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Synchronous Cultures from the Baby Machine: Anatomy of a Model
J. theor. Biol. (2001) 212, 391}398 doi:10.1006/jtbi.2001.2386, available online at http://www.idealibrary.com on
Synchronous Cultures from the Baby Machine: Anatomy of a Model N. B. GROVER*-, C. COUSTEE RE-YAKIR*
AND
C. E. HELMSTETTER?
*Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, ¹he Hebrew ;niversity Faculty of Medicine, PO Box 12272, Jerusalem 91120, Israel, and ?Department of Biological Sciences, Florida Institute of ¹echnology, 150 = ;niversity Blvd, Melbourne, F¸ 32901, ;.S.A. (Received on 27 February 2001, Accepted in revised form on 26 June 2001)
The baby-machine system, which produces newborn Escherichia coli cells from cultures immobilized on a membrane, was developed many years ago in an attempt to attain optimal synchrony with minimal disturbance of steady-state growth. In the present article, we describe in some detail a model designed to analyse such cells with a view to characterizing the nature and quality of the synchrony in a quantitative manner; it can also serve to evaluate the methodology itself, its potential and its limitations. The model consists of "ve elements, giving rise to "ve adjustable parameters (and a proportionality constant): a major, essentially synchronous group of cells with ages distributed normally about zero; a minor, random component from a steady-state population on the membrane that had undergone only very little age selection during the elution process; a "xed background count, to account for the signals recorded by the electronic particle counter produced by debris and electronic noise; a time-shift, to allow for di!erences between collection time and sampling time; and the coe$cient of variation of the interdivision-time distribution, taken to be a Pearson type III. The model is "tted by nonlinear least-squares to data from cells grown in glucose minimal medium. The standard errors of the parameters are quite small, making their estimates all highly signi"cant; the quality of the "t is striking. We also provide a simple yet rigorous procedure for correcting cell counts obtained in an electronic particle counter for the e!ect of coincidence. An example using real data produces an excellent "t. 2001 Academic Press
Introduction The baby machine is a device in which newborn cells are released from a growing population immobilized on a membrane (Helmstetter, 1969). It is a widely employed technique for the generation of synchronous cultures of Escherichia coli and other bacterial species. The output is close to optimal in terms of both quality and persistence - Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/01/190391#08 $35.00/0
of synchrony. More important, perhaps, the cells produced appear to be only minimally disturbed physiologically, an essential requirement for meaningful cell-cycle studies (Helmstetter et al., 2001). Recently, it has been adapted for use with animal cells (Thornton, M., Eward, K. L. & Helmstetter, C. E., manuscript in preparation). A model developed in order to characterize such cells in a quantitative manner was "rst put forward in 1995 (Grover & Helmstetter, 1995). The eluted cells were considered to consist of two 2001 Academic Press
392
N. B. GROVER E¹ A¸.
subpopulations: a major one of synchronous cells and a minor one of random cells. These, together with a background count and a proportionality constant, constituted the original form of the model. This description was quite successful as long as the data were limited to a single doubling (Grover & Helmstetter, 1995); it proved inadequate for longer measurements. A small time-shift was added to allow for the di!erences between collection time and sampling time that can arise in multi-sample experiments of the kind used. A more fundamental extension was the introduction of dispersion into the interdivision time, which we assumed to follow a Pearson type III distribution. The steady-state age distribution of the small random component on the membrane had to be modi"ed to re#ect this change. This model, which is presented formally in the following section, provides excellent "ts for measurements up to about 2.5 cell doublings (Helmstetter et al., 2001). Beyond that point, the experimental data show a clear tendency to drop further and further below the predicted curve as time increases. The reasons for this may be instrumental, methodological or biological (cell morbidity, for instance), and we have made no attempt to model the process in that range; we do, however, consider in some detail the question of coincidence within the ori"ce of an electronic particle counter at higher cell concentrations. The Model The purpose of the model is to provide an analysis of cell samples produced by the baby machine with a view to characterizing the nature and quality of the synchrony in a quantitative manner; it can also serve to evaluate the methodology itself, its potential and its limitations (Helmstetter et al., 2001). The cell populations eluted from the membrane were taken as being composed of two components. The age of the major component was assumed to be normally distributed about zero with a standard deviation p. The minor component, of relative size o, was considered to consist of cells that had undergone only very little age selection during the elution process: the number of these random cells was set proportional to 2? O ,
where a is the age of the eluted cell at the time of collection (t"0) and qN is the observed doubling time of the culture. These are bacteria released from the membrane for reasons other than division, such as weak binding. It was also necessary to include a "xed background b to account for the signals recorded by the electronic particle counter that were produced not by viable, replicating cells but by the debris and electronic noise that are invariably associated with such measurements. This, together with a proportionality constant k, constituted the original form of the model (Grover & Helmstetter, 1995); it proved inadequate once we began considering more than one cell cycle. First it became apparent that the initial doubling sometimes took place a little earlier than subsequent ones, sometimes a little later. We introduced a time-shift d to allow for the di!erences between collection time and sampling time that can arise in a multi-sample experiment of this type. The second addition to the original model was more fundamental. The dispersion p of the newborn subpopulation accounted rather well for the transition period as long as we limited ourselves to a single generation time. When we began studying several cycles, however, a value of p suitable for the "rst transition period systematically under-corrected for the second whereas a value suitable for the second overcorrected for the "rst. That was because the sharpness of the transition decreases from doubling to doubling and cannot be accommodated by a "xed p. In order to allow for such deterioration, we introduced dispersion into the interdivision times as well: q is distributed about qN with a coe$cient of variation q. The form of this distribution, for both cellular components, was assumed to be a Pearson type III (incomplete gamma function) rather than a normal distribution because of the widespread observation that f (q) is skewed to the right and the classic studies of Powell (1956). As experiments become longer, cell concentrations increase and with it the probability of coincidence within the ori"ce of the electronic particle counter. The easiest way to avoid this potential source of bias is to dilute the measuring medium periodically to keep the coincidence level negligible, and that is what we have done,
SYNCHRONY OF BABY-MACHINE CELLS*A MODEL
393
at intervals and diluted in Isoton II (BeckmanCoulter, Miami, FL, U.S.A.). The "rst 16 aliquots, spanning the initial 62 min, were diluted a 100-fold; the last ten aliquots, covering the next 40 min, were diluted 200-fold. Cell concentrations were determined with a model ZB electronic particle counter (Beckman-Coulter, Miami, FL, U.S.A.). Results FIG. 1. Cell concentration during growth of baby machine generated E. coli B/r A cells in glucose minimal medium, doubling time 41 min; semilogarithmic scale. Arrow indicates time at which dilution increases twofold.
