An Automata-theoretical Model of Meristem Development as Applied to the Primary Root ofZea maysL.

Annals of Botany 79 : 375–389, 1997

An Automata-theoretical Model of Meristem Development as Applied to the Primary Root of Zea mays L. J. L U$ C K*, P. W. B A R L OW†‡ and H. B. L U$ C K* * Laboratoire de Botanique analytique et Structuralisme ŠeU geU tal, FaculteU des Sciences et Techniques de St-JeU roW me, CNRS, URA 1152, aŠenue Escadrille Normandie-Niemen, 13397 Marseille cedex 13, France and † IACR—Long Ashton Research Station, Department of Agricultural Sciences, UniŠersity of Bristol, Long Ashton, Bristol BS18 9AF, UK Received : 9 January 1996

Accepted : 17 October 1996

Observations were made of the sequence of division within the cellular packets (groups of cells of common descent) which comprise the cell files that run the length of the central cortex of the primary root meristem of Zea mays. These sequences, and also the relative lengths of the cells within the packets recorded at various times during root growth, indicate that cell-file development can be expressed using one, or a limited number, of deterministic ‘ bootstrap ’ Lsystems which assign different lifespans to sister cells of successive cell generations. The outcome is a regular pattern of divisions from which daughter cells emerge usually with unequal, but definite, lengths. In the immediately postgermination stage of root growth, one division pathway is especially common in the cortex and generates sequences of unequal daughters having a particular basi-apical orientation. Later in root growth, the cellular pattern in the cortex indicates that this pathway is replaced by another where unequal divisions are not so marked, but which nevertheless continues to maintain a regular arrangement of differently sized cells. This latter pathway is characteristic of a zone close to the initial cells of the cortex. It is present at all stages of root growth and spreads along the length of the cortex as the descendants of these initials proliferate. The development of the whole cortical cell file can be simulated from knowledge of the growth functions of the bootstrap systems. The files so generated contain all the observed cell patterns. The growth functions also predict the sequence in which cells cease dividing near the proximal margin of the meristem, but for this it is necessary to incorporate a counter for the number of divisions that will be accomplished in the cell file. Cytological requirements for the propagation of unequal divisions, together with a consideration of the nature of the division counter, as well as the significance of the switch in division pathways encountered during early root growth, are discussed in the context of this deterministic model of cell division. # 1997 Annals of Botany Company Key words : Cell division, root meristem, L-systems, Zea mays.

INTRODUCTION Apical meristems are regions at the tips of plant organs (roots, shoots and sometimes leaves) where cells grow and divide, thereby producing the new cells required for continuous organ elongation. Within the root apical meristem, there is regional differentiation with respect to the rates of cell proliferation. Near the root tip there is a zone where the rate of proliferation is low ; this is the so-called quiescent centre (QC) (Clowes, 1961 a). In the remainder of the meristem [i.e. in the two zones distinguished as the proximal and distal meristems, whose initials lie, respectively, on the basiscopic and acroscopic faces of the QC (Torrey and Feldman, 1977)], the proliferation rate is significantly greater (Barlow and Macdonald, 1973). In roots of monocotyledonous plants, particularly of grasses with their ‘ closed ’ type of meristem (Clowes, 1961 b), cells in the proximal meristem participate in the differentiation of epidermis, cortex and stele, whereas the distal meristem is involved in root cap formation. The meristem also shows structuring of another kind which relates to the plane in which its cells divide with ‡ For correspondence.

0305-7364}97}040375­15 $25.00}0

respect to the principal axis of cell growth. Longitudinal (or formative) divisions usually occur just behind the summit of the meristem, especially in regions in and around the QC, whereas transversal (or proliferative) divisions predominate further away, in both the proximal and distal directions (Barlow, 1984). Inspection of sectioned root meristems reveals that there are consistent, and species-specific patterns in the way in which the longitudinal divisions are organized. Consistent patterns of transverse division are also recognizable in the various cell files that later become associated with the different root tissues (Luxova! , 1975). These patterns concern the relative lengths of cells, which in turn derive from the degree of inequality of successive divisions and the rates of cell extension (Barlow, 1987). All the above-mentioned features of root meristems occur so regularly that they suggest an underlying determinism of cellular development. With respect to cell division, determinism would mean that both the plane of division of a mitotic cell, and the sizes of its two daughter cells would be predictable, given the fulfilment of certain prescribed conditions. Indeed, determinism of plant cell division patterns has been demonstrated in filamentous algae (Lu$ ck and Lu$ ck, 1976), sheets of epidermal cells (Lu$ ck, Lindenmayer and Lu$ ck, 1988) and apical meristems of

