- Email: [email protected]
Deterministic dynamics control oscillations of bone marrow cell proliferation
Experimental Hematology 32 (2004) 822–827
Deterministic dynamics control oscillations of bone marrow cell proliferation Claire Wolfroma,*, Philippe Bourinb,c,*, Nguyen-Phong Chaud, Franc¸oise Cadepondd,e, and Jean Deschatrettea a INSERM U.347, le Kremlin-Biceˆtre, France; bE´tablissement Franc¸ais du Sang Pyre´ne´es-Me´diterrane´e, Service de the´rapie cellulaire, Toulouse, France; cCTSA, Laboratoire d’Immunologie Cellulaire, Clamart, France; dEquipe Biomath-Biostat, Universite´ Paris VII, Paris, France; eINSERM U.488, le Kremlin-Biceˆtre, France
(Received 2 April 2004; revised 4 June 2004; accepted 16 June 2004)
Objective. The production of blood cells in vivo, both normal and tumoral, displays oscillatory dynamics. Many cells in long-term cultures also show large amplitude oscillations of proliferative rate. Therefore we examined the proliferation dynamics of mouse bone marrow cells (MBM) and their clonogenic progenitor production (BMP), in order to characterize these dynamics. Methods. Five Dexter-type cultures of MBM cells and their clonogenic BMP production were examined for up to seven-months periods of time. The recorded time series exhibited a complex pattern of oscillations with variable amplitudes. We previously reported a method that allowed analysis of such nonlinear dynamics of hepatoma cell proliferation. We applied this method, based on the two-dimensional recurrent representation of data, to analyze the fluctuations of bone marrow cells proliferation. Results. The proliferation rate of mouse bone marrow cells shows large amplitude oscillations every 2 to 3 weeks. Mathematical analysis revealed a deterministic mechanism that controls all proliferation local maxima of MBM cells. Dynamics for progenitor production resembled that of parental cells. This reflects a predominant negative feedback on bone marrow cell proliferation. Conclusion. These dynamics were opposite of that previously described for hepatoma cells where the dominant control is applied to the local minima (troughs of proliferation). Therefore, the complex system of cell proliferation is controlled by a bipolar mechanism, with a predominant dampening command depending on the cell type. We propose that the dominant dampening control of local maxima in bone marrow cells protects the stock of stem cells. 쑖 2004 International Society for Experimental Hematology. Published by Elsevier Inc.
Oscillatory patterns are consistent with the reciprocal regulatory loops controlling cell growth, implying spontaneous self-maintained nonlinear behavior [1,2]. Such patterns are necessary for permanent adaptation of physiological functions and may be determinant for cell fate bifurcations such as commitment to differentiation. Episodic growth has been described for liver growth [3] and for tumor growth [4–6].
Offprint requests to: Claire M. Wolfrom, M.D., Inserm U.347, 80 rue du Gl Leclerc, 94276 le Kremlin-Biceˆtre, France; E-mail: wolfrom@kb. inserm.fr *Dr. Wolfrom and Dr. Bourin contributed equally to the research described in this article.
0301-472X/04 $–see front matter. Copyright doi: 1 0 .1 01 6 /j.ex p h e m.2 0 0 4. 06 .0 0 5
The proliferation rate of many mammalian cell populations of different cell types in long-term culture also displays such repeated oscillations [7–12]. The proliferation of blood cells, both normal and tumoral, oscillates every 3 to 5 weeks in vivo [13–17]. Circadian and seasonal [18] as well as weekly [19] variations in hematopoiesis are also known, possibly as harmonics of a main wavelength. In vitro, hemopoietic progenitors were found responsive to both positive and negative regulatory control mechanisms operating within cells of the feeder layer, with neoplastic progenitor cell types becoming insensitive to the negative arm of this control [20–22]. Elucidating the mechanisms which underlie these fluctuations has practical implications for understanding hematological diseases and for marrow stem cell engraftment.
쑖 2004 International Society for Experimental Hematology. Published by Elsevier Inc.
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
However, these complex mechanisms involve very different parameters, including population compartments, feedback loops, time delays, cell synchronization, and external disturbances [14]. Many mathematical models of these dynamics have been proposed, from spectral analysis of periodic or almost periodic oscillations [23] to models of stochastic [24] or chaotic processes [25]. We reasoned that to detect a regulation in such fluctuations, which may be periodic or almost periodic, nonlinear analysis could be needed. Mathematical models describing the long-term evolution of populations show a spectrum of complex dynamic behaviors, which may vary from periodic oscillations to almost periodic, possibly chaotic, fluctuations, through to disordered dynamics [26,27]. A topological analysis using geometric representation of proliferation data allows an overall analysis of the deterministic nature of such fluctuations, different from the usual spectral analysis, which mainly defines the presence of a periodicity. Using this approach, we previously demonstrated that in the proliferation of cultured rat hepatoma Fao cells, which showed oscillations over 3 to 5 weeks, the local minima are determined by a high-level attractor [17]. The present study used the same graphical method to analyze the aperiodic oscillations of cell proliferative rate in long-term cultures of mouse bone marrow cells (MBM) and in the clonogenic progenitor production (BMP). Methods Culture of mouse bone marrow cells and progenitors Five Dexter-type long-term cultures were conducted with female B6D2F1 mouse bone marrow over 29 weeks. Mice were killed by cervical dislocation during the first semester of the year. Bone marrow cells were seeded at a concentration of 2 × 107 cells per 25-cm2 flask in 10 mL of culture medium. The medium consisted of α-MEM medium containing 12.5% fetal calf serum, 12.5% donor horse serum, hydrocortisone sodium hemisuccinate (10⫺6 M), inositol (40 mg/L), 2-mercaptoethanol (10⫺4 M) and folic acid (10 mg/L). Each week, half of the medium was replaced by fresh medium. The total number of cells (MBM) in the supernatant was counted. MBM increase was the difference between N0 cells after the medium change and N counted cells at the end of the week. The rate of cell proliferation was expressed as the number of Population Doublings (PD ⫽ log2 N/N0), during the week (PD/ week). The progenitor (BMP) frequency (expressed as number of clones/104 cells) was evaluated by a clonogenic methylcellulose assay. At each time point, 104 cells from the supernatant were cultured in methylcellulose containing medium with mSCF (2.5 ng/ mL), mIL-3 (5 ng/mL), erythropoietin (5 U/mL), mGM-CSF (5 ng/mL), and mFlt3-L (25 ng/mL) in two 35-mm Petri dishes. Clones were counted at day 7. Data are means of duplicate determinations at each time point. The cell dynamics were similar in the five cultures, but only in the longest culture was the uninterrupted number of observations (n ⫽ 29; 10 local minima, 11 local maxima) sufficient for complete Monte Carlo statistical analysis, so we chose the data of that series to illustrate the present analysis.
