Vol. 153, No. 3JOURNAL OF BACTERIOLOGY, Mar. 1983, P. 1379-13870021-9193/83/031379-09$02.00/0Copyright 0 1983, American Society for Microbiology

Cell Elongation and Division Probability During theEscherichia coli Growth CycleH. E. KUBITSCHEKI* AND C. L. WOLDRINGH2

Division of Biological and Medical Research, Argonne National Laboratory, Argonne, Illinois 60439,1 andDepartment of Electron Microscopy and Molecular Cytology, University ofAmsterdam, Amsterdam, The

Netherlands2

Received 30 March 1982/Accepted 13 December 1982

A new method is presented for determining the growth rate and the probabilityof cell division (separation) during the cell cycle, using size distributions of cellpopulations grown under steady-state conditions. The method utilizes the cell life-length distribution, i.e., the probability that a cell will have any specific sizeduring its life history. This method was used to analyze cell length distributions ofsix cultures of Escherichia coli, for which doubling times varied from 19 to 125min. The results for each culture are in good agreement with a single model ofgrowth and division kinetics: exponential elongation of cells during growth phaseof the cycle, and normal distributions of length at birth and at division. Theaverage value of the coefficient of variation was 13.5% for all strains and growthrates. These results, based upon 5,955 observations, support and extend earlierproposals that growth and division patterns ofE. coli are similar at all growth ratesand, in addition, identify the general growth pattern of these cells to beexponential.

Studies of cell elongation of rod-shaped bacte-rial cells are limited by the small size of bacteria,whose dimensions approach the optical limits ofmicroscopic resolution. Electron microscopyprovides an alternate approach with sufficientresolution for growth studies. With this ap-proach, elongation patterns can be determinedfrom observed distributions of cell length byapplying the procedure of Collins and Richmond(1), which corrects the observed distribution tothat representing cell growth by removing thecontributions from birth and division (separa-tion) of cells.A major difficulty with the Collins-Richmond

procedure is that the distributions of cell size atbirth and at division are almost invariably un-known and must therefore be assumed or de-rived from the distribution of deeply constrictedcells (4). Moreover, the variability in cell lengthat birth and at division is usually so large that theassumed distributions for birth and division ex-tend across most of the cell division cycle. As aresult, errors in these assumed distributions canaffect the growth rates calculated throughout thecell cycle. In addition, the Collins-Richmondanalysis is strongly influenced by statistical vari-ation, and for this reason it is necessary toapproximate the observed data with a smoothlyvarying curve. The use of this smoothing tech-nique introduces further errors ofunknown mag-nitude.

We present here the results of a new methodof analysis of cell length or other growth param-eters that does not require any derivation or useof smoothing functions. This method, describedin detail in the Appendix, depends upon trans-formation of the observed population distribu-tion to obtain the life-length distribution, i.e.,the probability that a cell will have any specificsize at some time during its life history. Thetransformation used depends upon the particulargrowth pattern assumed-for example, whethercell growth is linear or exponential during thecycle-and results in a distribution in which theascending and descending limbs represent thecumulative transform distributions for birth anddivision, respectively. As described below, themethod then compares the observed life-length(cumulative) birth distribution with that calculat-ed from the cumulative division distribution.These are identical when the transformationrepresents the true growth pattern.Our method of analysis has two major advan-

tages over the Collins-Richmond procedure, atleast insofar as that method has been employed(1, 2, 5, 6). First, growth rate patterns can betested for the degree to which they describe theexperimental data. Second, birth and divisiondistributions are not assumed or derived but aredirectly obtained from the experimental data.

In the present study, we have identified thegrowth patterns for Escherichia coli, assuming

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1380 KUBITSCHEK AND WOLDRINGH

I-i L-

1 2 3 4 5 6

L, pm

FIG. 1. Cell length distribution for E. coli K-12 CR34 grown at a doubling time of 19 min.

equal daughter cell sizes at division. Earlierstudies by Trueba et al. (16) had already indicat-ed that the basic course of cell growth was verysimilar for different strains of E. coli. They alsofound that strains differed in their degrees ofinequality of size at division and in the durationof their cell constriction periods, and that thesecaused small differences in the growth pattern.Since our primary interest in this study was toexamine possible similarities of growth, ratherthan the differences arising from these moreminor effects, we excluded the effects of divisioninequalities.

