- Email: [email protected]
Physics models of centriole replication
Medical Hypotheses (2006) 67, 572–577
http://intl.elsevierhealth.com/journals/mehy
Physics models of centriole replication Kang Cheng *, Changhua Zou Science Research, 205 Hana Road, Edison, NJ 08817, USA Received 26 February 2006; accepted 28 February 2006
Summary Our previous pre-clinic experimental results have showed that the epithelialization can be enhanced by the externally applied rectangular pulsed electrical current stimulation (RPECS). The results are clinically significant for patients, especially for those difficult patients whose skin wounds need long periods to heal. However, the results also raise questions: How does the RPECS accelerate the epithelium cell proliferation? To answer these questions, we have previously developed several models for animal cells, in a view of physics, to explain mechanisms of mitosis and cytokinesis at a cellular level, and separation of nucleotide sequences and the unwinding of a double helix during DNA replication at a bio-molecular level. In this paper, we further model the mechanism of centriole replication during a natural and normal mitosis and cytokinesis to explore the mechanism of epithelialization enhanced with the externally applied RPECS at a biomolecular level. Our models suggest: (1) Centriole replication is an information flowing. The direction of the information flowing is from centrioles to centrioles based on a cylindrical template of 9 · 3 protein microtubules (MTs) pattern. (2) A spontaneous and strong electromagnetic field (EMF) force is a pushing force that separates a mother and a daughter centrioles in centrosomes or in cells, while a pulling force of interacting fibers and pericentriolar materials delivers new babies. The newly born babies inherit the pattern information from their mother(s) and grow using microtubule fragments that come through the centrosome pores. A daughter centriole is always born and grows along stronger EMF. The EMF mostly determines centrioles positions and plays key role in centriole replication. We also hypothesize that the normal centriole replication could not been disturbed in centrosome in the epithelium cells by our RPECS, because the centrioles have two non-conducting envelope (cell and centrosome membranes), that protect the normal duplication. The induced electric field by externally applied RPECS could be mild compared with the spontaneous and natural electric field of the centrioles. Therefore, the centriole replication during the epithelium cellular proliferation may be directly, as well as indirectly (e.g., somatic reflex) accelerated by the RPECS. c 2006 Elsevier Ltd. All rights reserved.
Introduction The idea to use externally imposed electrical stimulation to enhance skin wound healing probably came from the knowledge that there is an injury current around a fresh wound, which could play a * Corresponding author. Tel.: +732 248 0790. E-mail address: [email protected] (K. Cheng).
role in skin wound healing [1]. As early as 1860s, the founder of the science of bioelectricity, Du Bois–Reymond, discovered that an injured bleeding finger is electrically positive compared with an uninjured finger [2]. To our knowledge, the first modern application of electric current stimulation to a pre-clinic skin wound healing was reported in 1962 [3]. From that time on, electrical current stimulation has been frequently used in both
0306-9877/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mehy.2006.02.041
Physics models of centriole replication pre-clinic and clinic investigations of cutaneous regeneration. In our previous pre-clinic investigation of epithelialization enhanced with the externally applied rectangular pulsed electrical current stimulation (RPECS) and conforming electrodes, we observed, at a tissue level, that wounds treated with RPECS were healed about 15% faster than the control wounds [4]. The pre-clinical experimental results impelled us to measure in vitro conductivity [5] and in vivo 3D distributions of electric fields [6] on/in pig skin with RPECS. Our measured data indicate the externally applied electric field (EF) is about three times as strong on the skin surface under the electrode edges, where the bacteria could mostly invade the skin wound, as that in the skin. The results of our observation suggest that not only the epithelialization but also the infection/antiinfection could be related to the cell proliferation and the externally applied EF. However, the results also raise questions: How does the RPECS accelerate the epithelium cell proliferation? What are the relationships between the cellular proliferation and the spontaneous electromagnetic field (EMF)? What could be the driving force for the proliferation, in wild types of cells? To answer these questions, we have developed several models, in a view of physics, to explain mechanisms of mitosis and cytokinesis at a cellular level [7,8], and separation of nucleotide sequences and the unwinding of a double helix during DNA replication at a bio-molecular level [9]. In this investigation, with assistance of a model of RPECS for skin wound healing [10] and a model for mitosis and cytokinesis [7], we continue our exploration on centriole replication in wild types of animal cells in a perspective of physics. We believe centriole replication is one key procedure during the epithelium cell proliferation. In the research field of centriole replication, centrioles structures with cells from fibroblasts and stimulated lymphocyte were studied in 1970s [11] and the reproduction of centrosomes was investigated in 1980s [12]. Centrosome organization and centriole architecture and their sensitivity to divalent cations were investigated in early of 1990s [13]. For most animal cells, centrosomes are thought to ensure spindle bipolarity and thus correct chromosome segregation during mitosis [14]. Recently, centrioles and centrosomes positioned and duplicated in eucaryotes were observed and models were provided to depict the mechanism of centrosome positioning from physiological or biological perspective [15–17]. However, those models have likely focused on centrosome positioning, rather than a comprehen-
573 sive explanation of overall centrioles replication. We have not found any model in a view of physics to explain the mechanisms of this process, to answer the question: What is the kind or the source of the separating (pushing) force? And how does the force depart the centrioles and duplicate centrioles? In this paper, based on our previous models and published biological data, we are the first to develop an original model to approach the mechanism of centriole replication using the concept of inner spontaneous EF [7,18] in a view of physics. Then considering an externally applied EF, such as RPECS, to influence the natural and normal replication, we propose a hypothesis for the mechanisms of the overall centriole replication to explain the epithelium regeneration acceleration by RPECS based our previous pre-clinic experimental results [4].