both here (Fig. 1) and elsewhere (Helmstetter et al., 2001). Under certain conditions, however, this may not be feasible, as when high concentrations are required to keep the counting period short or other time constraints exist, and in such situations either the observed cell count can be adjusted upwards to compensate for coincidence loss or, equivalently, the predicted cell numbers can be adjusted downwards. Materials and Methods A culture of E. coli B/r A (ATCC 12407) was grown in minimal salts medium (Helmstetter & Pierucci, 1976) supplemented with 0.1% glucose, doubling time 41 min. The baby-machine technique was used to obtain synchronous cultures (Helmstetter, 1969). Brie#y, 100 ml of minimal medium was inoculated with bacteria from a fresh stationary-phase stock, diluted approximately 1:1000, and incubated in a shaking water bath at 373C until the cells were in steady-state growth. When a culture reached a concentration of approximately 5;10 cells/ml, it was "ltered onto the surface of a type GS 142 mm diameter membrane "lter (Millipore, Bedford, MA, U.S.A.) in an incubator at 373C. The "lter was then inverted and elution with fresh medium begun. An aliquot was collected from the e%uent and incubated in a 50 ml culture #ask (Bellco, Vineland, NJ, U.S.A.) at 373C in a shaking water bath. Samples were collected from the culture #ask
The raw data, part of a longer experiment (Helmstetter et al., 2001), are shown in Fig. 1 plotted on a semilogarithmic scale as a function of time t. The arrow just beyond t"62 min indicates a twofold increase in dilution, made in order to decrease the probability of coincidence within the ori"ce of the electronic particle counter. It may sometimes be preferable to correct for the coincidence instead, and a simple yet rigorous procedure for this is derived formally in Appendix A; since it is not, strictly speaking, an integral component of the baby-machine model, and has much broader application, we deal with it separately in the next section. For illustrative purposes, we present the model as a step-wise process (Fig. 2); in practice, all parameters are "t simultaneously using nonlinear least-squares. First (Fig. 2a) the doubling time of the culture qN is determined by observation; it is not an adjustable parameter of the model. The current model, cells doubling every 41 min, is inadequate (Fig. 2b): the doublings according to the model take place too early. We add a timeshift d to allow for the di!erences between collection time and sampling time that can arise in a multi-sample experiment of this type. The data are not horizontal prior to the "rst division (Fig. 2c). This suggests that the population eluted from the membrane includes a minor component of non-synchronous, randomly distributed cells. It is assumed that this component, of relative size o, consists of those cells from the steady-state population on the membrane that had undergone only very little age selection during the elution process: the number of these random cells was set proportional to 2? ON , where a is the age of the eluted cell at the time of collection (t "0). The "t during most of the "rst generation now seems adequate (Fig. 2d), but there is a clear
394
N. B. GROVER E¹ A¸.
FIG. 2. Schematic representation of model as step-wise process, in practice all parameters are "tted simultaneously using unconstrained nonlinear least squares; raw data and axes as in Fig. 1.
overshoot during the second, the implication being that not all particles are growing and dividing. We introduce a "xed background b to account for signals recorded by the electronic counter that were produced not by viable, replicating cells but by the debris and electronic noise that are invariably associated with such measurements. We now address the transition period over which the cells divide (Fig. 2e). The model as it stands describes a perfectly synchronous step for the major component of the population*the data clearly suggest otherwise. The age of the newborn population was assumed to be
normally distributed about zero with a standard deviation p. The introduction of dispersion in the newborn population seems to account rather well for the "rst transition period but undercorrects for the second; a value of p suitable for the second would overcorrect for the "rst. This is because the sharpness of the transition decreases from doubling to doubling and cannot be accommodated by a "xed p. In order to allow for such deterioration, we introduce dispersion into the interdivision times as well: q is distributed about qN with a coe$cient of variation q. The form of this f (q) distribution is assumed to be a Pearson type III
395
SYNCHRONY OF BABY-MACHINE CELLS*A MODEL
TABLE 1 Statistics of ,t of theoretical model to experimental data obtained from the baby machine for E. coli B/r cells growing in glucose minimal medium with a doubling time of 41 min Parameter
Value
Units
SE*
CV-
d o b pA q k
1.60 10.5 11.3 8.61 9.43 12.8
min % % % % 10/ml
0.27 3.3 1.2 3.0 1.8 0.24
16.8 31.6 10.6 34.7 19.0 1.86
p? (0.0001 0.0025 (0.0001 0.0046 (0.0001 (0.0001
* Standard error: standard deviation of parameter estimate. - Coe$cient of variation: standard error as a percentage of parameter estimate. ? Signi"cance: probability that true parameter value is zero, as determined by a two-tailed t-test. A Expressed as a percentage of mean doubling time qN .