MATERIALS AND METHODSThe bacterial strains used were E. coli strains B/r

A(ATCC 12407), B/r H266, B/r K, and K-12 CR34. Thecells were grown in a minimal salts solution containingdifferent supplements as indicated previously (17).Growth conditions, preparation of cells for electronmicroscopy, and measurements of cell dimensionshave been described elsewhere (16, 17).

RESULTSIn agreement with earlier observations (16),

observed distributions of cell length, A(L), weresimilar in shape for each of the six populations.The distribution for E. coli K-12 strain CR34 isshown as an example in Fig. 1. The mean cell

length L of this distribution was 3.340 ,m (n =

1,264). In the following, the life-length methodof analysis is presented in detail for this culture.We have followed the notation for the frequencyfunctions proposed by Koppes et al. (6) and thatof Painter and Marr (11), except that the sub-script e has been deleted. These symbols and therelationships of the frequency functions to thelife-length distributions are shown in Table 1.To construct the corresponding life-length dis-

tributions for linear growth, Ali, and for expo-nential growth, xp, the values in Fig. 1 weremultiplied by e or by L2, respectively. Theseand successive steps in the procedure are de-scribed in detail in the Appendix. The amplitudeof each distribution was normalized by multiply-ing by LO/A, where L" is the mean cell length atbirth and A is the area of the modified distribu-tion; that is, for exponential growth

Aexp = f A(L) dL

and for linear growth by

Alin = |eul/ A(L) dL

(1)

(2)

An initial estimate of the mean birth length was

TABLE 1. Symbols used and relationships between functions

Avg Extant Cumulative Life-lengthGrowth stage length' frequency frequency distnbutionfuinction" function"

Newly born cells L0 = fLoLdL WC - K T(L)tAt separation L. 4) = foL + dL (Ic = K T(L)41All cells E A Ac = K T(L) A(L)

a From Koppes et al. (6)b From Painter and Maff (11), except that the subunit e has been deleted.c K T(L) is the growth function (see the text).

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CELL ELONGATION IN E. COLI GROWTH CYCLE 1381

1000

A

50

Ea

0

2 3 4 5

L, Fm6

FIG. 2. Observed life-length distributions for E. coli CR34, assuming exponential or linear cell elongation.Symbols: -, exponential elongation; -----, linear elongation.

made from the observed mean cell length of theoriginal distribution by use of the relationshipgiven in reference 8,

Lo * 0.7 L (3)The value of Lo was 2.338 ,um. Values of Aexpand Alin were estimated from equations 1 and 2by Simpson's rule for areas and were 1,235 and297 ,um, respectively. The corresponding nor-

malized life-length distributions Aexp and Alin forexponential and for linear growth are shown inFig. 2.

Estimated values of P1(L), the transform dis-tributions of cell lengths at birth, were read

OltA299U.

T4116z

5

98

95

90

80706050

403020

105

2l

2 2.5 3

directly from the life-length distributions sinceT1(L) = A(L) on the ascending limb of thedistributions. Values of T2(L), the cumulativedistributions of birth lengths estimated fromdividing cells, were determined from T2(L) = 1- A(L) for values ofA on the descending limb ofthe distributions. These distributions for P1(L)and T2(L) are shown in Fig. 3, both for exponen-tial and for linear growth.The experimental distributions appear to be

fitted well with straight lines on arithmetic prob-ability coordinates (Fig. 3), suggesting that theyrepresent normal distributions. Therefore, eachof the four distributions in this figure was tested

2 2.5 3L, ,zm

FIG. 3. Observed and fitted cumulative length distributions for IV, and I2, assuming exponential or lineargrowth of E. coli CR34. Symbols: 0, experimental values for If,; 0, experimental values for AI2; -, linearregressions for A1; ------ linear regressions for P2.

