Lemna paucicostata Hegelm. 6746 - NCBI

Plant Physiol. (1980) 65, 913-923 0032-0889/80/65/0913/1 1/$00.50/0

Lemna paucicostata Hegelm. 6746 LIFE CYCLE AND CHARACTERIZATION OF THE COLONY TYPES IN A POPULATION Received for publication July 24, 1979 and in revised form November 29, 1979

ANNE H. DATKO, S. HARVEY MUDD, AND JOHN GIOVANELLI National Institute of Mental Health, Laboratory of General and Comparative Biochemistry, Bethesda, Maryland 20205 ABSTRACT The sequence offrond emergence and the intervals required for daughter colony separation have been determined for Lemna pauciostaa Hegelim. 6746 growing under standardized conditions. After separation of a new mother colony, the first daughter colony is produced from the left meristematic pocket and separates after approximately 60 hours, the second daughter, produced from the right pocket, separates after a further 30 hours, and the third daughter, again from the left, after a further 40 hours. The pattern of alternating longer and shorter intervals for separation of daughters continues throughout the life of the mother colony. Protein contents of fronds and whole colonies were determined either by chemical methods or by labeling to isotopic equilibrium with 35so42-. The smallest fronds measured were 5 to 6% of the final area they would attain when fuUly expanded and contained 10% of the protein they would finaly contain. Thus, most protein accumulation occurred during the phase of rapid frond expansion rather than in an earlier primordial stage. The specific rates of protein accumulation were 0.027 to 0.030 micrograms per microgram protein per hour during rapid frond expansion and relatively constant thereafter at 0.014 to 0.019 for whole colonies until at least four daughter colonies had separated. A substantial amount of 36S is transferred directly from the mother frond to its attached daughter fronds. The results are consistent with the transfer of soluble compounds arising through turnover of protein in the mother frond. Tree diagrams and a mathematical treatment were utilized to relate the distribution of colony types and the over-all doubling time in a large population of Lemna colonies to the doubling times and modes of division of the individual colonies within that population. It was shown that a population with the growth properties described increases exponentially and that within a relatively few generations reaches an equilibrium population distribution which is independent of the initial proportions of the members of the population. The measured distribution of colony types producing odd-numbered as compared to those producing even-numbered daughters was the same as that predicted by the theoretical analyses. Once a mass culture of Lemna paucicostata has attained its equilibrium distribution of colony types, under our standard conditions the proportions of colonies in which the indicated daughter will be next to separate will be: 60%, first; 15%, second; 11%, third; 5%, fourth; 8% fifth or higher. Thus, about 91% of colonies will have produced no more than four daughter colonies. For at least this period the specific rate of protein accumulation is relatively constant and senescent changes in mother fronds are rare. In the equilibrium distribution, 39% of colonies will themselves be first daughters, 25% will be second daughters, 14% will be third daughters, 9% fourth, and 13% greater than fourth daughters. These combined results suggest that obtaining representative samples of colonies from such a population for biochemical studies should be relatively simple.

microphyte Lemna advantageous (3, 5), and we plan to extend our metabolic, enzymic and labeling studies using this plant. To this end, we have established standard environmental conditions for the mixotrophic growth of Lemna paucicostata Hegelm. 6746' in a medium designed to meet the requirements of such studies

(4).

Here, we present observations concerning the vegetative life cycle of Lemna colonies and the types of colonies present in mass cultures grown under the standard conditions. Such information is important to establish the circumstances under which Lemna may validly be used for biochemical studies. Mass cultures of Lemna contain individual colonies with varying genealogies, of different ages and at nonuniform stages of their life cycles. Our observations provide evidence that such cultures reach a steadystate with respect to the proportional distribution of colony types, and that the different colony types accumulate protein at similar over-all specific rates. METHODS Growth Conditions. The standardized batch culture conditions for mixotrophic growth of L. paucicostata are given in the accompanying paper (4). Medium contained 20 ylM sulfate. Individual colonies were cultured in scintillation vials containing 10 ml of medium an4closed with silicone sponge plugs (Bellco Glass, Inc.) incubated in the same growth areas as batch cultures, but without aeration. To generate colonies labeled to equilibrium with 35S, 35So42(final specific radioactivity 2,600 dpm/nmol) was added to the medium before colonies were cultured in it for either 7 days (about four doublings) or 14 days (about eight doublings). Selection of Colonies. An individual Lemna colony may be designated by a daughter number, to indicate whether it is a first, second, or nth daughter of a frond, and by a mother number, to indicate whether it has itself given rise to zero, one, or n daughters. Thus, a DnMn colony is of unknown history, whereas a DlMO colony is the first daughter of its mother and has not yet given rise to any daughter colonies. To generate samples of DnMO colonies, DnMn colonies at late L-32 or early L-4 stage were selected from batch cultures, and maintained in vials, one or two colonies per vial. Preliminary experiments showed that 15-20%Yo of such colonies will give rise to daughters within the next 18-20 h, a further 3040%o will divide during the subsequent 10-12 h, and the remainder within another 10-15 h incubation. The specific requirements of each experiment determined the number of DnMn colonies initially selected, the length of the interval of collection of daughters, and the frequency of collection. Often, the DnMO colonies which

'This plant was previously designated Lemna perpusilla Torr. 6746 in work from this and other laboratories. 2 For description of nomenclature for fronds and colonies, see legend to Figure 12.

In our studies of sulfur metabolism and homocysteine biosynthesis in green plants, we have found the use of the aquatic 913

914

DATKO, MUDD, AND GIOVANELLI

Plant Physiol. Vol. 65, 1980

FIGS. 1-11. Photographs of L. paucicostata colonies showing stages of the vegetative growth cycle. Diagrams of the stages and description of the terminology are given in Figure 12. 1: Early L-3; 2: mature L-3; 3: L4; 4: L-5; 5: very early L-6; 6: later L-6; 7: late L-6, verging on L-7 (2-1 emerging from 2); 8: R-3 (early L-3 (not shown) has detached from left pocket); 9: R-4; 10: R-5; I 1: R-6, just prior to separation of early L-3 daughter colony from right pocket.