(incomplete gamma function), and therefore (Powell, 1956) so is the steady-state age distribution on the membrane l(a): qE\e\O Q and f (q)" sEC(g)
2ke\I? l(a)" qE\e\O Q dq, sEC(g) ? qN 2 E!1 1 . where g" , s" , k" g s q The "nal panel in the series (Fig. 2f ) presents the full model, with all six adjustable parameters (d, o, b, p, q and k, the overall proportionality coe$cient) "tted simultaneously by unconstrained, nonlinear least-squares (IMSL, 1995). The standard errors of the parameters are quite small, making their estimates all highly signi"cant (Table 1); they are also reasonable. The quality of the "t is striking: the sum of squares of the residuals is less than 0.1% of the total and they are distributed randomly (p"0.286), as determined by a one-tailed runs test, and normally (p"0.378), as determined by the Shapiro}Wilk test. Coincidence Using the Poisson distribution and a little algebra, we show in Appendix A that the true
TABLE 2 Observed cell concentration (millions/ml) at a series of relative dilutions Dilution 2 2 2 2 2 2 2
Observed counts 455.476 266.048 145.388 77.556 40.252 19.756 10.100
number of counts N can be expressed as N"c
and the observed counts M as M"c(1!e\(), where is the parameter of the Poisson distribution and c is the constant of proportionality. A series of measurements at known dilutions can be "tted by standard nonlinear least-squares techniques to provide estimates of and c; true count N is then obtained from observed count M via N"c ln[c/(c!M)]. We applied this procedure to the data listed in Table 2, obtained with the model ZB electronic particle counter. A nonlinear least-squares "t (IMSL, 1995) gave an estimate for c of 899$17 million cells/ml and for of 0.706$0.018; both are very highly signi"cant (p(0.0001), as determined by a two-tailed t-test. The complete correction curve computed from c is shown in Fig. 3 together with the statistics of the "t. The sum of squares of the residuals is exceedingly small, less
396
N. B. GROVER E¹ A¸.
discrepancy caused by coincidence, which is legitimate, it would a!ect the whole of it regardless of source, which is decidedly not. Such a design reduces the model from one in which every parameter has a clear and well-de"ned physical or biological meaning to one in which they serve merely to reduce the overall residual sum of squares. Discussion
FIG. 3. E!ect of coincidence on cell count in an electronic particle counter, together with the statistics of the "t of the theory to the observations listed in Table 2; log}log scale. (E) Measured; (*) Predicted; (---) No Coincidence.
than 0.005%, and they are distributed randomly (p"0.333), as determined by a one-tailed runs test. As stated in Appendix A, once c has been estimated from the least-squares "t to the calibration data, it is a simple matter to correct for coincidence loss all measurements obtained under the same experimental conditions (ori"ce and manometer volumes). Applied in reverse, these expressions are equally suitable for correcting the predicted cell number instead, by replacing N by c(1!e\, A), the true count reduced by coincidence. One may be tempted to correct predicted cell number in this way without "rst "tting independent calibration data for an estimate of c, by including c as an additional parameter of the model. It is a seductive idea, avoiding the need to perform a set of dilution experiments by giving up a degree of freedom*seductive and misguided. The experimental data tend to drop further and further below the predicted curve as time increases (Helmstetter et al., 2001), for reasons that are not at all clear. Using c as an adjustable parameter in the baby-machine model would thus not only decrease that part of the
A model to describe the behaviour of cells produced by the baby machine has been presented in some detail, and an example given of a "t to data from E. coli cells grown in glucose minimal medium with a doubling time of 41 min. A procedure for correcting the coincidence within the ori"ce of an electronic particle counter is also provided and shown to be quite accurate. The "t of the model is outstanding, as judged by the precision with which the various parameters are estimated, by the magnitude of the residuals, and by the runs test. Such attributes are an indication of what can be expected from a technique that yields high-quality synchrony of minimally disturbed cells. Results that di!er substantially from these usually occur because either the cells are less than optimally synchronous or they have been disturbed by the synchronization procedure, depending on the nature of the di!erence; in particular, data displaying a higher degree or a more extended duration of synchrony probably re#ect an arti"cial alignment of cellgrowth properties caused by cell perturbation, an altered physiology. Cell-cycle-dependent output generated in this way is always suspect and should be interpreted with great caution. It is clear that the information furnished by the model is useful, in the case of E. coli, both to quantify the properties of the synchrony and to characterize the baby machine itself (Grover & Helmstetter, 1995; Helmstetter et al., 2001). But what about animal cells? Will the model apply to them as well? Perhaps, but certainly not in its present form. At least one basic modi"cation will be required and at least one extension. Bacterial cells have an interdivision-time distribution f (q) that follows a Pearson type III, as shown by Powell (1956) many years ago; animal cells do not. There is still considerable
SYNCHRONY OF BABY-MACHINE CELLS*A MODEL
disagreement in the literature concerning even the most basic features of the division cycle in animal cells (Cooper, 1991), but if we adopt the pragmatic approach taken by Powell (1958), to use any frequency function that "ts adequately, then the most likely candidate distribution (Brooks, 1981) would seem to be that originally proposed by Kubitschek (1971): a reciprocalnormal. The reciprocal-normal distribution is not user-friendly; it cannot be expressed in closed form, there are no tables or approximations or asymptotic expressions, and its moments do not exist; calculations of the steady-state age distribution l(a) must be carried out numerically and iteratively, a rather tedious procedure. But there is a side bene"t: it allows one to compute mother}daughter and sister}sister correlations. It is a common observation (Powell, 1958) that in bacteria mother}daughter interdivision times are correlated (and, as a result, so are those between sisters). Unfortunately, the Pearson type III distribution does not allow the incorporation of such a relationship and consequently this feature is absent from the current model*a limitation we have not been able to overcome. But the normal (and hence reciprocal-normal) distribution does (Grover & Woldringh, 2001), quite easily. The extension concerns cell morbidity. It is well known (Powell, 1958) that bacteria grow and divide essentially without loss, at least over the time scale of a laboratory experiment, but such may not be the case with animal cells, which are far more delicate than free-living bacteria. Thus, when applied to animal cells, the model will have to allow for possible sterility, cells that continue to grow but stop dividing, as well as for outright cell death. Work along those lines is now in progress.