I ,II I0 /

Exponential Growth Linear Growth

'PI,0 0 *12

0 -

0

''

*2

i II

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1382 KUBITSCHEK AND WOLDRINGH

TABLE 2. Comparison of the slopes of If, and T2 for exponential and for linear cell elongation

Strain Doubling Exponential elongation Linear elongationtime (min) cvla CV2a p cv1 CV2 P

A 60 7.9 (3)b 12.9 (8) NSc 4.8 (3) 15.4 (7) <0.01A 125 11.2 (4) 9.9 (7) NS 7.5 (4) 15.9 (9) <0.001H266 72 15.6 (5) 15.7 (8) NS 12.7 (4) 26.4 (11) <0.05K 90 17.4 (4) 14.8 (8) NS 15.5 (4) 21.3 (8)K 106 16.7 (5) 15.4 (11) NS 5.3 (6) 7.4 (11) <0.01CR34 19 15.5 (5) 12.8 (9) NS 11.1 (4) 17.3 (8) <0.001

a CV, and CV2 are the coefficients of variation of I' and T2, and were approximately inversely proportional tothe slopes when the data were plotted on arithmetic probability coordinates.

b Numbers of population values fitted are shown in parentheses.c NS, Not significantly different.d Probability that the means were from the same population; the slopes were not significantly different in this

case.

by chi-square analysis for goodness of fit to thelinear regressions depicted in this figure. Proba-bility values for normality of T1 and T2 weregreater than 0.1 and 0.75 for exponential elonga-tion, respectively, and greater than 0.05 and0.975 for linear elongation. The data for all of theother experimental distributions also were con-sistent with the hypothesis of normality.To examine the assumptions of exponential or

linear growth, the slopes of TI and '2 werecompared by Student's t test for each growthpattern (18). The slopes were significantly differ-ent when linear elongation was assumed, exceptfor one culture, but for this the means weresignificantly different (8; Table 2). Consequent-ly, the hypothesis of linear elongation is reject-ed. The assumptiod of exponential elongationgave distributions for T1 and T2 that had thesame slopes and means within experimentalerror, so exponential elongation is consistentwith the experimental observations.Assuming exponential elongation, then, linear

regressions were fitted to the combined data forP1 and T2. These regressions provided im-proved values for the mean cell length at birthand for the coefficient of variation (CV) of eachdistribution (Table 3). Examination of the values

for CV and their 95% confidence limits showsthat the values for CV do not differ significantlyfrom one another. Thus, these results are consis-tent with a single value for CV for each culture,and therefore with a single growth pattern.To determine the mean value for CV for all of

the cultures, the results for each experimentwere normalized to unit length at birth, using thevalues for Lo shown in Table 3, and pooled toform a single composite distribution. Theselength-normalized composite data were groupedinto intervals of (L/Lo) = 0.1 and again fitted bylinear regression (Fig. 4), giving a value for CVof 13.46%o, with upper and lower 95% confidencelimits of 15X09 and 12.14%, respectively. Thesevalues are plotted in Fig. 5 together with thecorresponding values for each of the individualcultures.Of the 5,955 values comprising the composite

distribution of cell lengths for all experiments(Fig. 6), 5 values are not shown because they areoff the scale of this graph. One cell from theculture for strain K at 106 min was exceedinglyshort, with a relative length x = L/Lo of only0.34, well separated from its nearest neighbors.This may have been a dead cell or one of theDNA-less cells which are regularly observed in

TABLE 3. Estimates of Lo and CV from linear regressions to the combined data for *I and l2CV 95% confidence

Strain Doubling na Lo (Lm) CV M limitsLower Upper

A 60 670 11 1.897 11.75 8.87 17.21A 125 1,187 11 1.220 10.41 8.78 12.80H266 72 468 13 2.143 15.52 12.24 21.40K 90 493 12 1.895 13.31 10.64 17.75K 106 1,873 16 2.116 16.04 13.74 18.32CR34 19 1,264 14 2.439 13.75 12.23 15.68

a Total number of cells.b Number of points fitted.