Model development of centriole replication Experimental data show a centrosome is usually composed of two centrioles at proximately right angle. Each centriole is a cylindrical organelle organized with nine groups of three microtubules (MTs) [11,13]. MTs nucleating at the mother centriole were observed [19] and MTs are found to have functions to move cellular components [20]. In addition, experimental investigation have showed: at a activated state, at a certain point in G1 of a cell life cycle, the mother and daughter centrioles begin to departure from each other. During S phase, baby centrioles begin to grow near their mothers and MT always nucleate at a mother centriole [19]. We analyze those experimental data and provide our explanation: at a rest state, a mother centriole has much more positive charge than that of a daughter centriole has, a mother is almost saturated with positive charges, and a daughter is not. But, the EF interaction (pushing force), between a pair of centrioles is too weak to overcome the maximum resistance, including static friction. At the activated state, the daughter is saturated with positive charges too. When a pair of mother and daughter centrioles have enough positive charges, the interactive repulsing (pushing) EF force is strong enough to overcome the maximum resistance including the both static and dynamic friction (or viscosity), and to separate the pair of mother and daughter centrioles. We believe that biochemical or biophysical events provoke the spontaneous and strong interactive EMF, which is approximately a quasi-static EF [7,8].
574
Cheng and Zou
In our previous study [7], first, we consider the protoplasm as electrolyte with an uneven electric charge distribution as well as a runtime environment driven by biochemical or biophysical events. Second, we assume an aster MT complex (or a self-assembled MT complex) and a single MT has a positive net charge (including some bound ions, e.g., Ca2+, or other materials), a single MT is electrically, positively polarized at positive growing ends and negatively polarized at negative growing ends. The stronger the EF, the more microtubules grow. In this study, we still use these two assumptions as basic condition for modeling mechanism of centriole replication. Figs. 1 and 2 show our model of centriole replication: EF and a couple of centrioles at a rest state and at an active state, respectively, in systems of cylindrical coordinates (r, h, z) and laboratory reference. To develop applicable models, we consider a simple case that one mother and one daughter are separating along a line in an r–z plane because this is approximately a real world situation and it is easier to obtain analytical solutions mathematically. We assume this line is on z-axis for our modeling convenience. Using the superposition principle in physics, based on extended Coulomb’s law, a total EF force interacted between a mother and a daughter centrioles is approximately estimated with Z 27 X 27 Z X dqj dqi F md ¼ F dm ¼ d ij0 2 Qj Qi 4ped ij j i e ¼ permittivity
ð1Þ
where Qi and dqi, Qj and dqj, respectively, represent equivalent charges and a unit charge on a
- ends
daughter MT, on a mother’s MT. dij = dij(r, h, z) is a distance between dqi and dqj. dij0 is unit vector of dij, which represents EF direction. This is a repulsing force that separates the mother and the daughter. Mechanic equations for centrioles of a mother and a daughter are dP m dðmm v m Þ dv m dmm ¼ mm ¼ þ vm dt dt dt dt ¼ F dm þ F mf þ F mH
ð2Þ
dP d dðmd v d Þ dv d dmd ¼ md ¼ þ vd dt dt dt dt ¼ F md þ F df þ F dH
ð3Þ
Where Pm and Pd are momentum for the mother and the daughter, respectively, mm and md, vm and vd, Fmd and Fdm, Fmf, and Fdf are masses, velocities, EF forces, resistances (including dynamic friction or viscosity) for the mother and the daughter respectively. FmH and FdH are corresponding elastic force of interacting fibers and pericentriolar material on the centrioles of the mother and the daughter, respectively, and that could be pushing or pulling and be estimated with Hooke’s Law. Using the superposition principle in physics, at a moment, a mother’s EF at any point P around a mother (Fig. 2) is estimated with 27 Z X dqi Ep ¼ d ip0 ð4Þ 2 Q i 4ped ip i where Ep is a EF vector which aligns a microtubule fragment along EF direction, dip = dip(r, h, z) is a distance between dqi and point P, and dip0 is unit vector of dip, which represents EF direction. Other terms have the same meanings as that in Eq. (1).