823
Data analysis Recurrent representation of the proliferation data. (See Fig. 1.) We represented the long-term evolution of cell growth by displaying in rectangular coordinates xi⫺1 vs xi, where xi is the proliferation data at the i-th passage. Let Mi be the point of coordinates (xi, xi⫹1). Consecutive points are joined. In this representation, if xi is a local minima (a trough of proliferation rate), i.e., if xi ⬍ xi⫺1 and xi ⬍ xi⫹1, then the segment MiMi⫹1 goes southeast to northwest. Similarly, if xi is a local maxima (a peak of proliferation rate), i.e., if xi ⬎ xi⫺1 and xi ⬎ xi⫹1, then the segment MiMi⫹1 goes northwest to southeast. Analytical expression of the control mechanisms. In a previous work [17], analyzing the proliferation of a series of hepatoma cell data, we discovered the following control mechanisms: all the bisecting lines of the segments MiMi⫹1, corresponding to the different minima xi, converge to a fixed point situated on the diagonal of the coordinate axes. However, the bisecting lines corresponding to the local maxima did not converge. Let (a, a) be the coordinates of the fixed point A. We proved that the control mechanisms of the local minima can be expressed in the following analytical expression: if xi is a local minima, then (xi⫺1 ⫹ xi⫹1)/2 ⫽ a ⫽ constant (H). For the convenience of the reader, the proof is recalled in the Appendix below. The purpose of this study is to test if the local minima and/or the local maxima of bone marrow cell proliferation obey the same control mechanisms. Statistical test of the control mechanisms. We used a Monte Carlo procedure [28] to show that relationship (H) results from a deterministic control. Let yi ⫽ (xi⫺1 ⫹ xi⫹1)/2. We calculate the quantities y1, y2,…, yp corresponding to the different minima of the series and test the hypothesis that the set yi is a constant, i.e., is of variance (or standard deviation, std) zero. Let s0 be the std of the set (yi). We performed a random perturbation of the order of the series xi, then identified the local minima and calculated the quantities yk of the perturbed (uncontrolled) series. Let s1 be the std of these new yk. By repeating the same procedure, we calculated the new std’s s2, s3,…, of several new series, obtained each time from the initial series by randomly shuffling the order. Our hypothesis is highly probable if only a very small number of the si with i ⫽ 1, 2,…, are smaller than the observed value s0. The same calculus was performed for the local maxima. The same method could be applied for all series of data. However, we gave explicit calculations only for the 29-week series. The other series are either too short, or lacking one determination, and thus have only 8 or fewer local minima and 8 or fewer local maxima. We do not wish to compare the standard deviation between sets of so few numbers.
Results Proliferation of cultured mouse bone marrow cells displays almost periodic oscillations occurring every 2 to 3 weekly passages. In the longest series, these oscillations of proliferation rate persisted for 29 weeks, with very variable amplitudes. The series used for illustrating the data varied from ⫺2.47 to 3.17 PD/week. The negative data (expressed in
824
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
PD) recorded for some passages correspond to a net loss of cells during the passage, and are relevant to cell death and lowered proliferation. Peaks of cell numbers of MBM and of the frequency of BMP all coincided, and both parameters correlated well (r ⫽ 0.684, p ⬍ 0.01). The production of progenitors fell abruptly at passage 16 when the amplitude of the oscillations of MBM proliferation decreased (Fig. 2). Figure 2 shows the recurrent map for MBM cells, with bisecting lines of segments MiMi⫹1 corresponding to local maxima (northwest to southeast segments, Fig. 3A) and to local minima (southeast to northwest segments, Fig. 3B). In Figure 3A, the bisecting lines converge to a point on the diagonal with coordinates about (0.05, 0.05). This attractor is on the diagonal of the map (where xi ⫽ xi⫹1) and thus is an equilibrium fixed point. In calculating the quantities yk for the observed data (see above, statistical test), we obtain a mean value, a ⫽ 0.08 and a std s0 ⫽ 0.366. Using the data plus 4999 random perturbations of the original series, we obtained in total 5000 stds si, out of which only 40 are lower than s0. Therefore, our convergence hypothesis can be considered as plausible, with a probability of 40/5000 ⫽ 0.008. The same calculations on local minima show that the convergence hypothesis is not significant. Actually, we obtained for the local minima, s0 ⫽ 0.71. Of the 5000 si’s values (one is the observed s0 and 4999 are from random perturbations of the observed series), not fewer than 1653 are lower than or equal to s0. The probability that (H) may come from a random sequence is large, p ⫽ 1653/5000 ⫽ 0.33. In four other, shorter bone marrow cultures, the same coordination of growth peaks was observed (data not shown), but the smaller numbers of observations, including only 8 or fewer local minima, would not allow complete Monte Carlo statistical analysis. Although the recurrent plot of production of BMP clones also shows control of local extrema (coordinates 13.5/13.5),
Figure 1. Construction of a map using the proliferation data as phase-space coordinates. This technique analyses the evolution of the number of bone marrow cells (MBM) during successive weeks in culture. Top: proliferation curve; Bottom: proliferation map. For this representation of proliferation data on the bidimensional map, data are tranformed thus: each segment of the curve, from one passage to the following, is defined as a point on the map with coordinate x being the first data of the segment (⫽ xi), and coordinate y being the following data, at the end of the segment (⫽ xi⫹1). All points are joined together, showing the temporal evolution of the system in a succession of orbits. The different situations on the proliferation curve—ascending, peak, descending, trough—can be further defined on this map as vectors: proliferation peaks, or local maxima, are represented by vectors going southeast in the plane, as illustrated here by the vector in bold representing the peak of MBM cell proliferation between passages 10 and 12. The position and size of each vector is further studied by drawing its bisecting line.