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CELL ELONGATION IN E. COLI GROWTH CYCLE 1383

9998

9590

~3 10

a

l80

50

2

l3

0.8 0.9 1L/Lo

FIG. 4. Cumulative frequencies:data for E. coli. The data for eachments were normalized to unit birtcomposite frequency distribution waing values over intervals of IL/Lo=values for AP,; *, values for '2-

slowlv 2rowino cultures of E.

from strain CR34, and the remaining eight werefrom the Fil- B/r strains. One was from strain

*,V ~H266 (x = 2.75), one was from the 60-minculture of strain A (x = 3.25), and the remainingsix were from the 106-min culture of strain K. Itis not known why this culture of strain K con-tributed so many long cells, especially since the90-min culture of the same strain contributednone. Long forms are known to be produced,however, even in the Fil- B/r strains underconditions of stress (10), so accidental or sponta-neous triggering of the long-form response can-not be ruled out.The larger peak at smaller cell lengths, com-

1.1 1.2 posed of 5,940 observations, presumably repre-sents the usual growth mode of E. coli. These

for the composite data were fitted with the theoretical distributionof the six experi- for exponential elongation and normal distribu-h length, and the tions of lengths at birth and at division with CVis found by group- = 13.46% (smooth line). Comparison of the0.1. Symbols: 0, goodness of fit by chi-square analysis gave a

value of P > 0.05 for 0.7 c x c 2.7 (i.e., forexperimental frequencies >1%), demonstratingconsistency between the experimental and theo-

coli B/r K and retical distributions..3%PvIY 51FVVIII5 W . WI9v &. %,vsg "fl A`% "IM

H266 (15). At the other extreme, four cells werelonger than any shown in Fig. 6, and theselengths were scattered singly over the largerintervals. The relative length of the longest wasx = 4.26, and this cell was unconstricted, acharacteristic of filamentous cells. All of theseunusually long cells were from the only Fil+culture, E. coli K-12 CR34, a strain known to besusceptible to filamentation.The bulk of the remaining 5,950 data points

are centered about a peak at x = 1.5 (Fig. 6,smooth line), and the remaining 10 points areoutlying at values greater than x = 2.7. Again,two of these long cells (x = 2.75, 2.85) came

DISCUSSIONShape and average cell size change markedly

when cultures of E. coli are grown at differentrates (13, 14), but these changes are apparentlyunrelated to the pattern of growth and division.Nevertheless, the experimental results for thesix steady-state cultures of E. coli in this study,grown at doubling times from 19 to 125 min(Table 2), support a single pattern of growth anddivision for normal cell growth in these popula-tions (Fig. 6, smooth line). For the transformdistribution, this growth and division pattern ischaracterized by a normal distribution of cell

I -I

20

0 - I I woge

0 1 I

0 50 o0oDoubling Tinm, mnu

150 200

FIG. 5. Experimental values of CV for the transformed distributions as a function of culture doubling time.Individual values are shown with the extents of their 95% confidence internals. The mean value ( ) and the95% confidence limits (-----) were obtained from the regression to all of the data shown in Fig. 4.

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1384 KUBITSCHEK AND WOLDRINGH

50k

25r

0 1 1.5 2 2.5 3L/Lo

FIG. 6. Composite life-length distribution for all six cultures. Circles indicate experimental values, and thesmooth line shows the theoretical distribution expected for normal distributions of cell length at birth and atdivision, each with CV = 13.46%, and exponential cell elongation between divisions.