REFERENCES BROOKS, R. F. (1981). Variability in the cell cycle and the control of proliferation. In: ¹he Cell Cycle (John, P. C. L., ed), pp. 35}61. Cambridge, UK: Cambridge University Press. COOPER, S. (1991). Bacterial Growth and Division, pp. 391}419. San Diego, CA: Academic Press. GROVER, N. B. & HELMSTETTER, C. E. (1995). Characterization of cell-cycle-speci"c events in synchronous cultures of
397
Escherichia coli: a theoretical evaluation. Microbiology 141, 59}62. GROVER, N. B. & WOLDRINGH, C. L. (2001). Dimensional regulation of cell-cycle events in Escherichia coli during steady-state growth. Microbiology 147, 171}181. HELMSTETTER, C. E. (1969). Methods for studying the microbial division cycle. In: Methods in Microbiology (Norris, J. R. & Ribbons, D. W., eds), Vol. 1, pp. 327}363. London: Academic Press. HELMSTETTER, C. E. & PIERUCCI, O. (1976). DNA synthesis during the division cycle of three substrains of Escherichia coli B/r. J. Mol. Biol. 102, 477}486. HELMSTETTER, C. E., THORNTON, M. & GROVER, N. B. (2001). Cell-cycle research with synchronous cultures: An evaluation. Biochimie 83, 83}89. IMSL. (1995). Math and stat libraries. In: Fortran Powerstation Professional Edition version 4.0. Redmond, WA: Microsoft. KUBITSCHEK, H. E. (1971). The distribution of cell generation times. Cell ¹issue Kinet. 4, 113}122. POWELL, E. O. (1956). Growth rate and generation time of bacteria, with special reference to continuous culture. J. Gen. Microbiol. 15, 492}511. POWELL, E. O. (1958). An outline of the pattern of bacterial generation times. J. Gen. Microbiol. 18, 382}417.
APPENDIX A Let i be the number of bacteria within the e!ective measuring volume of the electronic particle counter. Then i is distributed according to the Poisson distribution: the probability P(i) of "nding exactly i cells within the ori"ce is P(i)"e\( G/i! where is the parameter of the distribution. It is easy to show that, as expected, P(i)"1, since P(i)" e\( G/i! G G G "e\([1# # /2!#2], and the quantity in brackets is just the series expansion of e(, which converges for all "nite values of . Furthermore, the mean value of i is, by de"nition, iP(i)/ P(i)" iP(i), since the G G G denominator is unity and the "rst term in the numerator is 0. But iP(i)" i (e\( G)/i! G G " e\( G/(i!1)!" (e\( G\)/(i!1)! G G " e\( H/j!" , where i!1 has been H replaced by j, making the expression following the "nal summation sign identical in form to the de"nition of P(i); its sum, therefore, is unity, and so the average value of i is given by , the parameter of the distribution. Consider now an actual measurement M. The true number of counts N will be proportional to P (1)#2P (2)#3P (3)#2" iP (i)" , G so that N"c , where c is the constant of proportionality and is, in e!ect, the ratio between the volume of the manometer and that of the
398
N. B. GROVER E¹ A¸.
ori"ce; because of coincidence, however, the value of M will only be proportional to P(1)# P (2)#P (3)#2" P (i)" P (i ) G G !P(0)"1!e\(, so that M"c(1!e\( ). If we make a series of measurements M , M , M ,2, M at known dilutions, then we have P r equations in two unknowns, c and , which can be "tted using standard nonlinear least-squares techniques. In particular, if successive dilutions are by a factor of 2, then M "c(1!e\(), M "c(1!e\ (),2, M "c(1!e\ (), where is the (unknown) average number of
cells in the measuring ori"ce at the highest concentration M . Once c has been estimated from the leastsquares "t to the calibration data, it is a simple matter to correct for coincidence loss all measurements obtained under the same experimental conditions (ori"ce and manometer volumes). The value of corresponding to any particular measurement M is obtained by solving M" c(1!e\() and then it is substituted into the expression for the true cell count N; in symbols, N"c "c ln[c/(c!M)].