Figure 2. Oscillatory pattern of mouse bone marrow cell proliferation. Ordinate: proliferative rate of mouse bone marrow cells (MBM; white dots) expressed in PD/week; and frequency of corresponding hematopoietic progenitors (BMP; black dots). Abscissa: number of weeks (passages) in culture.
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
825
Figure 4. Phase-space representation of the production of BMP clones. Abscissa: x ⫽ production of clones at week (n); ordinate: y ⫽ production of clones at week (n⫹1). Note the convergence of the bisecting lines for peaks.
Figure 3. Phase-space representation of the proliferation dynamics of MBM cells and of the production of BMP clones. The time series as phase-space coordinates. Abscissa: x ⫽ proliferation data at week (n); ordinate: y ⫽ proliferation data at week (n⫹1). (A) Peaks (local maxima) of proliferation. The “low” fixed point attracts bisecting lines. (B) Troughs (local minima) of proliferation. The convergence is not focused.
the convergence did not achieve statistical significance (p ⬎ 0.05). Local minima were not organized (Fig. 4).
Discussion Persistent oscillations of proliferation rate in MBM cells were almost periodic, with a frequency of one peak per 2 to 3 weeks, similar to the oscillatory pattern of various cells in vitro [17]. Although seasonal or hormonal disturbances may be a cause of in vivo variability in MBM cell proliferation,
the duration of stable in vitro conditions frees the proliferation dynamics of these influences. The vectors representing the peaks of proliferation rate converged on a nodal point on the diagonal line of the map. The diagonal of the map is the ensemble of theoretical fixed points (xi and xi⫹1 are equal) where growth is steady. Therefore this nodal point represents a fixed proliferation level. This “attractor” corresponds to the spontaneous optimal proliferative activity to which the cells tend to return in all cases, correcting the oscillations induced by the multiple feedback loops of proliferation inducers and inhibitors. The observed fluctuations nevertheless indicate that the system never reaches this center of maximum stability. Thus, a “low” command coordinated the peaks in MBM cells proliferation rate. By contrast, although the recurrent plot suggests a control of some local minima in MBM, particularly those of the first half period of the culture, irregular vectors as of week 17 make the convergence statistically not significant. Dynamics for the peaks of progenitor production resembled that of parental MBM cells. The local minima were not coordinated in BMP. The regulation was opposite in hepatoma Fao cells in which a “high” fixed point alone controlled all slow-downs of proliferation [17] in many series. However, both culture conditions and cell functions are different for the two cell types, as tumoral Fao cells are regularly passaged in new dishes at small seeding density. The oscillations of cultured bone marrow cell growth are very similar to the well-known fluctuations in the production of blood cells, both normal and tumoral in vivo [7–11]. These fluctuations are usually described as periodic in nature with some irregularity. Our
826
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
data suggest that the strong deterministic control of proliferation can result in different patterns of oscillations, from strictly periodic, approachable by spectral analysis, to almost periodic (and, strictly speaking, aperiodic), still exactly regulated to contain bone marrow cell growth within limits. This control of proliferation generating recurrent almost-periodic series may be explained in a dynamic system involving nonlinear interrelationships among several factors [29]. The major inhibitory factors are depletion of energetic compounds, contact inhibition, and erosion of telomeric DNA. Positive signals include autocrine growth factors and oncogene products, as well as exhaustion of inhibitory controls when nutrients are renewed and when telomeres are repaired. The resulting bipolar control of growth indicates that the cell population tends to recover a constrained proliferation rate. It is more precise than a single overall control, and is well adapted to different physiological purposes of the different cell types. The predominant dampening control on peaks in MBM cells reflects a dominant negative feedback that protects the stock of precursor cells. The coordinated oscillatory control for MBM cells and BMP is consistent with the stimulating effect of stromal cell–derived factor-1 on cycling of clonogenic progenitors [30]. It also fits well models of efflux in lineages of marrow cells depending on population size [10,31,32]. Additional evidence of such a determinism between the complex MBM cell proliferation dynamics and BMP clones production is illustrated by the synchrony between the loss of the two-week periodicity in MBM cell growth pattern at passage 14 and the drastic fall in progenitor number. Although requiring further analysis at the molecular level, this observation suggests that the mechanisms involved in commitment into BMP production are associated either with a periodicity or with a threshold level in MBM cells proliferation. Opposite clinical conditions like transient myeloproliferative syndrome [33] and transient erythroblastopenia of childhood [34] may reflect a transient imbalance in proliferation commands in marrow cells. Elucidating this deterministic dynamic pattern will help understand the evolution of hematological disorders and the development of grafted marrow stem cells. Acknowledgments This work was supported in part by l’Association “Biologie du Cancer et Dynamiques complexes” (L’ABCD). A part of this work was published as an abstract at the Meeting of the American Society of Hematology, New Orleans, LA, USA, Dec. 3–7, 1999.
References 1. Mackey MC, Glass L. Oscillation and chaos in physiological control systems. Science. 1977;197:287–289. 2. Laurent M, Kellershon N, Hervagault JF. Multistabilite´, oscillations et chaos dans les syste`mes biologiques. Me´d/Sciences. 1998;14:1240– 1251. 3. Khokha MK, Landini G, Iannacone PM. Fractal geometry in rat chimeras demonstrates that a repetitive cell division program may generate liver parenchyma. Dev Biol. 1994;165:545–555.