lengths at birth and at division, with a CV of13.5%, and exponential elongation of cells be-tween divisions (Table 3; Fig. 5 and 6).This is the first time that the cell length

distributions at birth and division have beenexperimentally identified. Heretofore, for pur-poses of analysis, it has been assumed that thepopulation distributions were normal distribu-tions at birth and at division (1, 2, 5, 6) or log-normal (4), rather than the normal distribution oflength-dependent probabilities of the division ofindividual cells that we have observed. Thus,the extant population distributions of celllengths are, in fact, normal distributions nega-tively skewed by the growth function.Our results confirm and extend earlier obser-

vations of Trueba et al. (16), who found thatgrowth patterns are similar for several strains ofE. coli growing as exponential-phase cultures orin chemostats. Since these authors found similargrowth patterns for a variety of different kinds ofrod-shaped bacteria, including E. coli, Salmo-nella typhimurium, and probably Bacillus mega-terium and Caulobacter crescentus, it may bethat all of these are characterized by exponentialcell elongation and a normal division distribu-tion.Although a single growth pattern is obtained

at the level of analysis employed here, we havenot included the effects of unequal cell division.The degree of inequality depends upon cellshape (15), which varies with strain and cultureconditions but has been shown to lead to onlyminor differences among observed populationdistributions (16). In every case, however, un-equal cell division would be expected to yield abirth length distribution that is slightly broaderthan that for the distribution of lengths at celldivision. The values in Table 1 seem to showsome evidence for the increase in the value of

CV, since CV, > CV2 for four of the six cultures(Table 2). However, since the two distributionsdo not differ significantly (for exponential elon-gation), the length distribution data we haveused cannot provide information on differencesbetween the two length distributions or upon thedegree of precision in cell size at division.

In addition, of course, the general growthpattern cannot account for the small proportion(0.25%) of unusually long cells, which are notpart of the normal cell length distribution (Fig.6). We suppose that these unusually long cellsarise through failure of some control(s) forgrowth or division, but they occur too rarely andperhaps too erratically to provide reliable infor-mation about their growth patterns.

Recently, evidence was presented for bilinearincrease in average cell volume and surface area(8), with rates of increase that doubled duringmidcycle. At first sight, the results in favor ofexponential elongation might appear to be in-compatible with bilinear increase in cell volume.This is not necessarily the case, however, be-cause the volume and area of a rod-shaped celldepend upon its diameter as well as its lengthand, as Trueba and Woldringh (17) have shown,cell diameters decrease during the first part ofthe E. coli cell cycle in all but the most rapidlygrowing cultures. Clearly, a decrease in diame-ter during the cell cycle would cause the cellvolume to increase more slowly than exponen-tially.More difficult to understand is the relative

shape constancy of bacterial volume distribu-tions observed with a Coulter Counter (7, 16).Differences in the shape of volume distributionshave been found by others (9, 14), but suchresults have been attributed to insufficient reso-lution of the apparatus, distortion by the use ofunduly short apertures, or both (7). Neverthe-

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CELL ELONGATION IN E. COLI GROWTH CYCLE 1385

less, if the relative cell age at which growth ratesdouble during the cycle shifts with culture dou-bling time, then the shape of the volume distri-bution would not be expected to remain con-stant, independent of culture growth rate. Asingle volume distribution could be expected,however, if rate doubling always occurred at thesame age during the cell cycle. With the life-length method of analysis, it may be possible toresolve this question.

APPENDIX AThe life-length method of analysis. We present here

an alternative to the Collins-Richmond (1) approach,based on the analysis of life-length distributions. Ourmethod requires that the growth function be assumed,but it has the advantage that the calculated distribu-tions can be compared for consistency with the ob-served results. In addition, this method determines thedistributions of cell lengths at birth and at divisionfrom the experimental data. Furthermore, althoughthe method is described here for length extensionduring the cell cycle, it is applicable to other measur-able cell parameters such as cell surface area, volume,and mass.The stable frequency distribution A(L) of cell length

L in a steady-state population depends upon thepattern of cell growth during the cycle and upon theextant distributions of cell length at birth, +j(L), and atseparation, 4(L). As presented here, the life-lengthmethod is a first approximation in that it neglects thecontribution of variability, usually minor, arising fromunequal cell division.