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# 1997 Annals of Botany Company

376

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment

cryptogams (Lindenmayer, 1984 ; De Boer and De Does, 1990 ; Lu$ ck and Lu$ ck, 1991) ; these are examples of systems where, respectively, one-, two-, and three-dimensional growth apply. To give this determinism a clearer conceptual definition, division ‘ rules ’, or algorithms, have been ascertained by which the pattern of insertion of new cell walls into a cellular ensemble is prescribed (Lu$ ck, 1975 ; Lindenmayer and Rozenberg, 1979 ; Lu$ ck and Lu$ ck, 1983 ; Barlow, 1991). Moreover, given additional knowledge about the growth of the walls of each cell, the shaping (morphogenesis) of the cellular ensemble during its development can also be simulated (Lu$ ck and Lu$ ck, 1983). In previous papers (Barlow, 1987 ; Lu$ ck, Barlow and Lu$ ck, 1994 a, b), we showed that the temporal sequence of transversal division can be followed in individual cell files of the maize root apical meristem. At the commencement of root growth upon germination, cell divisions result in discrete packets of cells, each packet being a spatially ordered collection of related cells. In the early stages of growth, divisions were found to be unequal, particularly in the root cortex (Barlow, 1987). The regular pattern of unequal cell lengths within the packets could be satisfactorily modelled through the use of a deterministic class of Lsystems. These were designated ‘ bootstrap ’ L-systems on account of the particular switching between two alternative patterns of growth by the daughter cells in each successive cell generation (Lu$ ck and Luck, 1976 ; Lu$ ck, 1977). Divisions were analysed only for the first few cell generations. However, it was suspected that the corresponding division algorithms might not apply in exactly the same form during later cell generations (Lu$ ck et al., 1994 a). In the present paper, the cell packets found at these later stages of root growth have been analysed and, thus, it is now possible to extend the description, in terms of bootstrap Lsystems, of the patterns of transversal division which occur in the central cortex of the actively growing maize root. The outcome is a more complete understanding of cellular behaviour in the root meristem than hitherto. Although other regions of the root could have been chosen for study, the cortical zone is convenient for analysis because it shows clear evidence of asymmetric division and is a relatively homogeneous tissue. The division pattern in the cortex is representative of other patterns elsewhere in the meristem and similar deterministic principles should apply there also. The type of analysis we present could also be useful in other situations where interdependence of successive interdivisional periods is suspected. MATERIALS AND METHODS The cell packets used for the present analysis were observed in primary roots of maize (Zea mays L., cv. LG11) whose conditions of growth and histological preparation have been described by Barlow (1987). In brief, after staining the cell walls of longitudinally sectioned roots by the periodic acid-Schiff’s reaction, groups of cells of common descent (cellular packets), clearly defined by the transverse endwalls of an original mother cell, can be observed in the longitudinal files of cells which comprise each of the tissues within the proximal root meristem. Each cell within a file is