4. Squartini F. Strain differences in growth of mammary tumors. J Natl Cancer Inst. 1961;206:8132–8138. 5. Sanders BG, Teplitz RL, Wilkie M. The mitogenic activity of a c57bl/ 10 mouse transplantable lymphoma. Cancer Res. 1970;30:447–454. 6. Moore VJ, Rowley R, Hopkins HA, Russell-Ritenour E, Looney WB. Cyclophosphamide as an adjuvant to X rays in treatment of a radioresistant solid tumor of the rat, hepatoma H4IIE. Int J Radiation Oncology Biol Phys. 1979;5:1471–1474. 7. Ikram Z, Norton T, Jat PS. The biological clock that measures the mitotic life span of mouse embryo fibroblasts continues to function in the presence of simian virus 40 large tumor antigen. PNAS USA. 1994;91:6448–6452. 8. Wolfrom C, Raynaud N, Maigne J, et al. Periodic fluctuations in proliferation capacity of SV40 transformed human skin fibroblasts with prolonged lifespan. Cell Biol Toxicol. 1994;10:247–254. 9. Scha¨chter F, Boucher N, Lesueur-Ginot L, et al. Cellular senescence and survival of T lymphocytes. C R Acad Sci Paris. 1995;318:563–572. 10. Chow M, Kong M, Rubin H. Unmasking large and persistent reductions in proliferation rate of aging cells. In Vitro Cell Dev Biol. 1997;33: 809–818. 11. Maigne´ J, Deschatrette J, Sarrazin S, Hecquet B, Guerroui S, Wolfrom C. The time-pattern of rises and falls in proliferation fades with senescence of mortal lines and is perpetuated in immortal rat hepatoma Fao cell line. In Vitro Cell Dev Biol. 1998;34:163–169. 12. Wolfrom C, Chau NP, Maigne´ J, et al. Evidence for deterministic chaos in aperiodic oscillations of proliferative activity in long-term cultured Fao hepatoma cells. J Cell Sci. 2000;113:1069–1074. 13. Moorley AA. A neutrophil cycle in healthy individuals. Lancet. 1966; dec 3:1220–1222. 14. Von Schulthess GK, Gessner U. Oscillating platelet counts in healthy individuals: experimental investigation and quantitative evaluation of thrombocytopoietic feedback control. Scan J Haematol. 1986;36:473– 479. 15. Horwitz M, Benson KF, Person RE, Aprikyan AG, Dale DC. Mutations in ela2, encoding neutrophil elastase, define a 21-day biological clock in cyclic hematopoiesis. Nat Genet. 1999;23:433–436. 16. Fortin P, Mackey MC. Periodic chronic myelogenous leukemia: spectral analysis of blood cell counts and etiological implications. Br J Haematol. 1999;104:336–345. 17. Bennett M, Grunwald AJ. Hydroxyurea and periodicity in myeloproliferative disease. Eur J Haematol. 2001;66:317–323. 18. Laerum OD. Hematopoiesis occurs in rhythms. Exp Hematol. 1995;23: 1145–1147. 19. Derer L. Rhythm and proliferation with special reference to the sixday rhythm of blood leukocyte count. Neoplasma. 1960;2:117–133. 20. Eaves AC, Eaves CJ. Maintenance and proliferation control of primitive hemopoietic progenitors in long-term cultures of human marrow cells. Blood Cells. 1988;14:355–368. 21. Sutherland HJ, Lansdorp PM, Henkelman DH, Eaves AC, Eaves CJ. Functional characterization of individual human hematopoietic cells cultured at limiting dilution on supportive marrow stromal layers. Proc Natl Acad Sci U S A. 1990;87:3584–3588. 22. Holyoake TL, Jiang X, Drummond MW, Eaves AC, Eaves CJ. Elucidating critical mechanisms of deregulated stem cell turnover in the chronic phase of chronic myeloid leukemia. Leukemia. 2002;16:549–558. 23. Fortin P, Mackey MC. Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications. Br J Haematol. 1999;104:336–345. 24. Abkowitz JL, Catlin SN, Guttorp P. Evidence that hematopoiesis may be a stochastic process in vivo. Nat Med. 1996;2:190–197. 25. Preisler H, Raza A. An overview of some studies of chronic myelogenous leukemia: biological-clinical observations and viewing the disease as a chaotic system. Leuk Lymphoma. 1993;11:145–150. 26. May R. Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science. 1974;186:645–647.
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827 27. Mackey M, Glass L. Oscillation and chaos in physiological control systems. Science. 1977;197:287–289. 28. Manly BFJ. Randomisation and Monte Carlo methods in biology. Boca Raton, FL: Chapman and Hall; 1991. 29. Glass L, Beuter A, Larocque D. Time delays, oscillations and chaos in physiological control systems. Math Biosci. 1988;90:111–125. 30. Lataillade JJ, Clay D, Bourin P, et al. Stromal cell–derived factor1 regulates primitive hematopoiesis by suppressing apoptosis and by promoting G0/G1 transition in CD34⫹ cells: evidence for an autocrineparacrine mechanism. Blood. 2002;4:1117–1129. 31. Abkowitz JL, Persik MT, Shelton GH, et al. Behavior of hematopoietic stem cells in a large animal. Proc Natl Acad Sci U S A. 1995;92: 2031–2035. 32. Haurie C, Dale DC, Mackey MC. Cyclical neutropenia and other periodic hematological disorders: a review of mechanisms and mathematical models. Blood. 1998;8:2629–2640. 33. Hitzler JK, Cheung J, Li Y, Scherer SW, Zipursky A. GATA1 mutations in transient leukemia and acute megakaryoblastic leukemia of Down syndrome. Blood. 2003;101:4301–4304. 34. Gustavsson P, Klar J, Matsson H, et al. Familial transient erythroblastopenia of childhood is associated with the chromosome 19q13.2 region but not caused by mutations in coding sequences of the ribosomal protein S19(RPS19) gene. Br J Haematol. 2002;119:261–264.
827
Appendix On rectangular coordinates (see Fig. 1), consider the points Mi⫺1(xi⫺1, xi) and Mi(xi, xi⫹1), where xi is a local minimum (xi ⬍ xi⫺1 and xi ⬍ xi⫹1). Let H be the midpoint of Mi⫺1Mi, then H has coordinates xH ⫽ (xi⫺1 ⫹ xi)/2, yH ⫽ (xi ⫹ xi⫹1)/2. We assume that the line orthogonal to Mi⫺1Mi cuts the diagonal line at a fixed point A(a, a). Then the slope of the line Mi⫺1Mi is s1 ⫽ (xi⫹1 ⫺ xi)/(xi ⫺ xi⫺1) and the slope of the line HA is s2 ⫽ (a⫺yH)/(a⫺xH). As HA and Mi⫺1Mi are orthogonal lines, s1 ⫽ ⫺1/s2. This relationship, with the above values of s1 and s2, gives, after developing the terms and simplifications, 2a(xi⫹1 ⫺ xi⫺1) ⫽ xi⫹12 ⫺ xi⫺12 As xi⫹12 ⫺ xi⫺12 ⫽ (xi⫹1 ⫺ xi⫺1)(xi⫹1 ⫹ xi⫺1), one obtains finally the relationship (H) in the text: 2a ⫽ (xi⫹1 ⫹ xi⫺1) (H). Relationship (H) is much simpler than the complete geometric construction, but the geometric construction leads to the idea of central control, and it is that construction that led us to discover the control mechanism.