separated during the 12-h period from 20 to 32 h after the initial selection were collected. In some experiments the colonies were collected each hour for the 12 h. As each DnMO colony was collected it was placed individually in a fresh vial and its mother was discarded. The DnMO daughter (an L-3 colony) was easily distinguished from its mother (which after separation was at the R-3 or R-4 stage) and from an undivided DnMn colony (which was at the L-5, L-6, or L-7 stage) (cf. Fig. 1 with Figs. 4-9). The first daughters of DnMO colonies, designated DIMO, are equivalent to the "first daughter" colonies often used in studies of Lemna (12). Measurement of Radioactivity. For measurement of total radioactivity in a single colony or in dissected single fronds, the tissue was placed in a scintillation vial containing 0.05 ml water and 0.4 ml Protosol (New England Nuclear). After 20 min, 10 ml LSC Complete phosphor (Yorktown Research) was added, and radioactivity in the sample was determined 30 h later. This procedure was adopted after it was found in preliminary experiments that the amount of radioactivity measured in such samples increased with time, reaching a plateau value 24-40 h after the tissue was placed in the Protosol-phosphor mixture. Neither maceration of

the tissue in Protosol nor prolonged ( 18 h) incubation of the tissue in Protosol before addition of the phosphor decreased the time required for the radioactivity to reach a constant value, nor did these treatments increase the final measured radioactivity compared to samples that were not macerated or that were incubated for 20 min in Protosol. For determination of radioactivity in the trichloroacetic acidsoluble and -insoluble fractions of single Lemna colonies, the colony to be fractionated, along with nine nonradioactive carrier colonies, was incubated in 0.1 ml cold 10% trichloroacetic acid containing 52 mm DTT for 10 min or longer. The material was homogenized in the cold, and 0.4 ml of the cold trichloroacetic acid-DTT mixture was added before the homogenate was centrifuged. Radioactivity in 0.2-ml aliquots of the supernatant solution was measured using 10 ml Ready-Solv HP (Beckman) phosphor. The remainder of the supernatant solution was decanted and the tube was allowed to drain well. The pellet was incubated in 0.4 ml Protosol at room temperature with occasional shaking until digestion was effected, usually 30 min or less. The solution was quantitatively transferred to a vial, with washes of LSC Complete phosphor, for determination of radioactivity.


915

LEMNA: LIFE CYCLE AND COLONY TYPES

R-5

L-3

R-4

2-2

R-3

L-3

L-3

5

~ .4S'

_12

FIG. 12. Diagram of the vegetative growth cycle of L. paucicostata. Nomenclature of fronds: The oldest, or mother, frond on the colony is denoted M, and its daughter fronds are numbered, in their order of appearance (1, 2, 3, etc.). As each daughter frond gives rise to its daughters, they also are numbered in order of appearance. For example, 1-1 and 1-2 indicate the first and second daughters of daughter frond 1. Nomenclature of colonies: a colony having the largest daughter frond on the left of the meristematic end of the mother frond is termed L, that with the largest on the right, R. The number indicates the total number of visible fronds on the colony. Growth cycle: in the diagram the growth cycle begins at the middle right of the figure. A recently detached L-3 colony (bearing fronds M, 1, and 2) is shown. After a period during which daughter fronds I and 2 expand, three new fronds emerge in the sequence: first daughter frond (1-1) of frond 1, forming an L4 colony; third daughter frond (3) of M, forming an L-5 colony; second daughter frond (1-2) of frond l, forming an L-6 colony. The early L-3 colony made up of fronds 1, 1-1, and 1-2 then detaches, and the original colony (now an R-3) subsequently gives rise to three more new fronds in the sequence: first daughter frond (2-1) of frond 2, forming an R4 colony; fourth daughter frond (4) of M, forming an R-5 colony; second daughter frond (2-2) of frond 2, forming an R-6 colony. The early L-3 colony made up of fronds 2, 2-1 and 2-2 then detaches, leaving the original frond M, bearing daughters 3 and 4 (an L-3 colony) to continue another cycle. Not shown on the diagram are stages L-7 and R-7. Occasionally daughter colony separation is slightly delayed so that, e.g. daughter 2-1 is formed on daughter 2 before daughter I separates. This results in the sequence L-6 to L-7 followed by separation to yield L-3 and R4. In this instance, a free R-3 does not occur. See Figure 7 for a photograph of a very early L-7 stage. Similarly, if frond 3-1 emerges on daughter 3 before daughter 2 separates, an R-7 colony results. This is usually only noted by dissection of the colony, since frond 3-1 grows under, and is hidden by, daughter 2.

For measurement of radioactivity in larger samples of Lemna, 10 radioactive whole colonies were added to 0.3 ml of 10%Yo trichloroacetic acid containing 10 carrier colonies. Fronds M,J1, and 2 dissected from 10 similar radioactive colonies were added to 0.3 ml of 10%o trichloroacetic acid containing 10, 18, or 20 carrier colonies, respectively, so that variation in the total protein/ ml of trichloroacetic acid was minimized. The tissue was homogenized, and 0.7 ml of cold trichloroacetic acid was added before the homogenate was centrifuged. The precipitate was resuspended in 1.0 ml cold trichloroacetic acid, and centrifuged again. Radioactivity in 0.5-ml aliquots of the combined trichloroacetic supernatant solutions was determined using Ready-Solv HP phosphor. The trichloroacetic acid-insoluble material was dissolved in 0.4 ml Protosol, diluted to 2.4 ml with Protosol, and radioactivity in 0.4ml aliquots was determined using LSC Complete phosphor. To prepare samples for determination of both radioactivity and protein content, the procedure given above was followed except that 10 or more radioactive colonies were added to trichloroacetic acid in the absence of carrier colonies, maintaining the ratio of 20 colonies/ml of trichloroacetic acid. Material insoluble in trichloroacetic acid was digested in I N NaOH and examined for protein as described in (4). Radioactivity in aliquots of the digest was measured using Ready-Solv HP phosphor.

RESULTS Vegetative Growth Cycle. A Lemna colony is made up of a mother frond and its daughter fronds, borne from meristematic pockets on both sides of one end of the mother frond (e.g. Fig. 2). In the vegetative reproduction of any Lemna strain, all mother fronds characteristically produce their first daughter fronds from

analogous pockets (6). In L. paucicostata Hegelm. 6746, this is the left pocket (13), that is, the one which occurs on the left side of the longitudinal axis of the mother frond when the end of the mother frond with the meristematic pockets is pointing away from the observer. The second daughter is produced in the pocket on the right side, the third in the left, and so on; Newly detached colonies of L. paucicostata are usually made up of three fronds, the mother frond and its first and second daughter fronds. In the experiments reported here, such colonies were observed at frequent intervals for extended periods to establish the sequence of events which led to formation of their daughter colonies. A reasonably reproducible sequence of frond emergence and daughter colony separation occurred. The sequence was cyclic since the formation of subsequeni pairs of daughter colonies followed the same pattern as the formation of the first pair of daughter colonies. Furthermore, each cycle could be divided into half- or hemicycles, on the basis of whether the daughter which would next separate was located in the left or the right pocket of the mother frond. The cycle could be further divided into stages on the basis of the size, position, and number of fronds on the colony. Figures 1 through 11 are photographs of colonies at the various stages. A diagram of a complete cycle is shown in Figure 12, and in the legend for Figure 12 a description of each event in the cycle is given, along with an explanation of the nomenclature used for fronds and colonies. In one cycle a three-fronded colony gives rise to six new fronds in a manner such that two new fronds are formed on each of the original three fronds, so that each of the original three fronds becomes a three-fronded colony. These colonies are the mother, ready to produce its next daughter colonies, and the two daughter colonies, ready themselves to produce