a member of a packet. Coincident with root growth, the packets lengthen and the number of cells contained within them increases ; the packets consequently move away from the root tip because of their elongation and the interpolation of new cells. Cell growth within the tip is coherent, as though symplastic growth applied, and hence the rate of movement of the packets along the cell files can be assessed by reference to the position of marker cell packets in files of developing metaxylem within the meristem (Barlow, 1987). In the present work, packet development was analysed in a region of the cortex (designated as region E), 1000–1300 µm from the root tip. This region had not been considered previously (Barlow, 1987). In roots " 40 mm long, the packets encountered in this region were those which had moved into it from locations which, in younger roots, had been closer to the tip. These more distal locations included the regions designated A and B by Lu$ ck et al. (1994 a) situated 0–50 µm and 225–275 µm, respectively, from the root tip. Details of the growth of packets originating in these two regions have already been described (Lu$ ck et al., 1994 a). The lengths of cells in packets in region E, and also in packets at a location 600 µm from the tip, were measured using an ocular micrometer. Standard statistical tests established the significance of differences between the lengths of cells at different positions within a packet. For the purposes of the present work, informative packets were those containing eight cells (octets). Depending on the stage of root development, an octet could either be comprised of eight cells still bearing the imprint of a familial relationship dating from an earlier, embryonic stage of root development, as was apparent in roots just commencing to germinate or, at the other extreme, the eight cells could be part of a larger packet of 16, 32, or even 64 cells, as were apparent in roots " 100 mm long. However, it was difficult to recognize the complete structure of 32- or 64-cell packets because the differential thickness of the transverse walls formed during different generations of cell division (Risuen4 o, Gime! nez-Martı! n and Lo! pez-Sa! ez, 1968) is not so easily recognized as it is in packets with fewer cells that were the product of fewer divisions (Barlow, 1987). The ordering of the cells in the packet with respect to their basi-apical orientation was always recorded. PATHWAYS OF CELL PACKET DEVELOPMENT ALONG THE ROOT MERISTEM The development of a cell packet in any of the cell files that run the length of the primary root meristem of maize follows one of a number of alternative sequences of division, or pathways, depending on the initial location of its mother cell (Lu$ ck et al., 1994 a). For cases where the respective mother cells were situated in regions A and B of the cortex, the most frequently encountered pathways (P) were those designated P and P , respectively (Table 1). Each of these $ ## pathways has been accurately described by a ‘ minimal bootstrap L-system ’ (Lu$ ck et al., 1994 a, b) in which the lifespans of the quartet of four cousin cells, spatially ordered within the packet according to their basi-apical positions

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment

377

T     1. Parameters of the most frequently encountered pathways, P, of cell packet deŠelopment within different regions of the cortex of primary maize roots Pathway

Region

P $ P ## PE " PE

A B E E

#

Lifespans (relative values) (4,5) (5,4) (9,6) (9,8)

Timestep (h)

(4,3,4,5) (6,5,5,4) (10,9,9,8) (12,11,11,10, 10,9,9,8)

Lifespans (actual values, h)

4±36 3±69 1±85 1±78

(17±4,21±8) (18±5,14±8) (16±7,11±1) (16±0,14±2)

Packet-size (no. cells)

(17±4,13±1,17±4,21±8) (22±1,18±5,18±5,14±8) (18±5,16±7,16±7,14±8) (21±4,19±6,19±6,17±8, 17±8,16±0,16±0,14±2)

1–8 1–8 8–16 8–16

The cellular lifespans pertain to four pathways (P) in three regions, A, B and E, initially located 0–50 µm, 225–275 µm and 1000–1300 µm from the root tip, respectively. The lifespans are presented (in parentheses) in basi-apical order of the cells. Both relative and actual (h) lifespan values are listed, the later being based on estimates of the actual duration of a timestep.

l

i

P3

1 2

0

4

5

4

5

10

5

4

3

4

5

5 4 4

4

3

5

5

4

5

10

4

5

3 4

3

4

5

5

6

5 4

16

15

3

4

8

4

0

4

4

5

4

i 1 2

4

5

3

4

l

P22

8

5

6

5

6

15

l

i 0

PE1

1 2

l

PE2

i 1 2

0

6 9

5

10

8

9

4

5

4

8 9 9

10

10

15

8

9

9

10

Timestep (i ) or cell number (l )

Timestep (i ) or cell number (l )