Deterministic dynamics control oscillations of bone marrow cell proliferation Claire Wolfroma,*, Philippe Bourinb,c,*, Nguyen-Phong Chaud, Franc¸oise Cadepondd,e, and Jean Deschatrettea a INSERM U.347, le Kremlin-Biceˆtre, France; bE´tablissement Franc¸ais du Sang Pyre´ne´es-Me´diterrane´e, Service de the´rapie cellulaire, Toulouse, France; cCTSA, Laboratoire d’Immunologie Cellulaire, Clamart, France; dEquipe Biomath-Biostat, Universite´ Paris VII, Paris, France; eINSERM U.488, le Kremlin-Biceˆtre, France
(Received 2 April 2004; revised 4 June 2004; accepted 16 June 2004)
Objective. The production of blood cells in vivo, both normal and tumoral, displays oscillatory dynamics. Many cells in long-term cultures also show large amplitude oscillations of proliferative rate. Therefore we examined the proliferation dynamics of mouse bone marrow cells (MBM) and their clonogenic progenitor production (BMP), in order to characterize these dynamics. Methods. Five Dexter-type cultures of MBM cells and their clonogenic BMP production were examined for up to seven-months periods of time. The recorded time series exhibited a complex pattern of oscillations with variable amplitudes. We previously reported a method that allowed analysis of such nonlinear dynamics of hepatoma cell proliferation. We applied this method, based on the two-dimensional recurrent representation of data, to analyze the fluctuations of bone marrow cells proliferation. Results. The proliferation rate of mouse bone marrow cells shows large amplitude oscillations every 2 to 3 weeks. Mathematical analysis revealed a deterministic mechanism that controls all proliferation local maxima of MBM cells. Dynamics for progenitor production resembled that of parental cells. This reflects a predominant negative feedback on bone marrow cell proliferation. Conclusion. These dynamics were opposite of that previously described for hepatoma cells where the dominant control is applied to the local minima (troughs of proliferation). Therefore, the complex system of cell proliferation is controlled by a bipolar mechanism, with a predominant dampening command depending on the cell type. We propose that the dominant dampening control of local maxima in bone marrow cells protects the stock of stem cells. 쑖 2004 International Society for Experimental Hematology. Published by Elsevier Inc.
Oscillatory patterns are consistent with the reciprocal regulatory loops controlling cell growth, implying spontaneous self-maintained nonlinear behavior [1,2]. Such patterns are necessary for permanent adaptation of physiological functions and may be determinant for cell fate bifurcations such as commitment to differentiation. Episodic growth has been described for liver growth [3] and for tumor growth [4–6].
Offprint requests to: Claire M. Wolfrom, M.D., Inserm U.347, 80 rue du Gl Leclerc, 94276 le Kremlin-Biceˆtre, France; E-mail: wolfrom@kb. inserm.fr *Dr. Wolfrom and Dr. Bourin contributed equally to the research described in this article.
0301-472X/04 $–see front matter. Copyright doi: 1 0 .1 01 6 /j.ex p h e m.2 0 0 4. 06 .0 0 5
The proliferation rate of many mammalian cell populations of different cell types in long-term culture also displays such repeated oscillations [7–12]. The proliferation of blood cells, both normal and tumoral, oscillates every 3 to 5 weeks in vivo [13–17]. Circadian and seasonal [18] as well as weekly [19] variations in hematopoiesis are also known, possibly as harmonics of a main wavelength. In vitro, hemopoietic progenitors were found responsive to both positive and negative regulatory control mechanisms operating within cells of the feeder layer, with neoplastic progenitor cell types becoming insensitive to the negative arm of this control [20–22]. Elucidating the mechanisms which underlie these fluctuations has practical implications for understanding hematological diseases and for marrow stem cell engraftment.
쑖 2004 International Society for Experimental Hematology. Published by Elsevier Inc.
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
However, these complex mechanisms involve very different parameters, including population compartments, feedback loops, time delays, cell synchronization, and external disturbances [14]. Many mathematical models of these dynamics have been proposed, from spectral analysis of periodic or almost periodic oscillations [23] to models of stochastic [24] or chaotic processes [25]. We reasoned that to detect a regulation in such fluctuations, which may be periodic or almost periodic, nonlinear analysis could be needed. Mathematical models describing the long-term evolution of populations show a spectrum of complex dynamic behaviors, which may vary from periodic oscillations to almost periodic, possibly chaotic, fluctuations, through to disordered dynamics [26,27]. A topological analysis using geometric representation of proliferation data allows an overall analysis of the deterministic nature of such fluctuations, different from the usual spectral analysis, which mainly defines the presence of a periodicity. Using this approach, we previously demonstrated that in the proliferation of cultured rat hepatoma Fao cells, which showed oscillations over 3 to 5 weeks, the local minima are determined by a high-level attractor [17]. The present study used the same graphical method to analyze the aperiodic oscillations of cell proliferative rate in long-term cultures of mouse bone marrow cells (MBM) and in the clonogenic progenitor production (BMP). Methods Culture of mouse bone marrow cells and progenitors Five Dexter-type long-term cultures were conducted with female B6D2F1 mouse bone marrow over 29 weeks. Mice were killed by cervical dislocation during the first semester of the year. Bone marrow cells were seeded at a concentration of 2 × 107 cells per 25-cm2 flask in 10 mL of culture medium. The medium consisted of α-MEM medium containing 12.5% fetal calf serum, 12.5% donor horse serum, hydrocortisone sodium hemisuccinate (10⫺6 M), inositol (40 mg/L), 2-mercaptoethanol (10⫺4 M) and folic acid (10 mg/L). Each week, half of the medium was replaced by fresh medium. The total number of cells (MBM) in the supernatant was counted. MBM increase was the difference between N0 cells after the medium change and N counted cells at the end of the week. The rate of cell proliferation was expressed as the number of Population Doublings (PD ⫽ log2 N/N0), during the week (PD/ week). The progenitor (BMP) frequency (expressed as number of clones/104 cells) was evaluated by a clonogenic methylcellulose assay. At each time point, 104 cells from the supernatant were cultured in methylcellulose containing medium with mSCF (2.5 ng/ mL), mIL-3 (5 ng/mL), erythropoietin (5 U/mL), mGM-CSF (5 ng/mL), and mFlt3-L (25 ng/mL) in two 35-mm Petri dishes. Clones were counted at day 7. Data are means of duplicate determinations at each time point. The cell dynamics were similar in the five cultures, but only in the longest culture was the uninterrupted number of observations (n ⫽ 29; 10 local minima, 11 local maxima) sufficient for complete Monte Carlo statistical analysis, so we chose the data of that series to illustrate the present analysis.