EF lines, weak: strong: MT: interacting fibers and pericentriolar material:
z
y θ
r x + ends
mother centriole
daughter centriole
Figure 1 At a rest state, a pair of centrioles in a centrosome. In this case, the EF around the mother’s sides is mostly parallel and much stronger than that around the ends. The draw is not in scale.
Physics models of centriole replication
575 microtubules: interacting fibers and pericentriolar material:
microtubules come into centrosome
force: EF lines , weak: strong
microtubule fragments migration
FEF
P
Fp
FEF
Fp new born baby centrioles
mother centriole
daughter centriole
Figure 2 At active state, a baby centriole (left) is separated and another baby centriole (right) is being separated from their mother centrioles respectively. The EF force, at P point, aligns a microtubule fragment that will be probably used by a new born baby centriole. FEF: EF pushing force to separate the centrioles. Fp: pulling force of interacting fibers and pericentriolar materials. The draw is not in scale.
Hypothesis of acceleration of epithelium regeneration with externally applied RPECS Fig. 3 elucidates our model of an epithelium cell exposed to externally applied EF. The EF induces extra polarized voltages across the membranes and extra inner EFs in centrosome, nucleus, and cell. Centrioles and centrosomes have two nonconducting membranes to protect their replications from external disturbs. Based on our previous pre-clinic experimental results [4] and the published data [19], we hypothesize that during the epithelialization, the externally imposed RPECS could be mild, compared with the natural/spontaneous EMF, for the centrioles in the epithelium cells, and the RPECS could not disturb the epithelium natural/spontaneous EMF at all, therefore
+ + + + + + + + + + + + + + +
-
centrioles
- +
- +
centrosome
externally applied electric field (strong)
could not suppress, but could stimulate and help, the normal centriole replication; and finally, the epithelium cellular proliferation might be directly as well as indirectly accelerated by the RPECS. RPECS could intensify neuron and muscle (somatic reflex) activities, therefore could increase blood circulation and supply more nutrients and chemical materials for needs of the epithelialization, because skin sensors receive strongest RPECS among skin cells [10].
Conclusion and discussion In this study, we have developed a physics model to explain the mechanism of centrioles replication. Our models suggest: when a mother and a daughter
+ + + a chromosome + + + + + - + - + + + + + + nucleus + +
electrically polarized charges: + and filds (mild):
-
an epithelium cell
Figure 3 Both centrioles and chromosomes in epithelium cells are protected by two layers of membranes from an externally applied EF. The draw is not in scale.