Synchronous Cultures from the Baby Machine: Anatomy of a Model N. B. GROVER*-, C. COUSTEE RE-YAKIR*
AND
C. E. HELMSTETTER?
*Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, ¹he Hebrew ;niversity Faculty of Medicine, PO Box 12272, Jerusalem 91120, Israel, and ?Department of Biological Sciences, Florida Institute of ¹echnology, 150 = ;niversity Blvd, Melbourne, F¸ 32901, ;.S.A. (Received on 27 February 2001, Accepted in revised form on 26 June 2001)
The baby-machine system, which produces newborn Escherichia coli cells from cultures immobilized on a membrane, was developed many years ago in an attempt to attain optimal synchrony with minimal disturbance of steady-state growth. In the present article, we describe in some detail a model designed to analyse such cells with a view to characterizing the nature and quality of the synchrony in a quantitative manner; it can also serve to evaluate the methodology itself, its potential and its limitations. The model consists of "ve elements, giving rise to "ve adjustable parameters (and a proportionality constant): a major, essentially synchronous group of cells with ages distributed normally about zero; a minor, random component from a steady-state population on the membrane that had undergone only very little age selection during the elution process; a "xed background count, to account for the signals recorded by the electronic particle counter produced by debris and electronic noise; a time-shift, to allow for di!erences between collection time and sampling time; and the coe$cient of variation of the interdivision-time distribution, taken to be a Pearson type III. The model is "tted by nonlinear least-squares to data from cells grown in glucose minimal medium. The standard errors of the parameters are quite small, making their estimates all highly signi"cant; the quality of the "t is striking. We also provide a simple yet rigorous procedure for correcting cell counts obtained in an electronic particle counter for the e!ect of coincidence. An example using real data produces an excellent "t. 2001 Academic Press
Introduction The baby machine is a device in which newborn cells are released from a growing population immobilized on a membrane (Helmstetter, 1969). It is a widely employed technique for the generation of synchronous cultures of Escherichia coli and other bacterial species. The output is close to optimal in terms of both quality and persistence - Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/01/190391#08 $35.00/0
of synchrony. More important, perhaps, the cells produced appear to be only minimally disturbed physiologically, an essential requirement for meaningful cell-cycle studies (Helmstetter et al., 2001). Recently, it has been adapted for use with animal cells (Thornton, M., Eward, K. L. & Helmstetter, C. E., manuscript in preparation). A model developed in order to characterize such cells in a quantitative manner was "rst put forward in 1995 (Grover & Helmstetter, 1995). The eluted cells were considered to consist of two 2001 Academic Press
392
N. B. GROVER E¹ A¸.
subpopulations: a major one of synchronous cells and a minor one of random cells. These, together with a background count and a proportionality constant, constituted the original form of the model. This description was quite successful as long as the data were limited to a single doubling (Grover & Helmstetter, 1995); it proved inadequate for longer measurements. A small time-shift was added to allow for the di!erences between collection time and sampling time that can arise in multi-sample experiments of the kind used. A more fundamental extension was the introduction of dispersion into the interdivision time, which we assumed to follow a Pearson type III distribution. The steady-state age distribution of the small random component on the membrane had to be modi"ed to re#ect this change. This model, which is presented formally in the following section, provides excellent "ts for measurements up to about 2.5 cell doublings (Helmstetter et al., 2001). Beyond that point, the experimental data show a clear tendency to drop further and further below the predicted curve as time increases. The reasons for this may be instrumental, methodological or biological (cell morbidity, for instance), and we have made no attempt to model the process in that range; we do, however, consider in some detail the question of coincidence within the ori"ce of an electronic particle counter at higher cell concentrations. The Model The purpose of the model is to provide an analysis of cell samples produced by the baby machine with a view to characterizing the nature and quality of the synchrony in a quantitative manner; it can also serve to evaluate the methodology itself, its potential and its limitations (Helmstetter et al., 2001). The cell populations eluted from the membrane were taken as being composed of two components. The age of the major component was assumed to be normally distributed about zero with a standard deviation p. The minor component, of relative size o, was considered to consist of cells that had undergone only very little age selection during the elution process: the number of these random cells was set proportional to 2? O ,
where a is the age of the eluted cell at the time of collection (t"0) and qN is the observed doubling time of the culture. These are bacteria released from the membrane for reasons other than division, such as weak binding. It was also necessary to include a "xed background b to account for the signals recorded by the electronic particle counter that were produced not by viable, replicating cells but by the debris and electronic noise that are invariably associated with such measurements. This, together with a proportionality constant k, constituted the original form of the model (Grover & Helmstetter, 1995); it proved inadequate once we began considering more than one cell cycle. First it became apparent that the initial doubling sometimes took place a little earlier than subsequent ones, sometimes a little later. We introduced a time-shift d to allow for the di!erences between collection time and sampling time that can arise in a multi-sample experiment of this type. The second addition to the original model was more fundamental. The dispersion p of the newborn subpopulation accounted rather well for the transition period as long as we limited ourselves to a single generation time. When we began studying several cycles, however, a value of p suitable for the "rst transition period systematically under-corrected for the second whereas a value suitable for the second overcorrected for the "rst. That was because the sharpness of the transition decreases from doubling to doubling and cannot be accommodated by a "xed p. In order to allow for such deterioration, we introduced dispersion into the interdivision times as well: q is distributed about qN with a coe$cient of variation q. The form of this distribution, for both cellular components, was assumed to be a Pearson type III (incomplete gamma function) rather than a normal distribution because of the widespread observation that f (q) is skewed to the right and the classic studies of Powell (1956). As experiments become longer, cell concentrations increase and with it the probability of coincidence within the ori"ce of the electronic particle counter. The easiest way to avoid this potential source of bias is to dilute the measuring medium periodically to keep the coincidence level negligible, and that is what we have done,
SYNCHRONY OF BABY-MACHINE CELLS*A MODEL
393
at intervals and diluted in Isoton II (BeckmanCoulter, Miami, FL, U.S.A.). The "rst 16 aliquots, spanning the initial 62 min, were diluted a 100-fold; the last ten aliquots, covering the next 40 min, were diluted 200-fold. Cell concentrations were determined with a model ZB electronic particle counter (Beckman-Coulter, Miami, FL, U.S.A.). Results FIG. 1. Cell concentration during growth of baby machine generated E. coli B/r A cells in glucose minimal medium, doubling time 41 min; semilogarithmic scale. Arrow indicates time at which dilution increases twofold.