823
Data analysis Recurrent representation of the proliferation data. (See Fig. 1.) We represented the long-term evolution of cell growth by displaying in rectangular coordinates xi⫺1 vs xi, where xi is the proliferation data at the i-th passage. Let Mi be the point of coordinates (xi, xi⫹1). Consecutive points are joined. In this representation, if xi is a local minima (a trough of proliferation rate), i.e., if xi ⬍ xi⫺1 and xi ⬍ xi⫹1, then the segment MiMi⫹1 goes southeast to northwest. Similarly, if xi is a local maxima (a peak of proliferation rate), i.e., if xi ⬎ xi⫺1 and xi ⬎ xi⫹1, then the segment MiMi⫹1 goes northwest to southeast. Analytical expression of the control mechanisms. In a previous work [17], analyzing the proliferation of a series of hepatoma cell data, we discovered the following control mechanisms: all the bisecting lines of the segments MiMi⫹1, corresponding to the different minima xi, converge to a fixed point situated on the diagonal of the coordinate axes. However, the bisecting lines corresponding to the local maxima did not converge. Let (a, a) be the coordinates of the fixed point A. We proved that the control mechanisms of the local minima can be expressed in the following analytical expression: if xi is a local minima, then (xi⫺1 ⫹ xi⫹1)/2 ⫽ a ⫽ constant (H). For the convenience of the reader, the proof is recalled in the Appendix below. The purpose of this study is to test if the local minima and/or the local maxima of bone marrow cell proliferation obey the same control mechanisms. Statistical test of the control mechanisms. We used a Monte Carlo procedure [28] to show that relationship (H) results from a deterministic control. Let yi ⫽ (xi⫺1 ⫹ xi⫹1)/2. We calculate the quantities y1, y2,…, yp corresponding to the different minima of the series and test the hypothesis that the set yi is a constant, i.e., is of variance (or standard deviation, std) zero. Let s0 be the std of the set (yi). We performed a random perturbation of the order of the series xi, then identified the local minima and calculated the quantities yk of the perturbed (uncontrolled) series. Let s1 be the std of these new yk. By repeating the same procedure, we calculated the new std’s s2, s3,…, of several new series, obtained each time from the initial series by randomly shuffling the order. Our hypothesis is highly probable if only a very small number of the si with i ⫽ 1, 2,…, are smaller than the observed value s0. The same calculus was performed for the local maxima. The same method could be applied for all series of data. However, we gave explicit calculations only for the 29-week series. The other series are either too short, or lacking one determination, and thus have only 8 or fewer local minima and 8 or fewer local maxima. We do not wish to compare the standard deviation between sets of so few numbers.
Results Proliferation of cultured mouse bone marrow cells displays almost periodic oscillations occurring every 2 to 3 weekly passages. In the longest series, these oscillations of proliferation rate persisted for 29 weeks, with very variable amplitudes. The series used for illustrating the data varied from ⫺2.47 to 3.17 PD/week. The negative data (expressed in
824
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
PD) recorded for some passages correspond to a net loss of cells during the passage, and are relevant to cell death and lowered proliferation. Peaks of cell numbers of MBM and of the frequency of BMP all coincided, and both parameters correlated well (r ⫽ 0.684, p ⬍ 0.01). The production of progenitors fell abruptly at passage 16 when the amplitude of the oscillations of MBM proliferation decreased (Fig. 2). Figure 2 shows the recurrent map for MBM cells, with bisecting lines of segments MiMi⫹1 corresponding to local maxima (northwest to southeast segments, Fig. 3A) and to local minima (southeast to northwest segments, Fig. 3B). In Figure 3A, the bisecting lines converge to a point on the diagonal with coordinates about (0.05, 0.05). This attractor is on the diagonal of the map (where xi ⫽ xi⫹1) and thus is an equilibrium fixed point. In calculating the quantities yk for the observed data (see above, statistical test), we obtain a mean value, a ⫽ 0.08 and a std s0 ⫽ 0.366. Using the data plus 4999 random perturbations of the original series, we obtained in total 5000 stds si, out of which only 40 are lower than s0. Therefore, our convergence hypothesis can be considered as plausible, with a probability of 40/5000 ⫽ 0.008. The same calculations on local minima show that the convergence hypothesis is not significant. Actually, we obtained for the local minima, s0 ⫽ 0.71. Of the 5000 si’s values (one is the observed s0 and 4999 are from random perturbations of the observed series), not fewer than 1653 are lower than or equal to s0. The probability that (H) may come from a random sequence is large, p ⫽ 1653/5000 ⫽ 0.33. In four other, shorter bone marrow cultures, the same coordination of growth peaks was observed (data not shown), but the smaller numbers of observations, including only 8 or fewer local minima, would not allow complete Monte Carlo statistical analysis. Although the recurrent plot of production of BMP clones also shows control of local extrema (coordinates 13.5/13.5),
Figure 1. Construction of a map using the proliferation data as phase-space coordinates. This technique analyses the evolution of the number of bone marrow cells (MBM) during successive weeks in culture. Top: proliferation curve; Bottom: proliferation map. For this representation of proliferation data on the bidimensional map, data are tranformed thus: each segment of the curve, from one passage to the following, is defined as a point on the map with coordinate x being the first data of the segment (⫽ xi), and coordinate y being the following data, at the end of the segment (⫽ xi⫹1). All points are joined together, showing the temporal evolution of the system in a succession of orbits. The different situations on the proliferation curve—ascending, peak, descending, trough—can be further defined on this map as vectors: proliferation peaks, or local maxima, are represented by vectors going southeast in the plane, as illustrated here by the vector in bold representing the peak of MBM cell proliferation between passages 10 and 12. The position and size of each vector is further studied by drawing its bisecting line.
Figure 2. Oscillatory pattern of mouse bone marrow cell proliferation. Ordinate: proliferative rate of mouse bone marrow cells (MBM; white dots) expressed in PD/week; and frequency of corresponding hematopoietic progenitors (BMP; black dots). Abscissa: number of weeks (passages) in culture.