16

15

8

20

8

10 10

9

9

9

9

9

8

10 11

25

12 9

8

8

20

9

10

25

11

8

9

30

16

10

30 16

F. 1. Genealogical trees (cellular descendances) of four cell populations that conform with the most frequent cell division pathways, P and P $ ## for 1–8 cell packets, and PE for 8–16 cell packets, which are encountered in cell files in the root cortex of Zea mays. They are simulated by minimal bootstrap devices based on ordered quartets of lifespans such as (4,5) (4,3,4,5), (5,4) (6,5,5,4) and (9,6) (10,9,9,8) for pathways P , P $ ## and PE , respectively, and an octet of lifespans such as (9,8) (12,11,11,10,10,9,9,8) for pathway PE . The left-to-right ordering of lifespans " # in the genealogies reflects their basi-apical orientation within the cell file which, in turn, is related to inequalities in the size of sister cells immediately following division of a mother cell. i, timestep ; l, number of cells in a file at timestep i.

378

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment

(b, a), were equal to (4,5) (4,3,4,5) for P and (5,4) (6,5,5,4) $ for P . In this notation, the pair of values (given in ## parentheses) preceding those of the quartet are the lifespans of the two daughter cells of the mother cell which initiated the cellular descendance. The lifespans are given in relative time units, but they can also be assigned absolute time units, as will be shown later. Up to this point, the notation describes, at the minimum, the first two divisions following germination ; further divisions would then continue according to the recurrent bootstrap principle, using the quartet of lifespans. However, when eight or 16 cells, rather than four cells, are the basis for constructing a bootstrap system, as has become necessary to describe the division pattern in older roots, a slightly more complex situation ensues (Lu$ ck et al., 1994 a). Cell packets which initially commenced development approx. 250 µm from the tip, in region B, for example, are found about 3 d later at a distance of 1000–1300 µm from the tip. Their most frequently observed pathway, PE, can be simulated by bootstrap systems (Lu$ ck et al., 1994 b) operating either over four lifespans equal to (9,6) (10,9,9,8) or over eight lifespans equal to (9,8) (12,11,11,10,10,9,9,8). These are designated by PE and PE , respectively (Table 1 and Fig. 1). " # The principle of a bootstrap L-system is highlighted by the characteristic cellular descendance which it predicts (Fig. 1). The growth functions (cell number increases over time) corresponding to the descendances for PE and PE are " # indicated in Table 2, and further details of the growth functions for other such systems, describing other pathways, are given by Lu$ ck et al. (1994 a, b).

TOWARDS A UNIQUE PATHWAY OF PACKET DEVELOPMENT WITHIN THE CORTICAL ZONE OF THE MAIZE ROOT MERISTEM The region of the functional initials surrounding the quiescent centre is where new cells are injected into each of the cell files of the cortex. Their descendants subsequently move through the three regions of observation, A, B, and E, at a rate previously estimated (Barlow, 1987). It is therefore necessary to discover a system, or systems, which will integrate, or couple, all three respective pathways, P , P $ ## and PE, which had previously been analysed independently (Lu$ ck et al., 1994 a, b), so that every cortical cell and cell file has a continuous development in the apico-basal direction. Comparing division systems between the three regions indicates that the lifespan values could increase with distance from the root tip, since the values of system P (6,5,5,4) ## in region B are greater than those of system P (4,3,4,5) in $ region A. For the minimal PE systems which occur in the proximal zone of the meristem, even longer relative lifespans [e.g. (10,9,9,8)] apply. In order to harmonize the three pathways, the lifespans defining each system should be given values that correspond to actual interdivisional times. This was done as follows. Assuming a mean cell length of 8 µm, six cells will occupy the length (50 µm) of region A. This region includes the functional initials. Within a 48 h period following germination, region A has produced a file