both here (Fig. 1) and elsewhere (Helmstetter et al., 2001). Under certain conditions, however, this may not be feasible, as when high concentrations are required to keep the counting period short or other time constraints exist, and in such situations either the observed cell count can be adjusted upwards to compensate for coincidence loss or, equivalently, the predicted cell numbers can be adjusted downwards. Materials and Methods A culture of E. coli B/r A (ATCC 12407) was grown in minimal salts medium (Helmstetter & Pierucci, 1976) supplemented with 0.1% glucose, doubling time 41 min. The baby-machine technique was used to obtain synchronous cultures (Helmstetter, 1969). Brie#y, 100 ml of minimal medium was inoculated with bacteria from a fresh stationary-phase stock, diluted approximately 1:1000, and incubated in a shaking water bath at 373C until the cells were in steady-state growth. When a culture reached a concentration of approximately 5;10 cells/ml, it was "ltered onto the surface of a type GS 142 mm diameter membrane "lter (Millipore, Bedford, MA, U.S.A.) in an incubator at 373C. The "lter was then inverted and elution with fresh medium begun. An aliquot was collected from the e%uent and incubated in a 50 ml culture #ask (Bellco, Vineland, NJ, U.S.A.) at 373C in a shaking water bath. Samples were collected from the culture #ask
The raw data, part of a longer experiment (Helmstetter et al., 2001), are shown in Fig. 1 plotted on a semilogarithmic scale as a function of time t. The arrow just beyond t"62 min indicates a twofold increase in dilution, made in order to decrease the probability of coincidence within the ori"ce of the electronic particle counter. It may sometimes be preferable to correct for the coincidence instead, and a simple yet rigorous procedure for this is derived formally in Appendix A; since it is not, strictly speaking, an integral component of the baby-machine model, and has much broader application, we deal with it separately in the next section. For illustrative purposes, we present the model as a step-wise process (Fig. 2); in practice, all parameters are "t simultaneously using nonlinear least-squares. First (Fig. 2a) the doubling time of the culture qN is determined by observation; it is not an adjustable parameter of the model. The current model, cells doubling every 41 min, is inadequate (Fig. 2b): the doublings according to the model take place too early. We add a timeshift d to allow for the di!erences between collection time and sampling time that can arise in a multi-sample experiment of this type. The data are not horizontal prior to the "rst division (Fig. 2c). This suggests that the population eluted from the membrane includes a minor component of non-synchronous, randomly distributed cells. It is assumed that this component, of relative size o, consists of those cells from the steady-state population on the membrane that had undergone only very little age selection during the elution process: the number of these random cells was set proportional to 2? ON , where a is the age of the eluted cell at the time of collection (t "0). The "t during most of the "rst generation now seems adequate (Fig. 2d), but there is a clear
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FIG. 2. Schematic representation of model as step-wise process, in practice all parameters are "tted simultaneously using unconstrained nonlinear least squares; raw data and axes as in Fig. 1.
overshoot during the second, the implication being that not all particles are growing and dividing. We introduce a "xed background b to account for signals recorded by the electronic counter that were produced not by viable, replicating cells but by the debris and electronic noise that are invariably associated with such measurements. We now address the transition period over which the cells divide (Fig. 2e). The model as it stands describes a perfectly synchronous step for the major component of the population*the data clearly suggest otherwise. The age of the newborn population was assumed to be
normally distributed about zero with a standard deviation p. The introduction of dispersion in the newborn population seems to account rather well for the "rst transition period but undercorrects for the second; a value of p suitable for the second would overcorrect for the "rst. This is because the sharpness of the transition decreases from doubling to doubling and cannot be accommodated by a "xed p. In order to allow for such deterioration, we introduce dispersion into the interdivision times as well: q is distributed about qN with a coe$cient of variation q. The form of this f (q) distribution is assumed to be a Pearson type III
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SYNCHRONY OF BABY-MACHINE CELLS*A MODEL
TABLE 1 Statistics of ,t of theoretical model to experimental data obtained from the baby machine for E. coli B/r cells growing in glucose minimal medium with a doubling time of 41 min Parameter
Value
Units
SE*
CV-
d o b pA q k
1.60 10.5 11.3 8.61 9.43 12.8
min % % % % 10/ml
0.27 3.3 1.2 3.0 1.8 0.24
16.8 31.6 10.6 34.7 19.0 1.86
p? (0.0001 0.0025 (0.0001 0.0046 (0.0001 (0.0001
* Standard error: standard deviation of parameter estimate. - Coe$cient of variation: standard error as a percentage of parameter estimate. ? Signi"cance: probability that true parameter value is zero, as determined by a two-tailed t-test. A Expressed as a percentage of mean doubling time qN .