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
825
Figure 4. Phase-space representation of the production of BMP clones. Abscissa: x ⫽ production of clones at week (n); ordinate: y ⫽ production of clones at week (n⫹1). Note the convergence of the bisecting lines for peaks.
Figure 3. Phase-space representation of the proliferation dynamics of MBM cells and of the production of BMP clones. The time series as phase-space coordinates. Abscissa: x ⫽ proliferation data at week (n); ordinate: y ⫽ proliferation data at week (n⫹1). (A) Peaks (local maxima) of proliferation. The “low” fixed point attracts bisecting lines. (B) Troughs (local minima) of proliferation. The convergence is not focused.
the convergence did not achieve statistical significance (p ⬎ 0.05). Local minima were not organized (Fig. 4).
Discussion Persistent oscillations of proliferation rate in MBM cells were almost periodic, with a frequency of one peak per 2 to 3 weeks, similar to the oscillatory pattern of various cells in vitro [17]. Although seasonal or hormonal disturbances may be a cause of in vivo variability in MBM cell proliferation,
the duration of stable in vitro conditions frees the proliferation dynamics of these influences. The vectors representing the peaks of proliferation rate converged on a nodal point on the diagonal line of the map. The diagonal of the map is the ensemble of theoretical fixed points (xi and xi⫹1 are equal) where growth is steady. Therefore this nodal point represents a fixed proliferation level. This “attractor” corresponds to the spontaneous optimal proliferative activity to which the cells tend to return in all cases, correcting the oscillations induced by the multiple feedback loops of proliferation inducers and inhibitors. The observed fluctuations nevertheless indicate that the system never reaches this center of maximum stability. Thus, a “low” command coordinated the peaks in MBM cells proliferation rate. By contrast, although the recurrent plot suggests a control of some local minima in MBM, particularly those of the first half period of the culture, irregular vectors as of week 17 make the convergence statistically not significant. Dynamics for the peaks of progenitor production resembled that of parental MBM cells. The local minima were not coordinated in BMP. The regulation was opposite in hepatoma Fao cells in which a “high” fixed point alone controlled all slow-downs of proliferation [17] in many series. However, both culture conditions and cell functions are different for the two cell types, as tumoral Fao cells are regularly passaged in new dishes at small seeding density. The oscillations of cultured bone marrow cell growth are very similar to the well-known fluctuations in the production of blood cells, both normal and tumoral in vivo [7–11]. These fluctuations are usually described as periodic in nature with some irregularity. Our
826
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827
data suggest that the strong deterministic control of proliferation can result in different patterns of oscillations, from strictly periodic, approachable by spectral analysis, to almost periodic (and, strictly speaking, aperiodic), still exactly regulated to contain bone marrow cell growth within limits. This control of proliferation generating recurrent almost-periodic series may be explained in a dynamic system involving nonlinear interrelationships among several factors [29]. The major inhibitory factors are depletion of energetic compounds, contact inhibition, and erosion of telomeric DNA. Positive signals include autocrine growth factors and oncogene products, as well as exhaustion of inhibitory controls when nutrients are renewed and when telomeres are repaired. The resulting bipolar control of growth indicates that the cell population tends to recover a constrained proliferation rate. It is more precise than a single overall control, and is well adapted to different physiological purposes of the different cell types. The predominant dampening control on peaks in MBM cells reflects a dominant negative feedback that protects the stock of precursor cells. The coordinated oscillatory control for MBM cells and BMP is consistent with the stimulating effect of stromal cell–derived factor-1 on cycling of clonogenic progenitors [30]. It also fits well models of efflux in lineages of marrow cells depending on population size [10,31,32]. Additional evidence of such a determinism between the complex MBM cell proliferation dynamics and BMP clones production is illustrated by the synchrony between the loss of the two-week periodicity in MBM cell growth pattern at passage 14 and the drastic fall in progenitor number. Although requiring further analysis at the molecular level, this observation suggests that the mechanisms involved in commitment into BMP production are associated either with a periodicity or with a threshold level in MBM cells proliferation. Opposite clinical conditions like transient myeloproliferative syndrome [33] and transient erythroblastopenia of childhood [34] may reflect a transient imbalance in proliferation commands in marrow cells. Elucidating this deterministic dynamic pattern will help understand the evolution of hematological disorders and the development of grafted marrow stem cells. Acknowledgments This work was supported in part by l’Association “Biologie du Cancer et Dynamiques complexes” (L’ABCD). A part of this work was published as an abstract at the Meeting of the American Society of Hematology, New Orleans, LA, USA, Dec. 3–7, 1999.
References 1. Mackey MC, Glass L. Oscillation and chaos in physiological control systems. Science. 1977;197:287–289. 2. Laurent M, Kellershon N, Hervagault JF. Multistabilite´, oscillations et chaos dans les syste`mes biologiques. Me´d/Sciences. 1998;14:1240– 1251. 3. Khokha MK, Landini G, Iannacone PM. Fractal geometry in rat chimeras demonstrates that a repetitive cell division program may generate liver parenchyma. Dev Biol. 1994;165:545–555.