of 41 cells which now lies between 0 and 330 µm from the QC zone. The growth function for the system which describes pathway P in this region (see Table 1 in Lu$ ck et $ al., 1994 b) allows the actual lifespans of cells to be estimated. The increase from six cells (at timestep i ¯ 8) to 41 cells (at timestep i ¯ 19) is achieved during 11 timesteps. Because these steps occur within a period of 48 h, one timestep represents, on average, 4±36 h. Accordingly, the lifespans (4,5) (4,3,4,5) pertaining to P have actual durations (Table $ 1) of (17±4 h,21±8 h) (17±4 h,13±1 h, 17±4 h,21±8 h), which conform approximately to mean interdivisional periods found in earlier studies of the maize root apex (Barlow, and Macdonald, 1973 ; Barlow, 1987). It should be noted that these time periods in the bootstrap system refer only to successive transverse divisions ; longitudinal divisions, which might occur in these regions, and which would have been included in the earlier studies of cell cycle durations, have been ignored. Similarly, a file of six cells located initially in region B, 225–275 µm from the root tip, produces, after 48 h, a file of 38±5 cells which is now located between 1000 µm and 1300 µm (see Fig. 2 in Lu$ ck et al., 1994 a). The growth function of the bootstrap system (5,4) (6,5,5,4) describing pathway P (see Table 1 in Lu$ ck et al., 1994 b), shows that ## the increase from six cells (at timestep i ¯ 9) to 37 cells (the nearest number to 38±5 at timestep i ¯ 22 in the Table referred to) requires 13 timesteps. Thus, one timestep lasts, on average, 3±69 h (Table 1). The actual lifespans of cells originating in region B and following pathway P ## are accordingly (18±5 h,14±8 h) (22±1 h,18±5 h,18±5 h,14±8 h), values which also conform to estimates of cycle durations in these more proximal cortical meristem regions (Barlow and Macdonald, 1973) where cell divisions are mainly transverse. Actual lifespan durations for the 8–16 cell packets can also be estimated according to system PE . Again, there are " six cells at timestep i ¯ 15 and these grow up to 40 cells at timestep i ¯ 41, as shown by the growth function in Table 2A. Thus, because 26 steps are accomplished in 48 h, one timestep is equivalent to 1±85 h, resulting (Table 1) in lifespans of (16±7 h,11±1 h)(18±5 h,16±7 h,16±7 h,14±8 h) for PE . For PE (where there are six cells at i ¯ 9 and 39 cells at " # i ¯ 36 (Table 2B), i.e. 27 steps in 48 h), one timestep is achieved in 1±78 h. All four pathways considered (Table 1) evidently have cellular lifespans of similar duration, although they had different relative values in their theoretical description. Moreover, it is inherent to the bootstrap device that there should be positionally determined variation of lifespans within the cell packets. Such variation is not incompatible with any of the experimental analyses of cell cycle durations, where quite large standard deviations were associated with the mean durations (see Barlow and Macdonald, 1973). Indeed, this variability may indicate some inherent and determinate feature of the lifecycle durations which could, in turn, be reflected in the observed distribution of cell sizes within the packets. Comparing the lifespans within the quartets, it is evident from Fig. 1 that pathways PE and PE describe patterns in " # which a wave of divisions starts at the distal (apical) end of a packet and travels towards its proximal end. This pattern

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment Ta b l e 2. Recursive growth functions for (A) a four-parameter bootstrap L-system and (B) an eight-parameter system that define pathways PE1 and PE2 (both in region E) with lifespans (9,6) (10,9,9,8) and (9,8)(12,11,11,10,10,9,9,8), the initial pair of lifespans being those that are found at the beginning of the descendance A

Quarter

B

PE1 (9,6) (10,9,9,8)

Octet

PE2 (9,8) (12,11,11,10,10,9,9,8)

Timestep Two partial complementary series

Four partial complementary series

i

g(i)

h(i)

f(i) = g(i) + h(i)

g(i)

0

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 5 6 7 8 8 8 8 8 9 10 12 14 15 16 16 16 17 18 21 24 27 30 31 32 33 34 38 42 48 54 58 62

1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 5 6 7 8 8 8 8 8 9 10 12 14 15 16 16 16 17 18 21 24 27 30 31 32 33 34 38 42 48 54 58 62 64 66 71 76

2 2 2 2 2 2 3 3 3 4 4 4 4 4 5 6* 6 6 7 8 8 8 9 10 11 12 13 14 15 16 17 18 20 22 24 26 28 30 32 34 37 40 * 44 48 52 56 60 64 69 74 81 88 96 104 112 120 129 138

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 5 6 7 8 8 8 8 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 . . .