(incomplete gamma function), and therefore (Powell, 1956) so is the steady-state age distribution on the membrane l(a): qE\e\O Q and f (q)" sEC(g)
2ke\I? l(a)" qE\e\O Q dq, sEC(g) ? qN 2 E!1 1 . where g" , s" , k" g s q The "nal panel in the series (Fig. 2f ) presents the full model, with all six adjustable parameters (d, o, b, p, q and k, the overall proportionality coe$cient) "tted simultaneously by unconstrained, nonlinear least-squares (IMSL, 1995). The standard errors of the parameters are quite small, making their estimates all highly signi"cant (Table 1); they are also reasonable. The quality of the "t is striking: the sum of squares of the residuals is less than 0.1% of the total and they are distributed randomly (p"0.286), as determined by a one-tailed runs test, and normally (p"0.378), as determined by the Shapiro}Wilk test. Coincidence Using the Poisson distribution and a little algebra, we show in Appendix A that the true
TABLE 2 Observed cell concentration (millions/ml) at a series of relative dilutions Dilution 2 2 2 2 2 2 2
Observed counts 455.476 266.048 145.388 77.556 40.252 19.756 10.100
number of counts N can be expressed as N"c
and the observed counts M as M"c(1!e\(), where is the parameter of the Poisson distribution and c is the constant of proportionality. A series of measurements at known dilutions can be "tted by standard nonlinear least-squares techniques to provide estimates of and c; true count N is then obtained from observed count M via N"c ln[c/(c!M)]. We applied this procedure to the data listed in Table 2, obtained with the model ZB electronic particle counter. A nonlinear least-squares "t (IMSL, 1995) gave an estimate for c of 899$17 million cells/ml and for of 0.706$0.018; both are very highly signi"cant (p(0.0001), as determined by a two-tailed t-test. The complete correction curve computed from c is shown in Fig. 3 together with the statistics of the "t. The sum of squares of the residuals is exceedingly small, less
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discrepancy caused by coincidence, which is legitimate, it would a!ect the whole of it regardless of source, which is decidedly not. Such a design reduces the model from one in which every parameter has a clear and well-de"ned physical or biological meaning to one in which they serve merely to reduce the overall residual sum of squares. Discussion
FIG. 3. E!ect of coincidence on cell count in an electronic particle counter, together with the statistics of the "t of the theory to the observations listed in Table 2; log}log scale. (E) Measured; (*) Predicted; (---) No Coincidence.
than 0.005%, and they are distributed randomly (p"0.333), as determined by a one-tailed runs test. As stated in Appendix A, once c has been estimated from the least-squares "t to the calibration data, it is a simple matter to correct for coincidence loss all measurements obtained under the same experimental conditions (ori"ce and manometer volumes). Applied in reverse, these expressions are equally suitable for correcting the predicted cell number instead, by replacing N by c(1!e\, A), the true count reduced by coincidence. One may be tempted to correct predicted cell number in this way without "rst "tting independent calibration data for an estimate of c, by including c as an additional parameter of the model. It is a seductive idea, avoiding the need to perform a set of dilution experiments by giving up a degree of freedom*seductive and misguided. The experimental data tend to drop further and further below the predicted curve as time increases (Helmstetter et al., 2001), for reasons that are not at all clear. Using c as an adjustable parameter in the baby-machine model would thus not only decrease that part of the
A model to describe the behaviour of cells produced by the baby machine has been presented in some detail, and an example given of a "t to data from E. coli cells grown in glucose minimal medium with a doubling time of 41 min. A procedure for correcting the coincidence within the ori"ce of an electronic particle counter is also provided and shown to be quite accurate. The "t of the model is outstanding, as judged by the precision with which the various parameters are estimated, by the magnitude of the residuals, and by the runs test. Such attributes are an indication of what can be expected from a technique that yields high-quality synchrony of minimally disturbed cells. Results that di!er substantially from these usually occur because either the cells are less than optimally synchronous or they have been disturbed by the synchronization procedure, depending on the nature of the di!erence; in particular, data displaying a higher degree or a more extended duration of synchrony probably re#ect an arti"cial alignment of cellgrowth properties caused by cell perturbation, an altered physiology. Cell-cycle-dependent output generated in this way is always suspect and should be interpreted with great caution. It is clear that the information furnished by the model is useful, in the case of E. coli, both to quantify the properties of the synchrony and to characterize the baby machine itself (Grover & Helmstetter, 1995; Helmstetter et al., 2001). But what about animal cells? Will the model apply to them as well? Perhaps, but certainly not in its present form. At least one basic modi"cation will be required and at least one extension. Bacterial cells have an interdivision-time distribution f (q) that follows a Pearson type III, as shown by Powell (1956) many years ago; animal cells do not. There is still considerable
SYNCHRONY OF BABY-MACHINE CELLS*A MODEL
disagreement in the literature concerning even the most basic features of the division cycle in animal cells (Cooper, 1991), but if we adopt the pragmatic approach taken by Powell (1958), to use any frequency function that "ts adequately, then the most likely candidate distribution (Brooks, 1981) would seem to be that originally proposed by Kubitschek (1971): a reciprocalnormal. The reciprocal-normal distribution is not user-friendly; it cannot be expressed in closed form, there are no tables or approximations or asymptotic expressions, and its moments do not exist; calculations of the steady-state age distribution l(a) must be carried out numerically and iteratively, a rather tedious procedure. But there is a side bene"t: it allows one to compute mother}daughter and sister}sister correlations. It is a common observation (Powell, 1958) that in bacteria mother}daughter interdivision times are correlated (and, as a result, so are those between sisters). Unfortunately, the Pearson type III distribution does not allow the incorporation of such a relationship and consequently this feature is absent from the current model*a limitation we have not been able to overcome. But the normal (and hence reciprocal-normal) distribution does (Grover & Woldringh, 2001), quite easily. The extension concerns cell morbidity. It is well known (Powell, 1958) that bacteria grow and divide essentially without loss, at least over the time scale of a laboratory experiment, but such may not be the case with animal cells, which are far more delicate than free-living bacteria. Thus, when applied to animal cells, the model will have to allow for possible sterility, cells that continue to grow but stop dividing, as well as for outright cell death. Work along those lines is now in progress.