576 of centrioles are separating by EMF pushing (repulsing) force, a new baby is delivered from each of them or the both of new babies are born from one of them by pulling forces of interacting fibers and pericentriolar materials. The both babies inherit the information of cylinder pattern with 3 · 9 microtubules from their mother(s). The babies grow using microtubule fragments that come through the centrosome pores. A daughter centriole is always born and grows along stronger EMF. The EMF mostly determines centrioles positions and plays key role in centriole replication. EMF is approximately a quasi static EF. Before the mother and the daughter separate, the pushing (repulsing) interacting EF force must overcome the mother’s or daughter’s maximum resistance including static friction. Then they begin to depart from each other. From Eqs. (2) and (3), we can obtain Eq. (5). The departure equation is approximately, dv r v d dmd v m dmm þ dt md dt mm dt 1 1 ðF md þ F df þ F dH Þ ðF dm þ F mf þ F mH Þ ¼ md mm ð5Þ where vr is a relative departure velocity. Other terms have the same meanings as that in Eqs. (2) and (3). Terms of dmm/dt and dmd/dt play roles when babies are delivered. They are or are not 0 when babies are not or are delivered. In Eq. (5), when the EF force is just equal to the sum of dynamic friction (or viscosity) and the elasticity in absolute values and no baby centriole is delivered, we obtain dvr/dt = 0. Therefore, vr has to be a constant (>0) mathematically, and we should observe a constant relative velocity of the separation. Otherwise, the relative velocity must be a variant. Generally, no naturally and normally doubled centrioles can occupy the same centrosome compartment envelope (centrioles exclusion), no naturally and normally doubled centrosomes can occupy the same eucaryotic compartment envelope (centrosomes exclusion) [7,19]. The spontaneous and strong pushing (repulsing) EF forces mostly drives pairs of centrioles to depart from each other and separates centrosomes into new cells. It obeys energy conservation that energies are transformed from biochemical energy to physical energy (electrical repulsion or attraction) in the autonomic process. Charges of other proteins are included in the MT or centrioles. Other natural forces in cells are neglected because they are either too weak or randomly distributed and balanced from each other.
Cheng and Zou Oscillations during centrioles separation [19] could be caused by periodic arrays of cellular skeletons blocks or by interactions between pushing (repulsing EF) force and pulling force of interacting fibers and pericentriolar materials. The principle of our models could be applied to abnormal centrioles replications as well as to a centrosome with multiple (>2) centrioles. The assumed EF lines directions (the arrows point to) are reversed if EF directions are exprimentally found to be against our assumption. This modification will not affect our electric analysis in principle. Our models or hypothsesis of centriole replication could be tested in future by using dynamic images and measuring the high accurate EF. The limitation of the model is to obtain the distributions of electric charges and permittivities in real world environments today. In summary, our models fit the published experimental data and explain how centrioles are positioned, separated, delivered and replicated, in a view of physics. A natural and normal EMF inside or around centrosomes can be alternated environmentally. A strong externally imposed EMF could induce aberrant centrioles and centrosomes that cause life disease. However, we enhanced skin wound healing with RPECS in our previous pre-clinical experiments [4]. The experimental results strongly suggested that our applied EF could not disturb the normal natural and normal EMF inside or around centrosomes, could not induce aberrant centrioles and centrosomes, and could not suppress, but could stimulate and help centriole and centrosomes replication during the epithelialization.
Acknowledgements We thank Dr. Cheng, Yaoting, Mrs. Li, Qiongqiu, Mr. Zou Zijia and Mrs. Zheng, Chaoying for financial support and Miss Vivien Cheng for helpful suggestions and comments, for this publication.
References [1] Weiss DS, Kirsner R, Eaglstein WH. Electrical stimulation and wound healing. Arch Dermatol 1990;126:222–5. [2] Barnes TC. Healing rate of human skin determined by measurement of the electrical potential of experimental abrasions. Am J Surgery 1945;69:82–8. [3] Burr HS, Harvey SC, Taffel M. Bioelectric correlates of wound healing. Yale J Biol Med 1938;11:104–7. [4] Cheng K, Cazzaniga AL, Mertz PM. Skin wound healing by rectangular pulse electrical stimulation (RPES) + vigilon dressing. In: Proceedings of 13th annual international
Physics models of centriole replication
[5]
[6]
[7] [8]
[9]
[10]
[11]
conference of the IEEE in medicine and biology 1991;13:1035–6. Cheng K, Tarjan PP, Mertz PM. Conductivities of pig dermis and subcutaneous fat measured with rectangular pulse electrical current. Bioelectromagnetics 1996;17(6): 458–66. Cheng K, Tarjan PP, Thio YC, Mertz PM. In vivo 3-D distributions of electric fields in pig skin with rectangular pulse electrical current stimulation (RPECS). Bioelectromagnetics 1996;17(4):253–62. Cheng K, Zou C. 3-D physical models of mitosis (with asters) and cytokinesis. Biomed Sci Instrum 2004;39:413–8. Cheng K, Zou C. Theoretical models of cytokinesis and mitosis (without asters). Am J Plant Physiol 2006;1(1): 21–7. Cheng K, Zou C. Electromagnetic field effect on separation of nucleotide sequences and unwinding of a double helix during dna replication. Med Hypotheses 2006;66(1): 148–53. Cheng K, Mertz PM, Tarjan PP. Theoretical study of rectangular pulse electrical stimulation (RPES) onskin cells (in vivo) under conforming electrodes. Biomed Sci Instrum 1993;29:349–54. Mcgrill M, Highfield DP, Monahan TM, Brinkley BR. Effects of nuclei acid specific dyes on centrioles of mammalian cells. J Ultrastructure Res 1976;57:43–52.
