&~ willappear across the cell membranes, because of their high resistivity. Thus,the maximum change in charge density on the cell membrane producedby the current will be AQ = CA&/2 = CBV*/2 = CBI/2u, where C is thecell membrane capacitance per unit area. This can be converted to an upperbound on the change in ion concentration by dividing by the Faradayconstant, F, to convert charge surface density to ion surface density, andby dividing by a lower bound on the distance from the cell surface withinwhich the ions are contained, 6. This gives CBII2Fub. Using C = 1 kf/cm,B = 10 pm, I= 1 kA/cm*, cr = 10-*/R cm, and b = 0.1 nm, gives a resultof the order of 10e4 mM.The relation between maximum ion flux and wave amplitude for an ionconcentration wave propagating in one dimension is J,,, = (27rfD)A,where J, is the maximum flux and A is the wave amplitude. At thefundamental resonance frequency, and for typical ligament widths, a


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RESONANCE FREQUENCY AND LIGAMENT WIDTH 485maximum flux of 10 pmol/cm set requires a wave amplitude of the orderof lop2 mM. After propagating across the ligament, the wave amplitudewill be attenuated by a factor of exp (-7~), becoming about 1O-4 mM.

APPENDIX BCurrent density, I, in the extracellular medium is given by

I = --(TV~ -F 1 tiDiVCi, (Al)allwhere u is the medium electrical conductivity, +, is electrical potential, Fis the Faraday constant, and ti, Di and Ci are valency, diffusion coefficientand concentration of the ith ionic species.The extracellular flux of the kth ion species, Jk, is given by

Jk = -ukVIJ//Fzk - DkVCk, (A21where (+k is the conductivity attributable to the kth species.Solving equation (Al) for VJI, substituting into equation (A2), andseparating some terms, results in

Jk =(Uk/U) 1 ZiDiVCi/Zk -[l -(Uk/U)]DkVCk +(Uk/U)I/FZL.all i#k (A31

Then, defining Bk = Cal, i+k ZiDiVCi/(Dk Cal, ;fk ZiVCi), and using theelectroneutrality condition, Cal, i ZiCi = 0, equation (A3) can be rewritten asJk = - (1 - [(T k/( T] [ 1 - &])Dkvck + (Uk/U )1/&k.

Note that Bk = 1 if Di =Dk for all i.(A4)

Equation (A4) expresses the important result that the flux of species kwill be carried almost entirely by diffusion if (+k


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486 J. FERRIERis given by 2D/L, which is in the range of 1 to 10 km/set. An upper boundon the convection velocity, uC, can be calculated by multiplying a fairlyhigh estimate for the endothelial hydraulic conductivity, say10 km/set MPa (which allows the tight junctions between the endothelialcells to be fairly leaky), by an upper bound on the possible hydrostaticpressure difference across the endothelium, say 1O-3 MPa, givinglo-* bm/sec. The result that od >>uC indicates that the effect of convectionon the K concentration wave should be negligible.It should also be mentioned in this context that the time-dependent iontransport equations of Ferrier (1981~) should be slightly modified by addingthe convection driven current density carried by the non-transportedspecies in the absence of membrane transport of the transported species,&r(O), to the right hand side of equations (2) and (3) and the equation inthe line preceding equation (3) in that paper. This term should also besubtracted from the right hand side of equation (5) in that paper. Essentially,this involves a change in the steady state result expressed in the paragraphpreceding equation (4) of Ferrier (1981a): the current density carried bythe non-transported species at steady state should be equal to &r(O). Thisfollows directly from the result discussed by Ferrier (1981b), obtained bytaking into account the possibility of convection driven fluxes of both thetransported and the non-transported ion species in the absence of anymembrane transport.

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A Mechanism for the Regulation of Ligament Frequency- Waves