5

10

15

20

25

30

35

40

45

50

55

h(i) (1) 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 5 6 7 8 8 8 8 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34 . . .

k(i)

l(i) f(i) = g(i) + h(i) + k(i) + l(i)

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 5 6 7 8 8 8 8 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34 . . .

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 5 6 7 8 8 8 8 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34 36 39 42 45 . . .

3 4 4 4 4 4 4 4 5 6* 7 8 8 8 8 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34 36 39 * 42 45 48 52 56 60 64 69 74 79 84 91 98 105 112 121 130 139 . . .

For the quartet in (A), the growth function f(i) is composed of two partial complementary series g(i) and h(i). For the octet in (B) there are four partial complementary series g(i), h(i), k(i), l(i). The recurrence indices for the quartet are (10,12,6,8) and for the octet (12,12,12,12,8,8,8,8). Arrows show the backsteps used for the two bootstrap systems. They represent the cell state derivation graph of the system (Lück et al., 1994b). The values in each column of the Table (except for the timesteps) form part of an infinite exponential series defined by the growth in cell numbers (or cell lengths) in complementary portions of the packet. The reading of these values from left to right in (A) and (B) corresponds to the basi-apical orientations of the corresponding sets of cells. Values denoted by the superscript * were used to evaluate actual timestep durations.

repeats at each cell generation. The many other, but less frequently encountered, pathways in this region (Lu$ ck et al., 1994 a) can be considered as deviations from this scheme.

379

On the other hand, the division pattern in region A (P , Fig. $ 1) is inverted with respect to those of the other regions, the lifespans being apparently longer in the more distal cells of a packet. This may partly be the result of unobserved longitudinal radial cell divisions that accompany the increasing tangential growth in this zone. The two PE pathways are able to simulate the development described by P found in the more distal region. The ## lifespan values of PE are obtained by adding four units to " all the values of P , though here the lifespans of the initial ## pair of ancestor cells have a different relation (four and two units are added). This slight inconsistency can be avoided by using an eight parameter system, such as PE . The two non# overlapping quartets of lifespans which compose the octet of PE are obtained by adding six and four units to the # lifespans of P , respectively (Table 1). However, these ## additions of an integer to all lifespans of a quartet or an octet will not generate quite the same distribution of cell lengths as that for P (Fig. 1). Notwithstanding this small ## discrepancy, we will take for the moment pathway P as a $ descriptor of cellular development of region A, and PE as " representing the development of the rest of the meristematic cell file. VARIATION IN DIVISION PATHWAYS DURING ROOT GROWTH In the pre-emergent primary maize root apex there are about 125 potentially meristematic cells at the distal end of each cortical cell file. At germination, they begin to divide again. Each of these cells is a mother cell for a pre-formed cell packet in the emerging radicle, so-called because the mother cell was formed prior to germination. Cells in the packets derived by divisions after germination become, in their turn, mother cells for the formation of new cell packets ; we call these new packets post-formed packets. The question now is whether the post-formed packets, particularly those formed from mothers arising at the second round of post-germination divisions and subsequently, behave in the way described for the pre-formed packets. The behaviour of post-formed packets was inferred from their observation at a fixed location in the centre of the cortical meristem, 600 µm from the root tip. At this point, the length of each cell in an octet was measured at seven different times following germination. The octets appear to flow past this point, having been initiated from either pre- or post-formed mother cells located in progressively more distal locations (Fig. 2). Two additional observations were made on octets of cortical cells : in region A before germination and in a proximal region 1000 µm from the tip in older, postgerminated roots (Table 3). The structure of the octets is described by comparing the lengths of each pair of sister cells, b and a, contained within. These pairs reflect each of the three previous rounds of cell division that generated the eight cells : that is, the doublets (b,a) of generation one become the quartets (bb,ab,ba,aa) of generation two, which then become the octets (bbb,abb,bab, aab,bba,aba,baa,aaa) of generation three, the subscripts indicating the apical (a) or basal (b) characteristic of the mother or grandmother cell. As a consequence of this