DATKO, MUD'), AND GIOVANELLI

916

Table I. Time Requiredfor Each Stage of the 1st and 2nd Vegetative Growth Cycles of L. paucicostata Individual colonies were observed each 2 h and the time that each colony required to complete each stage was noted. The experiment was begun with several series of colonies, some of which were sacrificed for determination of protein content as they reached the appropriate stage. The total sample sizes (number of series shown in parentheses) were: time of separation to L-4, 152 colonies (5 series); L-4 to the end of the 1st left hemicycle, 102 (3 series); 1st right hemicycle, 68 colonies (2 series); and 2nd cycle (both left and right hemicycles), 34 (I series). For the 1st cycle, all values given are the means of the values of the series. For the 2nd cycle, the values are the means of the observed times for the individuals in the single series of colonies. Time Stage 2nd Cycle 1st Cycle ± h(± SD SE) 6.8 (+ 4.8 ± 0.8) 34.5 (+ 3.6 ± 1.6) L-3 16.4 (+ 4.8 ± 0.8) 4.6 (+ 0.2 ± 0.1) L-4 3.2 (+ 3.0 ± 0.5) 9.5 (+ 0.7 ± 0.4) L-5 11.5 (+ 5.4 ± 0.9) 5.9 (± 1.4 ± 0.8) L-6 4.8 (± 5.1 ± 1.0) 2.8 (± 1.1 ± 0.6) L-7 42.7 56.6 (± 5.0 ± 2.9) Total, left hemicycle 0.1 R-2 1.7 (± 4.9 ± 1.7) 2.3 (± 1.6 ± 1.2) R-3 9.6 (± 5.2 ± 0.9) 7.3 (± 0.2 ± 0.2) R4 5.1 (± 5.0 ± 0.9) 5.3 (+ 0.8 ± 0.6) R-5 20.5 (+ 6.4 ± 1.1) 16.9 (± 3.3 ± 2.3) R-6 37.0 31.6 (+ 0.7 ± 0.5) Total, right hemicycle 79.7 90.2 (5.8 ± 4.1) Total cycle

100 75 50 cc

25 0

0s

ul)

18 16 14 12 10 DAYS FIG. 13. Intervals between separation of daughter colonies. Forty three Lemna colonies (designated DnMO) were collected within a 12-h period as they detached from their mothers. They were maintained individually in vials. At 12-h intervals they were examined and separation of their daughter colonies was recorded (A). When the DnMO colonies in A gave rise to their first daughters, becoming themselves DnM 1, the first daughter colonies (DIMO) were retained, placed singly in vials, and treated as a second series of colonies (B). All colonies were subcultured after separation of each daughter. By microscopic examination, it was noted that occasionally, among older colonies especially, a frond failed to emerge, or, after emerging, failed to grow. Such colonies were thereafter omitted from the calculations. In the figure, the per cent of colonies separated for each series in each 12-h interval was plotted versus time. For DnMO colonies (A), their detachment from their mother, as well as their production of eight daughters, is shown. For DI MO colonies (B), their detachment from their mother as well as their production of six daughters, is shown. Values for the intervals are given in Table II. 0

2

4

6

8


Table II. Intervals Between Separation of Daughter Colonies The experiment is that described in Figure 13. For method l, the number of hours required for the separation of 50%o of successive daughters for the DnMO and DI MO series were determined from the graph shown in Figure 13, and these values were averaged. For method 2, the interval between daughters is the mean of the measured number of hours (± SD ± SE) for separation of successive daughters for all colonies. The decrease in sample size for daughters I through 6 reflects the number of abnormal colonies excluded from the calculations, whereas for daughters 7 and 8, only the DnMO colonies were observed. Intervals between Daughters Daugher No. Sample Size Method 2 Method I h( SD ± SE) h 67.5(± 9.9± 1.1) 63.0 82 1 7.8 ± 0.9) 29.3 30.6 82 2 9.2 ± 1.0) 43.7 81 42.0 3 33.6 11.5 ± 1.3) 4 75 34.5 52.9 14.5 ± 1.7) 69 50.1 5 40.2(+ 16.4± 2.1) 43.8 63 6 7 59.4 58.2 (20.0 ± 3.9) 26 32.4 35.0 (± 22.2 ± 4.4) 25 8

Table III. Distribution of Colonies in Each Stage in Cultures of L. paucicostata Four cultures were examined to determine the vegetative stage of the 900 colonies they contained. The values given are the means (± SD ± SE) of the per cent of colonies in each stage for the four cultures. Included in "Other' are three colonies which had fewer than three visible fronds and 11 which had more than seven fronds. Distribution Stage % (+ SD ± SE) 50.3 (+ 5.5 ± 2.7) L-3 10.5 (± 2.6 ± 1.3) L-4 9.9 (+ 3.0 ± 1.5) L-5 6.9 (4.0 ± 2.0) L-6 1.2 (+ 1.6 ± 0.8) L-7 78.8 (± 2.3 ± 1.1) Total, left hemicycle 2.3 (± 1.9 ± 0.9) R-3 4.8 (+ 2.5 ± 1.2) R4 5.4 (0.7 ± 0.4) R-5 8.1(±4.7±2.3) R-6 20.6 (+ 2.2 ± 1.1) Total, right hemicycle 0.6(± 0.1 ± 0.1) Other

daughters. Newly detached individual colonies of Lemna were observed to ascertain not only the sequence of frond emergence, but also the time required to pass through each stage of the growth cycle. The times for each stage (i.e. the times between the appearance of new fronds) and the times for completion of each hemicycle (i.e. the times between separations of daughter colonies) are shown in Table I. Of the individual stages, the first L-3 stage required about 35 h, far longer than any other stage. For comparison, the second cycle L-3 stage lasted only 7 h. The transition from L-3 to L-4 for a first cycle colony occurs after a period during which its two attached daughter fronds expand markedly (cf. Figs. I and 2), whereas on a second cycle L-3 colony the daughter fronds are closer to full expansion at the time of separation of the colony. The results of this experiment also showed that the separation of the first daughter required more time than the separation of other daughters, and suggested that the separation of subsequent oddnumbered daughters required more time than the separation of even-numbered daughters. The results of a separate experiment (Fig. 13) confirmed these