577 [12] Sluder G, Miller FJ, Rieder CL. The reproduction of centrosomes: nuclear versus cytoplasmic controls. JCB 196;103:1873–81. [13] Paitrand M, Mopudjou M, Delacroix H, Brornens M. Centrosome organization and centriole architecture: their sensitivity to divalent cations. J Struct Biol 1992;108:107–28. [14] Hyman AA. Centrosomes: SiC transit gloria centri. Curr Biol 2000;10(7):R276–8. [15] Burakov A, Nadezhdina E, Slepchenko B, Rodionov V. Centrosome positioning in interphase cells. JCB 2003;162:963–9. [16] O’Toole ET, McDonald KL, Ma ¨ntler J, McIntosh JR, Hyman AA, Mu ¨ller-Reichert T. Morphologically distinct microtubule ends in the mitoyic centrosome of Caenorhabditis elegans. JCB 2003;163:451–6. [17] Bahe S, Stierhof YD, Wilkinson CJ, Leiss F, Nigg EA. Rootletin forms centriole-associated filaments and functions in centrosome cohesion. JCB 2005;171(1):27–33. [18] Meggs WJ. Electric fields determine the spatial organization of microtubules and actin filaments. Med Hypotheses 1988;26:165–70. [19] Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P. Molecular biology of the cell. 4th ed. New York (NY): Garland Science; 2002. [20] Howard J, Hyman AA. Dynamics and mechanics of the microtubule plus end. Nature 2003;422(6933):741–5.
http://intl.elsevierhealth.com/journals/mehy
Physics models of centriole replication Kang Cheng *, Changhua Zou Science Research, 205 Hana Road, Edison, NJ 08817, USA Received 26 February 2006; accepted 28 February 2006
Summary Our previous pre-clinic experimental results have showed that the epithelialization can be enhanced by the externally applied rectangular pulsed electrical current stimulation (RPECS). The results are clinically significant for patients, especially for those difficult patients whose skin wounds need long periods to heal. However, the results also raise questions: How does the RPECS accelerate the epithelium cell proliferation? To answer these questions, we have previously developed several models for animal cells, in a view of physics, to explain mechanisms of mitosis and cytokinesis at a cellular level, and separation of nucleotide sequences and the unwinding of a double helix during DNA replication at a bio-molecular level. In this paper, we further model the mechanism of centriole replication during a natural and normal mitosis and cytokinesis to explore the mechanism of epithelialization enhanced with the externally applied RPECS at a biomolecular level. Our models suggest: (1) Centriole replication is an information flowing. The direction of the information flowing is from centrioles to centrioles based on a cylindrical template of 9 · 3 protein microtubules (MTs) pattern. (2) A spontaneous and strong electromagnetic field (EMF) force is a pushing force that separates a mother and a daughter centrioles in centrosomes or in cells, while a pulling force of interacting fibers and pericentriolar materials delivers new babies. The newly born babies inherit the pattern information from their mother(s) and grow using microtubule fragments that come through the centrosome pores. A daughter centriole is always born and grows along stronger EMF. The EMF mostly determines centrioles positions and plays key role in centriole replication. We also hypothesize that the normal centriole replication could not been disturbed in centrosome in the epithelium cells by our RPECS, because the centrioles have two non-conducting envelope (cell and centrosome membranes), that protect the normal duplication. The induced electric field by externally applied RPECS could be mild compared with the spontaneous and natural electric field of the centrioles. Therefore, the centriole replication during the epithelium cellular proliferation may be directly, as well as indirectly (e.g., somatic reflex) accelerated by the RPECS. c 2006 Elsevier Ltd. All rights reserved.