If all cells were born with exactly the same length Loand separated at length 2L0, then the frequency distri-bution of such an idealized population would dependonly upon growth rate. Idealized population distribu-tions of this kind have been determined (see, forexample, reference 3) for cells growing at a constantrate (linear elongation),

Ni, = 4 (In 2) 2--/L4/o= (4/L) e-L/L, Lo < L < 2Lo(A-1)

as well as for cells growing at rates proportional totheir size (exponential elongation),

Nexp = 2LOWL2= L/(L2 In 2), Lo < L < 2Lo(A-2)

Here, L is the mean cell length of the population, andthese distributions vanish for L < Lo or L > 2Lo. Forlinear elongation

Lo =LIn 2 (A-3)and for exponential elongation

Lo = L/(2 In 2) (A-4)Extant length distributions for the two modes ofgrowth are shown in Fig. 7A along with a distributionA(L) for a population in which cell lengths vary at birthand at division.The corresponding cell life-length distributions in

Fig. 7B give the probability that a cell will have alength L at some point during its life cycle. Life-lengthdistributions are obtained by transformation of each

population distribution with its appropriate growthfunction. For linear and exponential growth distribu-tions, these growth functions are reciprocals of Nlinand Ne,p, respectively. Thus, for linear growth, thecorresponding life-length distribution is

Alin(L) =4

{±4 eL/L} = 1 for Lo S L < 2Lo

=0 for L < Lo or L > 2Lo (A-5)

with the rectangular shape shown in Fig. 7B. The life-length distribution for idealized exponential growthhas exactly the same rectangular shape.

Similarly, the life-length distribution AC(L) in Fig.7B is the transform of the population distribution X(L)in Fig. 7A, given by

AC(L) = K T(L) A(L) (A-6)

where, for convenience, numerical factors have beengathered into the constant K, and the subscript c hasbeen added to distinguish the transform function fromthe frequency function (Table 1). As above, for linearelongation, T(L) = e-L/L; for exponential growth,T(L) = L2. Because the area of each normalizeddistribution is Lo (see Fig. 7A), the value of K can bedetermined from the area A of the unnormalizeddistribution:

K T(L) X(L) dL = KA = Lo

Therefore,

AC(L) = (LoJA) T(L) X(L) (A-7)In general, the probability that a cell will have some

length L during the growth cycle is determined by twoparameters, the probability that the cell was born withlength less than or equal to L, and the probability thatit did not divide before reaching length L, or

AC(L) = 'Ic(L) [1 - (L)] (A-8)

NV,(L) and b(DL) are the cumulative transform distri-butions for birth and division, respectively (Fig. 7):

LLAP,(L) = K T(L) | (L) dL

Jo

and

OD(L) = K T(L) 4 (L) dL

Life-length distributions are of two qualitativelydifferent types, depending upon the degree of variabili-ty in cell lengths at birth and at division. If thisvariability is small compared with mean cell birth size,and if birth and division distributions of cell length donot overlap, then the life-length distribution will reacha maximum probability amplitude of unity (P = 1) atsome point during the cell cycle, or even a plateau atthis value, as shown for each of the distributions inFig. 7B. However, if the variability is broad, then P <

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1386 KUBITSCHEK AND WOLDRINGH

2 N

U-~ ~ ~ u

1. F---

CT

1.0 Alin, A

1 1.5 2 2.5L

FIG. 7. Cell length distributions. (A) Frequency distributions. Symbols: -----, NeX, exponentiallyelongating cells without variability at birth or division; ---, N13,,, linearly elongating cells without variability atbirth or division; -, X(L), exponentially growing cells with CV = 10%o at both birth and division. (B) Life-length transform distributions corresponding to the population distributions in A. Symbols: ---, AeX or A1,,,without variability at birth or division;-, exponential growth with a value ofCV = 10%o. (C) Decomposition ofa life-length distribution. The life-length distribution AC(L) in B is resolved into the difference of the distributionof cell lengths at birth 'I'(L) and the distribution of cell lengths at division 4c(L).