917

Table IV. Content of Protein in Fronds of Recently Detached L-3 Colonies in Relation to Their Area Batch cultures were grown in medium containing 'SO42- for 13 days. Growth of DnMn colonies selected at the L-3 stage was continued in vials containing medium of the same specific radioactivity until collection of the daughter DnM0 colonies (begun 17 h after the initial selection). DnM0 colonies were collected at hourly intervals for 15 h and randomly distributed into eight samples of 10 colonies each. Colonies of three of these samples were placed as they were collected into 0.3 ml 10%o trichloroacetic acid containing carrier colonies. Colonies of three samples were dissected into individual fronds M, 1, and 2 as they were collected. The length and width of each frond was measured before it also was placed in 0.3 ml 10%Yc trichloroacetic acid containing carrier colonies. Colonies of two samples were placed as they were collected into 0.3 ml 10%o trichloroacetic acid, but with no carrier colonies added. An additional three samples of colonies were selected at random from batch cultures kept in medium of the same specific radioactivity. These samples contained 20 colonies each, 10 of which were collected 3.5 h and 10 of which were collected 9.5 h after the DnMO colony collection was begun. These colonies were added to 0.3 ml 10%o trichloroacetic acid in the absence of carrier colonies. Radioactivity and protein contained in the samples was measured as described under "Methods." Specific radioactivity of protein from the two samples of DnMO colonies at the L-3 stage and from the samples of randomly selected colonies was the same, and the mean of the values, 713 dpm/,ug protein, was used to convert radioactivity in the trichloroacetic acid insoluble fraction of whole and dissected fronds into protein. Areas of fronds were calculated assuming the frond is elliptical. Values given in the table are the means (± SD ± SE) of the values for the three replicates of each sample. Protein Area Protein/Area Insoluble ;'S mm'2 % ,g ,g/mm2 Dissected M 13.7 (± 0.4 ± 0.2) 7.55 (± 0.13 ± 0.08) 1.81 (± 0.03 ± 0.02) 62.1 (± 1.3 ± 0.8) 1 4.3 (± 0.3 ± 0.2) 1.92 (± 0.09 ± 0.05) 2.24 (± 0.06 ± 0.04) 65.3 (± 1.4 ± 0.8) 2 1.3 (± 0.1 ± 0.1) 0.43 (± 0.08 ± 0.05) 3.14 (± 0.25 ± 0.14) 68.7 (± 0.6 ± 0.4) M+ I+2 19.3a 9.90a 1.95 Whole 19.0 (± 0.7 ± 0.4) _h 64.3 (± 0.6 ± 0.4) a Arithmetical sum. b Not determined. Table V. Protein Content of L. paucicostata Colonies at Different Stages in Growth Cycle Individual colonies were collected within 2 h after the transition indicated, and stored in ethanol until all colonies had been collected. The ethanol was removed by evaporation with a stream of N2. After trichloroacetic acid (10%o) was added, the procedure for protein determination described in (4) was followed. Values are the means of analyses of two or three samples containing 17 colonies each, except the value for DI MO L4 which was obtained from analysis of one such sample. Each sample was analyzed at least twice. Values have been arranged in the table, insofar as the appropriate samples were available, according to the developmental sequence: a recently detached MO colony at the L-3 stage, after its transition to L4 stage, and after its transition to M I (i.e. separation of Dl), to M2 (separation of D2) and to M4 (separation of D4). (No value was obtained after transition to M3.) Values for protein content of daughters D2, D3, and D4 are also given. The value for Dl would be the same as that of the initial recently detached D I MO colony. Some of the M 1, M2, and M4 colonies had progressed to the four frond stage (either during the 2 h between collections or before separation occurred) so that the samples analyzed were made up of both three- and four-fronded colonies. Transition Protein Colony Stage ,ug/colony Detachment from parent L-3 18 DIMO 30 Appearance of frond 1-1 L4 D2M I R-3 or R4 Separation of DI 27 L-3 or L-4 Separation of D2 24 D1M2 DnM4 L-3 or LA Separation of D4 25 D2MO Detachment from parent L-3 21 Detachment from parent D3MO L-3 20 Detachment from parent D4MO L-3 25

latter observations. Lemna colonies that detached from their mother within a 12-h period were observed each subsequent 12 h, and the times of separation of their daughter colonies were re-

corded until eight daughters had been produced (Fig. 13A). Their first daughter colonies were also observed each 12 h until six daughters had been produced (Fig. 13B). Intervals for separation of 50%Yo of the colonies (determined from the curves in Fig. 13) and measured mean intervals (calculated from the actual times) are given in Table II. The values determined by the two methods are very close and are in good agreement with the results of the experiment described in Table I for the separation of the first four daughters. For both series, the mean interval between detachment of a colony from its mother to the separation of its first daughter was longer than the interval between separation of any other two daughters and the intervals for separation of odd-numbered daughters (1, 3, 5, etc.) were longer than the intervals for separation of even-numbered daughters (2, 4, 6, etc.). The separation of daughter colonies remained in good synchrony (Fig. 13), particularly during production of the first four daughters. For example, both series of colonies gave rise to their fourth daughters within no more than a 60-h period beginning about 6.5 days after the initiation of the particular series. Even for the eighth daughter in the DnMO series, all daughters were separated within only a 72h period beginning 14 days after the initiation of the experiment. The mean number of daughters which colonies could produce was also determined. Two series of colonies containing altogether 49 individuals were examined at 12 h intervals and scored for the separation of daughters. All colonies were retained even though some may have had an abnormal sequence of frond emergence. The mean number of daughters separated by this strain of Lemna growing under our standard conditions was 18, a value close to that observed by others (1, 1 1). Furthermore, up to the separation of the 18 daughters, the interval to separate odd-numbered daughters was in almost all cases longer than the interval to separate the next even-numbered daughter. Of the 18 intervals involved (two series of colonies, each having nine such odd to even intervals), 15 showed this pattern, two were of equal length, and only one showed a reversal of the pattern. Also, up to the separation of 18 daughters, there was not a progressive lengthening or shortening


918


Table VI. Specifc Rates of Protein Accumulation and Protein-Doubling Times during Several Periods of Growth Cycle Protein-doubling times were calculated: lo& protein (final stage) - log} protein (initial stage) . T (final stageinitial stage), using values for protein content from Tables IV and V and values for time through stages from Table I. For example, for frond I, log2 protein (fronds M + I + 2) - log& protein (frond 1) . T = logs (19.3) log2 (4.3) + 56.6 = 26. For colony DIMO from detachment until appearance of frond 1-1, log& protein (D IMO at L4) -log2 protein (DIMO at L-3) + T(L-3) = log2 (30) - log2 (18) + 34.5 =. 45. Note that for the remaining colony calculations, the protein (final stage) is the sum of that in the mother colony and the daughter colony. The mean of the protein in D2MI, DIM2, and DnM4 (25 jig/colony) was used as the value for the mother (D I M I through Dl M4). Specific rates of protein accumulation were calculated using the general formula relating specific

rate of increase to DT (7): specific rate of increase

-

Dlg ProteinDoubling Time h

Specific Rate of Protein Accumulation ig/,ug protein * h

On newly separated L-3 until its separation as D IMO On newly separated L-3 until its separation as D2MO