Introduction The idea to use externally imposed electrical stimulation to enhance skin wound healing probably came from the knowledge that there is an injury current around a fresh wound, which could play a * Corresponding author. Tel.: +732 248 0790. E-mail address: [email protected] (K. Cheng).
role in skin wound healing [1]. As early as 1860s, the founder of the science of bioelectricity, Du Bois–Reymond, discovered that an injured bleeding finger is electrically positive compared with an uninjured finger [2]. To our knowledge, the first modern application of electric current stimulation to a pre-clinic skin wound healing was reported in 1962 [3]. From that time on, electrical current stimulation has been frequently used in both
0306-9877/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mehy.2006.02.041
Physics models of centriole replication pre-clinic and clinic investigations of cutaneous regeneration. In our previous pre-clinic investigation of epithelialization enhanced with the externally applied rectangular pulsed electrical current stimulation (RPECS) and conforming electrodes, we observed, at a tissue level, that wounds treated with RPECS were healed about 15% faster than the control wounds [4]. The pre-clinical experimental results impelled us to measure in vitro conductivity [5] and in vivo 3D distributions of electric fields [6] on/in pig skin with RPECS. Our measured data indicate the externally applied electric field (EF) is about three times as strong on the skin surface under the electrode edges, where the bacteria could mostly invade the skin wound, as that in the skin. The results of our observation suggest that not only the epithelialization but also the infection/antiinfection could be related to the cell proliferation and the externally applied EF. However, the results also raise questions: How does the RPECS accelerate the epithelium cell proliferation? What are the relationships between the cellular proliferation and the spontaneous electromagnetic field (EMF)? What could be the driving force for the proliferation, in wild types of cells? To answer these questions, we have developed several models, in a view of physics, to explain mechanisms of mitosis and cytokinesis at a cellular level [7,8], and separation of nucleotide sequences and the unwinding of a double helix during DNA replication at a bio-molecular level [9]. In this investigation, with assistance of a model of RPECS for skin wound healing [10] and a model for mitosis and cytokinesis [7], we continue our exploration on centriole replication in wild types of animal cells in a perspective of physics. We believe centriole replication is one key procedure during the epithelium cell proliferation. In the research field of centriole replication, centrioles structures with cells from fibroblasts and stimulated lymphocyte were studied in 1970s [11] and the reproduction of centrosomes was investigated in 1980s [12]. Centrosome organization and centriole architecture and their sensitivity to divalent cations were investigated in early of 1990s [13]. For most animal cells, centrosomes are thought to ensure spindle bipolarity and thus correct chromosome segregation during mitosis [14]. Recently, centrioles and centrosomes positioned and duplicated in eucaryotes were observed and models were provided to depict the mechanism of centrosome positioning from physiological or biological perspective [15–17]. However, those models have likely focused on centrosome positioning, rather than a comprehen-
573 sive explanation of overall centrioles replication. We have not found any model in a view of physics to explain the mechanisms of this process, to answer the question: What is the kind or the source of the separating (pushing) force? And how does the force depart the centrioles and duplicate centrioles? In this paper, based on our previous models and published biological data, we are the first to develop an original model to approach the mechanism of centriole replication using the concept of inner spontaneous EF [7,18] in a view of physics. Then considering an externally applied EF, such as RPECS, to influence the natural and normal replication, we propose a hypothesis for the mechanisms of the overall centriole replication to explain the epithelium regeneration acceleration by RPECS based our previous pre-clinic experimental results [4].
Model development of centriole replication Experimental data show a centrosome is usually composed of two centrioles at proximately right angle. Each centriole is a cylindrical organelle organized with nine groups of three microtubules (MTs) [11,13]. MTs nucleating at the mother centriole were observed [19] and MTs are found to have functions to move cellular components [20]. In addition, experimental investigation have showed: at a activated state, at a certain point in G1 of a cell life cycle, the mother and daughter centrioles begin to departure from each other. During S phase, baby centrioles begin to grow near their mothers and MT always nucleate at a mother centriole [19]. We analyze those experimental data and provide our explanation: at a rest state, a mother centriole has much more positive charge than that of a daughter centriole has, a mother is almost saturated with positive charges, and a daughter is not. But, the EF interaction (pushing force), between a pair of centrioles is too weak to overcome the maximum resistance, including static friction. At the activated state, the daughter is saturated with positive charges too. When a pair of mother and daughter centrioles have enough positive charges, the interactive repulsing (pushing) EF force is strong enough to overcome the maximum resistance including the both static and dynamic friction (or viscosity), and to separate the pair of mother and daughter centrioles. We believe that biochemical or biophysical events provoke the spontaneous and strong interactive EMF, which is approximately a quasi-static EF [7,8].