4. Squartini F. Strain differences in growth of mammary tumors. J Natl Cancer Inst. 1961;206:8132–8138. 5. Sanders BG, Teplitz RL, Wilkie M. The mitogenic activity of a c57bl/ 10 mouse transplantable lymphoma. Cancer Res. 1970;30:447–454. 6. Moore VJ, Rowley R, Hopkins HA, Russell-Ritenour E, Looney WB. Cyclophosphamide as an adjuvant to X rays in treatment of a radioresistant solid tumor of the rat, hepatoma H4IIE. Int J Radiation Oncology Biol Phys. 1979;5:1471–1474. 7. Ikram Z, Norton T, Jat PS. The biological clock that measures the mitotic life span of mouse embryo fibroblasts continues to function in the presence of simian virus 40 large tumor antigen. PNAS USA. 1994;91:6448–6452. 8. Wolfrom C, Raynaud N, Maigne J, et al. Periodic fluctuations in proliferation capacity of SV40 transformed human skin fibroblasts with prolonged lifespan. Cell Biol Toxicol. 1994;10:247–254. 9. Scha¨chter F, Boucher N, Lesueur-Ginot L, et al. Cellular senescence and survival of T lymphocytes. C R Acad Sci Paris. 1995;318:563–572. 10. Chow M, Kong M, Rubin H. Unmasking large and persistent reductions in proliferation rate of aging cells. In Vitro Cell Dev Biol. 1997;33: 809–818. 11. Maigne´ J, Deschatrette J, Sarrazin S, Hecquet B, Guerroui S, Wolfrom C. The time-pattern of rises and falls in proliferation fades with senescence of mortal lines and is perpetuated in immortal rat hepatoma Fao cell line. In Vitro Cell Dev Biol. 1998;34:163–169. 12. Wolfrom C, Chau NP, Maigne´ J, et al. Evidence for deterministic chaos in aperiodic oscillations of proliferative activity in long-term cultured Fao hepatoma cells. J Cell Sci. 2000;113:1069–1074. 13. Moorley AA. A neutrophil cycle in healthy individuals. Lancet. 1966; dec 3:1220–1222. 14. Von Schulthess GK, Gessner U. Oscillating platelet counts in healthy individuals: experimental investigation and quantitative evaluation of thrombocytopoietic feedback control. Scan J Haematol. 1986;36:473– 479. 15. Horwitz M, Benson KF, Person RE, Aprikyan AG, Dale DC. Mutations in ela2, encoding neutrophil elastase, define a 21-day biological clock in cyclic hematopoiesis. Nat Genet. 1999;23:433–436. 16. Fortin P, Mackey MC. Periodic chronic myelogenous leukemia: spectral analysis of blood cell counts and etiological implications. Br J Haematol. 1999;104:336–345. 17. Bennett M, Grunwald AJ. Hydroxyurea and periodicity in myeloproliferative disease. Eur J Haematol. 2001;66:317–323. 18. Laerum OD. Hematopoiesis occurs in rhythms. Exp Hematol. 1995;23: 1145–1147. 19. Derer L. Rhythm and proliferation with special reference to the sixday rhythm of blood leukocyte count. Neoplasma. 1960;2:117–133. 20. Eaves AC, Eaves CJ. Maintenance and proliferation control of primitive hemopoietic progenitors in long-term cultures of human marrow cells. Blood Cells. 1988;14:355–368. 21. Sutherland HJ, Lansdorp PM, Henkelman DH, Eaves AC, Eaves CJ. Functional characterization of individual human hematopoietic cells cultured at limiting dilution on supportive marrow stromal layers. Proc Natl Acad Sci U S A. 1990;87:3584–3588. 22. Holyoake TL, Jiang X, Drummond MW, Eaves AC, Eaves CJ. Elucidating critical mechanisms of deregulated stem cell turnover in the chronic phase of chronic myeloid leukemia. Leukemia. 2002;16:549–558. 23. Fortin P, Mackey MC. Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications. Br J Haematol. 1999;104:336–345. 24. Abkowitz JL, Catlin SN, Guttorp P. Evidence that hematopoiesis may be a stochastic process in vivo. Nat Med. 1996;2:190–197. 25. Preisler H, Raza A. An overview of some studies of chronic myelogenous leukemia: biological-clinical observations and viewing the disease as a chaotic system. Leuk Lymphoma. 1993;11:145–150. 26. May R. Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science. 1974;186:645–647.
C. Wolfrom et al. / Experimental Hematology 32 (2004) 822–827 27. Mackey M, Glass L. Oscillation and chaos in physiological control systems. Science. 1977;197:287–289. 28. Manly BFJ. Randomisation and Monte Carlo methods in biology. Boca Raton, FL: Chapman and Hall; 1991. 29. Glass L, Beuter A, Larocque D. Time delays, oscillations and chaos in physiological control systems. Math Biosci. 1988;90:111–125. 30. Lataillade JJ, Clay D, Bourin P, et al. Stromal cell–derived factor1 regulates primitive hematopoiesis by suppressing apoptosis and by promoting G0/G1 transition in CD34⫹ cells: evidence for an autocrineparacrine mechanism. Blood. 2002;4:1117–1129. 31. Abkowitz JL, Persik MT, Shelton GH, et al. Behavior of hematopoietic stem cells in a large animal. Proc Natl Acad Sci U S A. 1995;92: 2031–2035. 32. Haurie C, Dale DC, Mackey MC. Cyclical neutropenia and other periodic hematological disorders: a review of mechanisms and mathematical models. Blood. 1998;8:2629–2640. 33. Hitzler JK, Cheung J, Li Y, Scherer SW, Zipursky A. GATA1 mutations in transient leukemia and acute megakaryoblastic leukemia of Down syndrome. Blood. 2003;101:4301–4304. 34. Gustavsson P, Klar J, Matsson H, et al. Familial transient erythroblastopenia of childhood is associated with the chromosome 19q13.2 region but not caused by mutations in coding sequences of the ribosomal protein S19(RPS19) gene. Br J Haematol. 2002;119:261–264.
827
Appendix On rectangular coordinates (see Fig. 1), consider the points Mi⫺1(xi⫺1, xi) and Mi(xi, xi⫹1), where xi is a local minimum (xi ⬍ xi⫺1 and xi ⬍ xi⫹1). Let H be the midpoint of Mi⫺1Mi, then H has coordinates xH ⫽ (xi⫺1 ⫹ xi)/2, yH ⫽ (xi ⫹ xi⫹1)/2. We assume that the line orthogonal to Mi⫺1Mi cuts the diagonal line at a fixed point A(a, a). Then the slope of the line Mi⫺1Mi is s1 ⫽ (xi⫹1 ⫺ xi)/(xi ⫺ xi⫺1) and the slope of the line HA is s2 ⫽ (a⫺yH)/(a⫺xH). As HA and Mi⫺1Mi are orthogonal lines, s1 ⫽ ⫺1/s2. This relationship, with the above values of s1 and s2, gives, after developing the terms and simplifications, 2a(xi⫹1 ⫺ xi⫺1) ⫽ xi⫹12 ⫺ xi⫺12 As xi⫹12 ⫺ xi⫺12 ⫽ (xi⫹1 ⫺ xi⫺1)(xi⫹1 ⫹ xi⫺1), one obtains finally the relationship (H) in the text: 2a ⫽ (xi⫹1 ⫹ xi⫺1) (H). Relationship (H) is much simpler than the complete geometric construction, but the geometric construction leads to the idea of central control, and it is that construction that led us to discover the control mechanism.