REFERENCES BROOKS, R. F. (1981). Variability in the cell cycle and the control of proliferation. In: ¹he Cell Cycle (John, P. C. L., ed), pp. 35}61. Cambridge, UK: Cambridge University Press. COOPER, S. (1991). Bacterial Growth and Division, pp. 391}419. San Diego, CA: Academic Press. GROVER, N. B. & HELMSTETTER, C. E. (1995). Characterization of cell-cycle-speci"c events in synchronous cultures of
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Escherichia coli: a theoretical evaluation. Microbiology 141, 59}62. GROVER, N. B. & WOLDRINGH, C. L. (2001). Dimensional regulation of cell-cycle events in Escherichia coli during steady-state growth. Microbiology 147, 171}181. HELMSTETTER, C. E. (1969). Methods for studying the microbial division cycle. In: Methods in Microbiology (Norris, J. R. & Ribbons, D. W., eds), Vol. 1, pp. 327}363. London: Academic Press. HELMSTETTER, C. E. & PIERUCCI, O. (1976). DNA synthesis during the division cycle of three substrains of Escherichia coli B/r. J. Mol. Biol. 102, 477}486. HELMSTETTER, C. E., THORNTON, M. & GROVER, N. B. (2001). Cell-cycle research with synchronous cultures: An evaluation. Biochimie 83, 83}89. IMSL. (1995). Math and stat libraries. In: Fortran Powerstation Professional Edition version 4.0. Redmond, WA: Microsoft. KUBITSCHEK, H. E. (1971). The distribution of cell generation times. Cell ¹issue Kinet. 4, 113}122. POWELL, E. O. (1956). Growth rate and generation time of bacteria, with special reference to continuous culture. J. Gen. Microbiol. 15, 492}511. POWELL, E. O. (1958). An outline of the pattern of bacterial generation times. J. Gen. Microbiol. 18, 382}417.
APPENDIX A Let i be the number of bacteria within the e!ective measuring volume of the electronic particle counter. Then i is distributed according to the Poisson distribution: the probability P(i) of "nding exactly i cells within the ori"ce is P(i)"e\( G/i! where is the parameter of the distribution. It is easy to show that, as expected, P(i)"1, since P(i)" e\( G/i! G G G "e\([1# # /2!#2], and the quantity in brackets is just the series expansion of e(, which converges for all "nite values of . Furthermore, the mean value of i is, by de"nition, iP(i)/ P(i)" iP(i), since the G G G denominator is unity and the "rst term in the numerator is 0. But iP(i)" i (e\( G)/i! G G " e\( G/(i!1)!" (e\( G\)/(i!1)! G G " e\( H/j!" , where i!1 has been H replaced by j, making the expression following the "nal summation sign identical in form to the de"nition of P(i); its sum, therefore, is unity, and so the average value of i is given by , the parameter of the distribution. Consider now an actual measurement M. The true number of counts N will be proportional to P (1)#2P (2)#3P (3)#2" iP (i)" , G so that N"c , where c is the constant of proportionality and is, in e!ect, the ratio between the volume of the manometer and that of the
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ori"ce; because of coincidence, however, the value of M will only be proportional to P(1)# P (2)#P (3)#2" P (i)" P (i ) G G !P(0)"1!e\(, so that M"c(1!e\( ). If we make a series of measurements M , M , M ,2, M at known dilutions, then we have P r equations in two unknowns, c and , which can be "tted using standard nonlinear least-squares techniques. In particular, if successive dilutions are by a factor of 2, then M "c(1!e\(), M "c(1!e\ (),2, M "c(1!e\ (), where is the (unknown) average number of
cells in the measuring ori"ce at the highest concentration M . Once c has been estimated from the leastsquares "t to the calibration data, it is a simple matter to correct for coincidence loss all measurements obtained under the same experimental conditions (ori"ce and manometer volumes). The value of corresponding to any particular measurement M is obtained by solving M" c(1!e\() and then it is substituted into the expression for the true cell count N; in symbols, N"c "c ln[c/(c!M)].