26 23

0.027 0.030

From detachment until appearance of frond I - 1 From appearance of frond 1-1 until separation of Dl From separation of Dl until separation of D2 From separation of D2 until separation of D3 From separation of D3 until separation of D4

47 43

0.015 0.016 0.019 0.014 0.018

Period

Frond I 2 Colony

DIMO DIMI D1M2 D1M4

36 50 38

Table VII. Distribution of 35S in Fronds of Recently Detached L-3 Colonies, and of Similar Fronds after Growth to Separation Colonies were labeled to isotopic equilibrium by growth in medium containing 3'SO42- for 7 days. The radioactive medium was aseptically aspirated, and the colonies were rinsed twice with nonradioactive medium. DnMn colonies at late L-3 stage, selected from this culture, were incubated in vials in nonradioactive medium. The DnMO colonies they produced beginning 20 h later were collected at hourly intervals for 10 h and randomly distributed into four groups. Individual colonies (number given in parentheses) of three of the groups were placed immediately upon collection into Protosol or trichloroacetic acid-DTT for measurement of: total radioactivity (20 colonies); proportion of total radioactivity which was soluble and insoluble in trichloroacetic acid (23 colonies); and radioactivity in dissected individual fronds M, 1, and 2 (20 colonies). Individuals of the fourth group (20 colonies) were transferred immediately upon collection to fresh nonradioactive medium and allowed to continue growth. These colonies were examined at 12 hourly intervals and when their first daughter colonies, D 1, separated (on the average, 66 h later), the radioactivity they contained was determined. When their second daughters, D2, separated (on the average, 31 h later), the radioactivity in both the D2 and M colonies was determined. The mean value for the proportion of the total 5S which was soluble in trichloroacetic acid for the 23 colonies was 0.17 (± SD 0.05 ± SE 0.01). Values given in the Table are the means (± SD + SE) of the dpm found in each group of colonies. Radioactivity Initial Frond Whole Dissected After Growth to Separation (± ± dpm SD SE) M 11,083 (±922 + 206) 7,255 (± 1,008 ± 225) 1 2127 (± 620 ± 139) 2,783 (± 404 ± 90) 2 483 (± 187 ± 42) 2,037 (± 293 ± 66) M+ I+2 13,218 (± 2,036 ± 455) 13,693a (± 1,488 ± 333) 12,075a (± 1,580 ± 353) 8 Arithmetical sum.

of the intervals.

Population Distribution of Cultures. To ascertain the proportion of colonies in various stages, four standard batch cultures were selected at random over a period of 5 months, and the number of colonies in each culture at each stage of the vegetative growth cycle was counted (Table III). Colonies in the left hemicycle (i.e. those which would next separate an odd-numbered daughter) were preponderant (78.8%), while colonies in the right hemicycle (i.e. those which would next separate an even-numbered daughter) were many fewer (20.6%). Of those colonies in the left hemicycle, the largest proportion were in the L-3 stage. As shown under "Discussion," the population distribution found in these cultures

closely approaches the theoretical proportional distribution predicted from the characteristics of the growth cycle established in experiments described above. Protein Contents and Specific Rates of Protein Accumulation. Visible fronds of Lemna sp. contain many embryonic fronds, detected microscopically (2, 14). To determine the extent to which protein accumulation takes place preferentially in such primordia, recently separated DnMO L-3 colonies (similar to the one shown in Fig. 1) were obtained from cultures which had been labeled to isotopic equilibrium by growth in medium containing 35SO4 2- for more than eight doublings. The fronds M, 1, and 2 making up such colonies were dissected and their areas and the amounts of



60h

A. MO

MO LM21ml MO

B. Ml, M3,

30 h

f..

C. M2, M4,

....

40h

MM2. -

Vtd

MO

M3, M5,..

FIG. 14. Three idealized types of colonies of Lemna. DT and products formed at each doubling are indicated.

trichloroacetic-soluble and -insoluble 35S they contained were measured. Similar analyses of samples of undissected whole L-3 colonies proved that recovery of 5S, both trichloroacetic acidsoluble and -insoluble, was quantitative during dissection. Other samples of similar L-3 colonies and of random colonies were analyzed both for radioactivity and for protein content. From the results of the latter analyses a value for trichloroacetic acidinsoluble 35S/,ug protein was derived, and used to convert the values for trichloroacetic acid-insoluble 35S content in the dissected fronds into protein contents (Table IV). The smallest fronds recovered (frond 2) were 0.43 mm2 in area, or 5-6% of the area they would attain at the time of their separation as M fronds. As shown in a separate experiment, M fronds expand only minimally after separation from their own mother, by that time having attained 93% of their final area. With respect to protein, the smallest fronds (frond 2) contained only about 10%7o of the protein they would contain at the time of their separation as M fronds clearly indicating that the major portion of the protein of a frond accumulates during frond expansion, rather than having been accumulated at some earlier primordial stage. To investigate the rates of protein accumulation during later stages of growth, colonies were collected shortly after major transitions in the growth cycle and their protein contents were measured (Table V). The measured protein content of fronds and colonies (Tables IV and V) and the times required for passage through the stages in the vegetative cycle (Table I), were used to calculate the approximate specific rates of accumulation of protein for several periods of the growth cycle. Protein doubling times were 23-26 h during rapid frond expansion, and thereafter, for whole colonies,

919

to continue growth in nonradioactive medium until daughter colonies 1 and 2 had separated from the mother. Radioactivity in these two daughter colonies, as well as that in the mother colony, was measured. The sum of the radioactivity in these three colonies was not significantly different from that in the initially examined whole colonies, but it was now distributed differently. The radioactivity in the mother colony had decreased to 601% of the total, whereas that in daughter colonies 1 and 2 had concomitantly increased to 23 and 17%. These results indicate that a substantial portion of 3S initially present in the mother frond had been transferred to daughters 1 and 2 during their growth. To examine the transfer of3S from mother to later daughters, Lemna colonies were again labeled to isotopic equilibrium in medium containing35SO4 2-. Colonies, most of which were in the L-3 stage, were rinsed with nonradioactive medium and then matched samples were either examined immediately for radioactivity or transferred to vials of nonradioactive medium and allowed to produce daughters which were examined for radioactivity. About 70%91 of the radioactivity in the initial colonies was trichloroacetic acid-insoluble,'and this value increased to about 90% for the daughter colonies, indicating that during the growth period 35S had been "chased" from soluble to insoluble material. All radioactivity could be accounted for in the colonies. No 35S was lost to the medium or to the atmosphere. All of the daughters examined (that is, up to the fourth) produced by radioactive mothers contained considerable 35S. The mean values for radioactivity in the daughters were: first colonies, 24% of that in the mother colonies; second colonies, 20% of the total remaining in the mother colony after separation of the first colony; third, 19% of the total remaining in the mother colony after separation of the second; and fourth, 17% of the total remaining in the mother colony after separation of the third. The values for transfer of 35S from mother to the first two daughter colonies in this experiment were similar to those observed in the previous experiment. This experiment further demonstrates that transfer of 35S continues with relatively little diminution at least through separation of the third and fourth daughters.