574
Cheng and Zou
In our previous study [7], first, we consider the protoplasm as electrolyte with an uneven electric charge distribution as well as a runtime environment driven by biochemical or biophysical events. Second, we assume an aster MT complex (or a self-assembled MT complex) and a single MT has a positive net charge (including some bound ions, e.g., Ca2+, or other materials), a single MT is electrically, positively polarized at positive growing ends and negatively polarized at negative growing ends. The stronger the EF, the more microtubules grow. In this study, we still use these two assumptions as basic condition for modeling mechanism of centriole replication. Figs. 1 and 2 show our model of centriole replication: EF and a couple of centrioles at a rest state and at an active state, respectively, in systems of cylindrical coordinates (r, h, z) and laboratory reference. To develop applicable models, we consider a simple case that one mother and one daughter are separating along a line in an r–z plane because this is approximately a real world situation and it is easier to obtain analytical solutions mathematically. We assume this line is on z-axis for our modeling convenience. Using the superposition principle in physics, based on extended Coulomb’s law, a total EF force interacted between a mother and a daughter centrioles is approximately estimated with Z 27 X 27 Z X dqj dqi F md ¼ F dm ¼ d ij0 2 Qj Qi 4ped ij j i e ¼ permittivity
ð1Þ
where Qi and dqi, Qj and dqj, respectively, represent equivalent charges and a unit charge on a
- ends
daughter MT, on a mother’s MT. dij = dij(r, h, z) is a distance between dqi and dqj. dij0 is unit vector of dij, which represents EF direction. This is a repulsing force that separates the mother and the daughter. Mechanic equations for centrioles of a mother and a daughter are dP m dðmm v m Þ dv m dmm ¼ mm ¼ þ vm dt dt dt dt ¼ F dm þ F mf þ F mH
ð2Þ
dP d dðmd v d Þ dv d dmd ¼ md ¼ þ vd dt dt dt dt ¼ F md þ F df þ F dH
ð3Þ
Where Pm and Pd are momentum for the mother and the daughter, respectively, mm and md, vm and vd, Fmd and Fdm, Fmf, and Fdf are masses, velocities, EF forces, resistances (including dynamic friction or viscosity) for the mother and the daughter respectively. FmH and FdH are corresponding elastic force of interacting fibers and pericentriolar material on the centrioles of the mother and the daughter, respectively, and that could be pushing or pulling and be estimated with Hooke’s Law. Using the superposition principle in physics, at a moment, a mother’s EF at any point P around a mother (Fig. 2) is estimated with 27 Z X dqi Ep ¼ d ip0 ð4Þ 2 Q i 4ped ip i where Ep is a EF vector which aligns a microtubule fragment along EF direction, dip = dip(r, h, z) is a distance between dqi and point P, and dip0 is unit vector of dip, which represents EF direction. Other terms have the same meanings as that in Eq. (1).
EF lines, weak: strong: MT: interacting fibers and pericentriolar material:
z
y θ
r x + ends
mother centriole
daughter centriole
Figure 1 At a rest state, a pair of centrioles in a centrosome. In this case, the EF around the mother’s sides is mostly parallel and much stronger than that around the ends. The draw is not in scale.
Physics models of centriole replication
575 microtubules: interacting fibers and pericentriolar material:
microtubules come into centrosome
force: EF lines , weak: strong
microtubule fragments migration
FEF
P
Fp
FEF
Fp new born baby centrioles
mother centriole
daughter centriole
Figure 2 At active state, a baby centriole (left) is separated and another baby centriole (right) is being separated from their mother centrioles respectively. The EF force, at P point, aligns a microtubule fragment that will be probably used by a new born baby centriole. FEF: EF pushing force to separate the centrioles. Fp: pulling force of interacting fibers and pericentriolar materials. The draw is not in scale.
Hypothesis of acceleration of epithelium regeneration with externally applied RPECS Fig. 3 elucidates our model of an epithelium cell exposed to externally applied EF. The EF induces extra polarized voltages across the membranes and extra inner EFs in centrosome, nucleus, and cell. Centrioles and centrosomes have two nonconducting membranes to protect their replications from external disturbs. Based on our previous pre-clinic experimental results [4] and the published data [19], we hypothesize that during the epithelialization, the externally imposed RPECS could be mild, compared with the natural/spontaneous EMF, for the centrioles in the epithelium cells, and the RPECS could not disturb the epithelium natural/spontaneous EMF at all, therefore
+ + + + + + + + + + + + + + +
-
centrioles
- +
- +
centrosome
externally applied electric field (strong)
could not suppress, but could stimulate and help, the normal centriole replication; and finally, the epithelium cellular proliferation might be directly as well as indirectly accelerated by the RPECS. RPECS could intensify neuron and muscle (somatic reflex) activities, therefore could increase blood circulation and supply more nutrients and chemical materials for needs of the epithelialization, because skin sensors receive strongest RPECS among skin cells [10].