1 for the entire distribution. The method of analysisdeveloped in this paper is for distributions of the firsttype.When the distributions in cell length at birth and at

division do not overlap (Fig. 7B), the life-length distri-bution can be separated into three regions: an ascend-ing limb, a flat top, and a descending limb. Then, since

(L) = 0 in the first and second regions and TP(L) =1 in the third region, equation A-8 becomes

AC(L) = 'P.(L) - (L) (A-9)

An analytical derivation of this equation was givenearlier for the case of exponential elongation (refer-

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CELL ELONGATION IN E. COLI GROWTH CYCLE 1387

ence 11, equation 16), but equation A-9 is valid for anypattern of growth K T(L) between divisions.

In addition, in the first and second regions

NQ(L) = A<(L) (A-10)

and in the third region

SD(L) = 1 - AC(L) (A-11)

Assuming equal lengths of newly formed daughtercells,

'l'(L) = SQ(2L) (A-12)

Then from equations A-11 and A-12, we may obtain asecond estimate of (L):

TQ(L) = 1 - Aj(2L) (A-13)

We designated values of T'(L) obtained from equationA-10 as TI and designated values of T'(L) obtainedfrom equation A-13 as T2. P, and T2 provide indepen-dent estimates of the transform distribution of celllengths at birth. When the curves generated by the twosets of data are indistinguishable, the assumed growthrate function is consistent with the data (see Fig. 3, forexample).

In practice, the method is iterative. The value of LOis estimated from equation 3 or from the graph of thelife-length distribution. The area A of the distributionis estimated from the experimental data by usingSinmpson's rule. Values of A1 and T2 are calculated,and curves are fitted to the data for these distributions.If the curves do not differ significantly, the averagevalue of the means for T, and T2 is calculated as animproved estimate for Lo, and that value is used forthe second round of iteration, and so on until thedesired accuracy is obtained. Convergence is rapid,and in our study, usually no more than a second roundof iteration was needed.

ACKNOWLEDGMENTS

We thank F. J. Trueba for providing access to his computer-filed data bank, W. N. Kubitschek for advice on statisticalprocedures, and A. L. Koch and W. Voorn for helpfuldiscussions.

This work was supported by the U.S. Department of Energyunder contract no. W-31-109-ENG-38.

LITERATURE CITED

1. Collins, J. F., and M. H. Richmond. 1962. Rate of growthof Bacillus cereus between divisions. J. Gen. Microbiol.28:15-33.

2. Harvey, R. J., A. G. Marr, and P. R. Painter. 1967.Kinetics of growth of individual cells of Escherichia coliand Azotobacter agilis. J. Bacteriol. 93:605-617.

3. Koch, A. L., and M. Schaechter. 1962. A model forstatistics of the cell division process. J. Gen. Microbiol.29:435-454.

4. Koppes, L. J. H., and N. Nannlnga. 1980. Positive correla-tion between size at initiation of chromosome replicationin Escherichia coli and size at initiation of cell constric-tion. J. Bacteriol. 143:89-99.

5. Koppes, L. J. H., N. Overbeeke, and N. Nanninga. 1978.DNA replication pattern and cell wall growth in Esche-richia coli PAT 84. J. Bacteriol. 133:1053-1061.

6. Koppes, L. J. H., C. L. Woldringh, and N. Nanninga.1978. Size variations and correlation of different cell cycleevents in slow-growing Escherichia coli. J. Bacteriol.134:423-433.

7. Kubitschek, H. E. 1969. Growth during the bacterial cellcycle: analysis of cell size distributions. Biophys. J.9:792-809.

8. Kubitschek, H. E. 1981. Bilinear cell growth of Escherich-ia coli. J. Bacteriol. 148:730-733.

9. Marr, A. G., P. R. Painter, and E. H. Nilson. 1969.Growth and division of individual bacteria. Symp. Soc.Gen. Microbiol. 19:237-259.

10. Mon, R. J., and F. I. McLean. 1961. The nature andradiation sensitivity of the long fonhis of Escherichia colistrain B/r. J. Gen. Microbiol. 25:29-39.

11. Painter, P. R., and A. G. Marr. 1968. Mathematics ofmicrobial populations. Annu. Rev. Microbiol. 22:519-548.

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