DISCUSSION The observations presented here provide the first complete description of the vegetative growth cycle of Lemna (Figs. 1-12) under standardized growth conditions. In a previous study of Lemna minor (9, 10) the time of emergence of daughter fronds from the maternal frond, but not the emergence of new fronds on were 36-50 h during periods from detachment of a colony from its mother through production of its first four daughters (Table the attached daughter fronds, was measured. The alternation of VI). The values for specific rates of accumulation of protein were long and short intervals between separation of daughter colonies, noted here for L. paucicostata, was also noted in this study (9, 10). 0.014-0.030 ILg/tg protein-h. Is 3S Transferred from Mother Fronds to Daughter Fronds? That a similar pattern was observed in a different species, growing The extent to which 35S-containing material was supplied directly under different conditions (a light/dark cycle of 12 h each on agar by a mother frond to its daughters was measured. Colonies were medium) suggests that the rhythmicity of reproduction is a general property of Lemna. grown for several generations in medium containing 35s42Occasionally, the usual sequence of frond emergence or colony Individual L-3 or L-4 colonies were then transferred to nonradioactive medium and their next daughters collected within I h of separation described here varied. Sometimes, in older colonies their separation. This chase period in nonradioactive medium was especially, a frond failed to emerge, or even if it emerged, failed added to ensure that a major portion of the 35S would be in the to grow. The production of daughters from the opposite pocket trichloroacetic acid-insoluble fraction (i.e. presumptive protein). usually continued in the normal manner. This behavior has been Indeed in the newly separated L-3 colonies collected for this noted earlier (11). Also separation of a colony from its mother experiment 83% of the 35S was in the trichloroacetic acid-insoluble colony was sometimes premature or delayed. If separation was fraction (legend, Table VII) as compared to a mean of 64% for premature, colonies having only one or two fronds were produced. similar colonies collected fresh from radioactive medium (Table If separation was delayed, then the stages of L-7 and R-7 (Fig. 12) IV). Radioactivity was measured in whole L-3 colonies and in were encountered. The growth sequence was also found to be fronds M, 1, and 2 from a sample of similar colonies which had influenced by culture conditions. Colonies growing in medium been dissected (Table VII). In agreement with the previous exper- with very low sulfate had very delayed separation, so they came iment, radioactivity in the whole colonies was not significantly to contain 30 or more fronds per colony. As such multifronded different from that in the sum of the dissected parts. Frond M colonies eventually became further depleted of sulfur, separation contained 8%o of the radioactivity, daughter frond 1, 16%, and occurred, leading to the formation of colonies with only one or daughter frond 2, 4%. A third sample of L-3 colonies was allowed two fronds (4). In medium containing 28.8 mm sucrose L. pauci-

920



Table VIII. Theoretical and Measured Population Distributions and Doubling Times of Lemna Cultures For the theoretical situations, the population distributions and DT values are those predicted when the individuals in the growth cycle have the characteristics indicated in Figure 14. Tree diagrams were constructed by noting, when starting with a single colony of each type, the numbers of the three types of colonies which would be present at appropriate intervals. The means of such values from the three diagrams are given for 280 and 560 h. DT was calculated from the number of colonies which would be present between 280 and 560 h. The mathematical model, described in the Appendix, gives both population distribution and DT. The experimental values for population distribution are those percentages measured in four typical batch cultures (Table 111), and the DT is the mean of the DT values of those cultures. Population Distribution

MO MO

Ml,M3,...

M2,M4,...

M2,M4,.

%

Theoretical Tree diagram 280 h Tree diagram 560 h Mathematical Model

59.1 60.0 59.9

Experimental

23.3 21.7 22.4 20.6

Table IX. Distribution of Colonies of Indicated Daughter (D) and Mother (M) Number in an Equilibrium Population Tree diagrams were constructed starting with single colonies which were themselves of unknown daughter number and which had produced either zero, one, or two daughters (i.e. DnMO, DnM 1, and DnM2). The numbers of colonies of the indicated D and M number contained in the three cultures at 330 h were combined and are expressed here as percentages of the total number of colonies (377). The sums of the values in both the "Total" column and row do not add up quite to 100%c because of the presence of a few colonies with M greater than 7 or D unspecified or greater than 8. D

M

0 1 2 3 4 5 6 7 Total

1 23.4 5.9 4.2 2.1 1.5 0.7 0.6 0.2 39.0

2 15.0 3.8 2.7 1.3 0.9 0.5 0.4

3 8.4 2.1 1.5 0.8 0.5 0.3

4 5.3 1.3 1.0 0.5 0.3

5 3.2 0.8 0.6 0.3

6 1.9 0.5 0.4

1.0 0.2

24.9

14.1

8.8

5.3

3.2

1.6

7

8 Total 0.6 60.0 15.1 10.9 5.3 3.7 1.9 1.6 0.5 1.1

costata grows slightly faster than in the medium containing 9.6 mm sucrose used for the present experiments (4). Colonies grown in 28.8 mm sucrose have a higher frond to colony ratio, and the reproductive sequence shown here was not as easily observed in

such colonies. In the present study, protein accumulation in fronds and colonies was also measured. While protein accumulated in very young fronds to a somewhat higher concentration (per unit area) than that attained finally, most of the total frond protein accumulated during the period of rapid frond expansion (Table IV). During such expansion the specific rate of protein accumulation is higher than the rates measured for whole colonies at later stages (Table VI), suggesting that the expanding fronds contain the dominant protein synthesizing regions of the Lemna colony. For whole colonies, there were not large differences in the specific rate of protein accumulation up through the production of four daughters, and the protein doubling times for the colonies were comparable to the over-all DT3 of the organism (Table VI). 3Abbreviation: DT: doubling time.