Conclusion and discussion In this study, we have developed a physics model to explain the mechanism of centrioles replication. Our models suggest: when a mother and a daughter
+ + + a chromosome + + + + + - + - + + + + + + nucleus + +
electrically polarized charges: + and filds (mild):
-
an epithelium cell
Figure 3 Both centrioles and chromosomes in epithelium cells are protected by two layers of membranes from an externally applied EF. The draw is not in scale.
576 of centrioles are separating by EMF pushing (repulsing) force, a new baby is delivered from each of them or the both of new babies are born from one of them by pulling forces of interacting fibers and pericentriolar materials. The both babies inherit the information of cylinder pattern with 3 · 9 microtubules from their mother(s). The babies grow using microtubule fragments that come through the centrosome pores. A daughter centriole is always born and grows along stronger EMF. The EMF mostly determines centrioles positions and plays key role in centriole replication. EMF is approximately a quasi static EF. Before the mother and the daughter separate, the pushing (repulsing) interacting EF force must overcome the mother’s or daughter’s maximum resistance including static friction. Then they begin to depart from each other. From Eqs. (2) and (3), we can obtain Eq. (5). The departure equation is approximately, dv r v d dmd v m dmm þ dt md dt mm dt 1 1 ðF md þ F df þ F dH Þ ðF dm þ F mf þ F mH Þ ¼ md mm ð5Þ where vr is a relative departure velocity. Other terms have the same meanings as that in Eqs. (2) and (3). Terms of dmm/dt and dmd/dt play roles when babies are delivered. They are or are not 0 when babies are not or are delivered. In Eq. (5), when the EF force is just equal to the sum of dynamic friction (or viscosity) and the elasticity in absolute values and no baby centriole is delivered, we obtain dvr/dt = 0. Therefore, vr has to be a constant (>0) mathematically, and we should observe a constant relative velocity of the separation. Otherwise, the relative velocity must be a variant. Generally, no naturally and normally doubled centrioles can occupy the same centrosome compartment envelope (centrioles exclusion), no naturally and normally doubled centrosomes can occupy the same eucaryotic compartment envelope (centrosomes exclusion) [7,19]. The spontaneous and strong pushing (repulsing) EF forces mostly drives pairs of centrioles to depart from each other and separates centrosomes into new cells. It obeys energy conservation that energies are transformed from biochemical energy to physical energy (electrical repulsion or attraction) in the autonomic process. Charges of other proteins are included in the MT or centrioles. Other natural forces in cells are neglected because they are either too weak or randomly distributed and balanced from each other.
Cheng and Zou Oscillations during centrioles separation [19] could be caused by periodic arrays of cellular skeletons blocks or by interactions between pushing (repulsing EF) force and pulling force of interacting fibers and pericentriolar materials. The principle of our models could be applied to abnormal centrioles replications as well as to a centrosome with multiple (>2) centrioles. The assumed EF lines directions (the arrows point to) are reversed if EF directions are exprimentally found to be against our assumption. This modification will not affect our electric analysis in principle. Our models or hypothsesis of centriole replication could be tested in future by using dynamic images and measuring the high accurate EF. The limitation of the model is to obtain the distributions of electric charges and permittivities in real world environments today. In summary, our models fit the published experimental data and explain how centrioles are positioned, separated, delivered and replicated, in a view of physics. A natural and normal EMF inside or around centrosomes can be alternated environmentally. A strong externally imposed EMF could induce aberrant centrioles and centrosomes that cause life disease. However, we enhanced skin wound healing with RPECS in our previous pre-clinical experiments [4]. The experimental results strongly suggested that our applied EF could not disturb the normal natural and normal EMF inside or around centrosomes, could not induce aberrant centrioles and centrosomes, and could not suppress, but could stimulate and help centriole and centrosomes replication during the epithelialization.
Acknowledgements We thank Dr. Cheng, Yaoting, Mrs. Li, Qiongqiu, Mr. Zou Zijia and Mrs. Zheng, Chaoying for financial support and Miss Vivien Cheng for helpful suggestions and comments, for this publication.
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