DT

h

17.6 18.2 17.7

76.7 78.2 77.6 78.8

43.1 45.5 39.5

The transfer of 35S from mother to daughter fronds occurred to a considerable degree. The 35S, which ended chiefly in the presumptive protein fraction of daughter fronds, probably originated to a large extent in protein of the mother fronds, but could have been transferred in a soluble form. Some soluble 3S was present in colonies even after prolonged growth in nonradioactive medium. The protein of growing L. minor turns over with a half-life of 7.1 ± 0.5 days (8). If protein in L. paucicostata mother fronds turns over at a similar rate, during the 40 h required to form a new colony about 15% of the protein of the mother frond would be broken down to amino acids. If protein turnover does indeed provide most of the 35S transferred from mother to daughter fronds, under the conditions of our experiments the gradual decrease in the per cent of 3S transferred to successive daughters could be due to time-dependent preferential retention of S in more slowly turning over proteins in the mother. From the time course of the growth cycle (Tables I and II; Fig. 13), and with some rounding off of DT values, an idealized Lemna culture can be considered to consist of three colony types (Fig. 14): A. colonies (MO) with a DT of 60 h, which have not yet produced a daughter; B. colonies (Ml, M3, .. .), with a DT of 30 h, which have produced one or subsequent odd numbered daughters; C. colonies (M2, M4, .. .), with a DT of 40 h, which have produced second or subsequent even numbered daughters. Two approaches were used to relate the distribution of colony types and the over-all DT in a large population to the DT and modes of division of the individual colonies in that population. In the first, tree diagrams were constructed by noting the number and type of individuals which would be contained in a culture at intervals when a single colony of each of the three types was the progenitor. These diagrams showed that, irrespective of the type of colony with which a culture was begun, by 280 h the culture would reach an equilibrium state in which the proportions of the three types was constant (Table VIII). The calculated DT for such cultures was 43.1 h. In the second approach it was proven mathematically that a population with the growth properties described here increases exponentiaily and that it reaches an equilibrium population distribution which is independent of the initial concentrations of the members of the population (Appendix). The equilibrium concentration and DT values predicted by this method (Table VIII) are very close to those predicted by use of the tree

diagrams. For experimental batch cultures, the proportions of MO plus M2, M4, ... colonies (colonies in the left hemicycle) and of Ml, M3, ... colonies (colonies in the right hemicycle) were in good


921


agreement with those predicted by the theoretical methods (Table VIII). That the DT of the experimental cultures differed from that predicted by the models by a few hours may reflect the extent to which the real DTs of the colonies differed from the approximated DTs used in the idealized model. These results show that the experimental cultures used in our work, which were begun with eight to 10 randomly selected colonies, attained the predicted equilibrium distribution of colony types while undergoing four to five doublings in 7 days (when experimental observations were usually made). Cultures used by other workers have often been started with one to three "threefronded" colonies. Such cultures may not attain an equilibrium distribution in the 7 days they are usually allowed to grow. For example, the tree diagrams showed that whereas MO colonies always comprised more than half the population at 7 days, the proportions of the other two types varied between 12 and 25% depending on the type of single colony used as progenitor. The results of the tree diagrams and the mathematical analysis indicate that, once a mass culture of L. paucicostata has attained its equilibrium distribution of colony types, under our standard conditions the proportions of colonies will be (Table IX): 60%1o producing the first daughter; 15% producing the second daughter; 11% the third; 5% the fourth; and 8% the fifth, or higher. Thus, 75% of the colonies will be in the first cycle and 91% of the colonies will be passing through one of the first two cycles. The same analyses indicate that 39% of colonies will themselves be first daughters, 25% will be second daughters, 14%, third daughters, 9% fourth, and 13% greater than fourth daughters. A major consideration in planning many types of experiments using various organisms is that of determining the extent to which any sample taken is representative of the entire population. In the equilibrium Lemna cultures described here most colonies will be traversing their first two cycles during which the specific rates of protein accumulation do not vary widely. Therefore, obtaining representative samples from such a population for biochemical studies should be relatively simple. LITERATURE CITED 1. CLAUS WD 1972 Lifespan and budding potential of Lemna as a function of age of the parent-a genealogic study. New Phytol 71: 1081-1095 2. CLELAND CF, WR BRIGGS 1967 Flowering responses of the long-day plant Lemna gibba G3. Plant Physiol 42: 1553-1561 3. DATKO AH, SH MUDD, J GIOVANELLI 1977 Homocysteine biosynthesis in green plants. Studies of the homocysteine-forming sulfhydrylase. J Biol Chem 252: 3436-3445 4. DATKO AH, SH MUDD, J GIOVANELLI 1980 Lemna paucicostata Hegelm. 6746: development of standardized growth conditions suitable for biochemical experimentation. Plant Physiol 65: 906-912 5. DATKO AH, SH MUDD, J GIOVANELLI, PK MACNICOL 1978 Sulfur-containing compounds in Lemnaperpusilla 6746 grown at a range of sulfate concentrations. Plant Physiol 62: 629-635 6. HILLMAN WS 1961 The Lemnaceae, or duckweeds. A review of the descriptive and experimental literature. Bot Rev 27: 221-287 7. HERBERT D, R ELSWORTH, RC TELLING 1956 The continuous culture of bacteria: a theoretical and experimental study. J Gen Microbiol 14: 601-622 8. HUMPHREY TJ, DD DAVIES 1976 A sensitive method for measuring protein turnover based on the measurement of 2-3H-labelled amino acids in protein. Biochem J 156: 561-568 9. KAsINOV VB 1973 Handedness in Lemnaceae. On the determination of left and right types of development in Lemna clones and on its alteration by means of external influences. Beitr Biol Pflanz 49: 321-337 10. KASINOV VB, GV KASINOVA 1974 The reproduction rhythm in Lemnaceae: a possible link with right and left handedness. Int J Chronobiol 2: 47-52 11. POSNER HB 1962 Permanent and temporary effects of x-rays on the reproduction and aging of Lemna perpusilla. PhD thesis. Yale University, New Haven 12. POSNER HB 1%7 Aquatic vascular plants. In FH Wilt, NK Wessels, eds, Methods in Developmental Biology. Thomas Y. Crowell Co., New York, pp 301-317 13. POSNER HB, WS HILLMAN 1960 Effects of X irradiation on Lemna perpusilla. Am J Bot 47: 506-511 14. WITZTUM A 1979 Morphogenesis of asymmetry and symmetry in Lemna perpusilla Torr. Ann Bot 43: 423-430

APPENDIX4 AN APPLICATION OF LINEAR ALGEBRA TO A BIOLOGICAL GROWTH PROCESS

In this note we present an application of the theory of positive matrices to the modeling ofcertain multispecies population growth processes. As an example, we will apply this model to the situation in the preceding paper, which can be symbolized by A A A

C 4

A

B 1C BThe interpretation is that after 6 time units (1 time unit = 10 h) a member of type A splits into one member of A and one member of B. The new member of A repeats this process while the new member of B splits after 3 time units into a new member of A and a new member of C and so forth. More generally, suppose there are n types S1, ..., S. which grow interactively in a way symbolized by n diagrams, the jth diagram of which is

kj1Sl Sj tj kjnSn' This indicates that after tj (an integer) time units, one member of Sj transforms into kji members of Si (i = 1, ..., n). We would like our model to answer the following questions. (1) Does the total population grow (at least approximately) exponentially, and if so, what is the DT? (2) Is there an equilibrium population distribution? (3) Is the equilibrium distribution independent of the initial concentrations? Theorem 1 (see ref. Al). Suppose A is a square non-negative matrix so that some power of A is positive. Then the following hold. (a) A has a positive eigenvalue Xo such that Xo > IAI for all other eigenvalues A. (b) The left and right eigenvectors of A corresponding to Xo can be chosen to be positive and are unique up to multiplication by scalars. (c) Let X and Y be left and right eigenvectors of A corresponding to Xo which have been normalized to satisfy XY = 1. Let P be the matrix YX. Then -n -- P as n -- oo.

Theorem 2 (see ref. 2A). Let A be a non-negative n x n matrix. Denote by Alfp the ij'th entry of Aq where q is an integer 2 0. Suppose that

alXn,

atAnt = o (n > ni > > nt; a,1 0, ..., at 6 0) is the characteristic polynomial of A. Let g be the greatest common An +

+ ..+

divisor of the differences {n - ni, ni - n2 . nt- - nt). The following conditions are equivalent. (a) An is positive for some n. (b) g = 1 and for each ij there exists a q > 0 so that ai,1q' > 0. Turning now to the construction of our model, we define a population vector to a row vector of the form

(Al) (Sio, *--, si(tt-1)- *--, Sno. Sn(tn-1)) where sij is the number of individuals of type si present at the ... 9

'George Benke, Department of Mathematics, Georgetown University, Washington, D. C. 20057

current time who are j time units old. Thus the absolute population distribution broken down by types and ages at any time may be represented by such a row vector. We will define a population growth matrix A so that if S is a population vector then SA is the population vector which results from S after one time unit. A has the structure shown in equation 2 (Fig. Al). The growth of the population is then described by the behavior of Am as m increases. Let us assume for the moment that AN > 0 for some N. Then, according to theorem 1, there exists a A (dropping the subscript 0) Am and P so that Am - P . In other words, Am is close to AmP for large m. Therefore the population grows, in total numbers, exponentially with base A, and the population distribution is determined by P. Suppose S is a population vector and P is written as YX as in theorem 1. Then after a long time m, the population vector is SAm which is very close to Am SP = Am SYX. Since SY is a scalar, we see that the long time population distribution is described by the vector X and is independent of the original distribution. The condition that AN > 0 for some N is generally not easy to check directly. This is where theorem 2 is useful. Checking that the greatest common divisor g = 1 is easy. We therefore focus on checking that ai(f) > 0 for some q (depending on ij). Suppose that the ijth entry of Sq lies within the block K,q) and has indices rs with respect to that block. Then aff) is the number of individuals of age s in type m which result from one individual of age r in type 1 after a time interval of length q. Therefore, it is required to show that, given any two individuals of any two ages, by starting with only the first individual after some time the resultant population contains the second individual. This condition is fulfilled if every type can have ancestors from every other type. This property is generally easy to check. t1 columns

01 01 01

t2 columns

0

0o

= E kijXi(tj_l) AXo-

AXjr =

E BijXi(r-l) = Xj(,i)

For notational convenience, let zj = A6 imply that

0 10_

(A6)

(1 c r s tj-,).

Xj(t11-). Then the equations

Xj = zj (X ', AX-2, ... , A, l)

(A7)

and z; ==. kizi.

(A8)

The system of equations A8 has a nontrivial solution if and only if

(k,- A") det

k12 k22-A

k2l

kl.

k2

t2)

(A9)

=

(knn-Atn)

E j=O

t,-I

( t2 rows

xlj

E -...- j=0

Xj

(A 10)

I

which by virtue of equation A7 and the well known formula

(2)

k2n

l+A+

o:

A2 + ..

1- Ar + Ar-l= 1-A

gives

.

*

tn rows kn2

1kn l

in

we can rewrite equations A4 as

t-1

.

k22

O

(A5) XiKij = (kijXi(tj-l), SiXDo, ..., SijXi(t,-l ) where Sij = 0 unless i = j in which case Sii = 1. Using equation A5

0

0 k2l

Using the structure of the Kij as shown in equation A2 it is clear that

The left side of equation A9 can be shown to be the characteristic polynomial of the original eigenvalue problem A3. Once this polynomial has been solved for the maximum positive A, this ( t1 rows value is used in equation A8, which is then solved for zi, ..., z. The age structure of each type is then given by equation A7. If we ignore age, then the population distribution is given by the vector

0

01 01

(A4)

(l ' jsn).

AXj = XiKij

kl

k12

0

kll

where the Kij are the blocks in A from equation 2 (Fig. Al). Equation A3 can be rewritten as

tn columns

0

0

We



922

now

more Xj(tj-1))

I

look at the eigenvalue problem XA = AX

detail. Let X

=

X.) where Xj (XI, ...,

(A3) =

(Xjo,

...,

(All) ll A KtI the)zl,...tt , l e men th We will now apply this theory to the example mentioned at the beginning. In this case t, = 6, t2 = 3, t3 = 4 and K= [ 0 1 -010The characteristic equation A9 is -1-_A6 11-

det[l

and

Kil

...

Kin

A-

LKnl

Knnj

-0

-A3 l

=0

A4_

which is easily seen to reduce to A'3-A7-A6 - A4 = 0. By using Newton's method we can solve this equation for its



largest positive root and get A = 1.1644 which then gives a DT of 4.554 as determined by the formula DT = log 2/log A. We now check that the growth matrix satisfies the hypothesis of theorem 1 by verifying the condition (a) in theorem 2. The greatest common divisor of the set of numbers {13 - 7, 7 - 6, 6 - 4) = (6, 1, 2) is 1. Note that A can come from A, B, or C directly; B can come from A and C directly, and from B by passing through C; C can come from B directly, and from A or C by passing through B. Therefore by our earlier remarks, condition (a) of theorem 2 is fulfilled. It follows from theorem 1 that there will be an equilibrium distribution. To find this distribution it is necessary to solve the system of equations A8 which in this example become (1-A6)zs + Z2 + zl-A3z2 +

Z3 =

0

z3 =

0

Z2- A4z3

=

0

Since this system is singular there are infmitely many solutions of

923

the form ZI = (A7 -_ )C Z2 = A4C Z3 = C where C is arbitrary. According to eqsuation Al 1 the distribution is given (up to a scale factor) by [(A - 1) (A7 - 1), (A3 -1)4, (A - 1)]. Evaluating this vector numerically and then normalizing, we get

(0.5988, 0.2244, 0.1768) as the equilibrium population distribution for the types A, B, C. LITERATURE CITED Al. GANTMACHER FR 1959 Applications of the Theory of Matrices. Interscience Publ. Inc., New York, p 478 A2. KARLIN S 1973 A First Course in Stochastic Processes. Academic Press, New York, p 96