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment

380

Germination 1000

AABA

900

Distance from root tip (ím)

800 700 ABAB

BBBB

ABBB

BBBB

AAAA

AAAA

AAAA

600 500

AA

400 300

AB

AA

BB

BB

AB

AB

AA

A 200 A

A

A

B B

B

100

A

AAAA

0 –40

–20

0

20 40 Time of observation (h)

60

80

F. 2. The estimated position of cellular packets at different times during root growth together with indications of their types of cell division behaviour. Packets were observed, always as octets, at a fixed location 600 µm from the tip, although one group of packets was observed at 1000 µm. From measurements of the lengths of the individual cells and their position within the octets, it is possible, with knowledge of the growth functions for the division pathways which the packets adopt (see Table 2 for PE , and Table 1 in Lu$ ck et al. (1994 b) for P ), to estimate their " $ position at earlier times and at earlier states of development (i.e. when the packets were quartets, doublets and mother cells). The estimated trajectories, defined by the growth functions, of the various developing packets is indicated by curved lines. Some of these trajectories extend back to a time before germination and can therefore indicate packet behaviour in the embryonic radicle. That the packets change their developmental pathway is evident from the changing patterns of cell sizes in the packets passing the 600 µm reference point with increasing times following germination. A and B refer to cells with A-type or B-type behaviours of the various pairs of sister cells in the growing packets (see Fig. 3).

Table 3. Mean cell lengths and the division sequences in octets located 600 lm from the root tip, and at two other locations, all within the cortex of the root tip of Zea mays at different times following germination Root

Octet Mean length of cells (lm)

Structure of pairs of sister cells

Length (mm)

Age (h)

Position (lm)

bbb

abb

bab

aab

bba

aba

baa

aaa

0 10 30 60 80 105 130–140 70 105

dormant 14·7 33·6 52·5 62·6 74·2 85·3–89·9 57·7 74·2

600 600 600 600 600 600 600 Above QC 1000

10·42 10·42 08·44 08·98 09·34 10·60 09·34 17·42 12·39

09·88 11·14 09·34 08·26 08·44 09·52 08·62 16·52 12·21

08·98 08·80 09·88 08·98 08·62 09·88 08·80 15·09 12·03

09·70 10·42 10·96 09·16 08·44 09·16 08·44 14·37 11·85

09·70 09·16 08·26 08·80 08·62 10·06 08·80 11·14 11·31

09·16 10·06 09·34 09·16 08·44 10·06 08·44 10·96 11·85

08·80 08·44 09·70 09·34 08·98 10·42 08·62 09·34 11·49

10·42 10·60 10·96 09·70 08·98 09·88 08·44 09·16 11·31

Octets

Quartets

Doublets

The right-hand side of the table shows the packet structure in the different cell generations. The longer cell of each pair of sister cells is marked in black.

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment T     4. Times required for packets to acquire the status of octets, quartets and doublets according to different pathways, P, of deŠelopment specified by the respectiŠe bootstrap systems Packet structure Octets Pathway P $ P ## PE " PE

#

Steps 13 18 30 32

or or or or

16 16 26 28

Time (h) 63±22 62±73 51±80 51±30

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or or or or

11 10 16 18

43±60 40±59 33±30 32±49

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or or or or

6 5 7 9

Time (h) 23±98 20±30 15±73 16±25

According to the bootstrap devices (see Fig. 1), there are two possible times to achieve the various packet configurations. The actual times represent the average of these possibilities. The order in which the steps are presented (e.g. 32 or 28) reflects the basi-apical orientation of the packets.

temporal-spatial development, each octet contains a memory of the physical structuring (e.g. the cell lengths) of each preceding generation. The structured cell length distribution along an octet (Table 3) emphasizes the inequalities, due to asymmetric cell division, in consecutive, but related, pairs of cells along the packet in the basi-apical direction. Octets observed at 600 µm from the tip can mostly be generated only with parameters of a bootstrap system that defines one of the possible pathways cited in Table 1. From their respective growth functions (see Table 1 in Lu$ ck et al., 1994 b, and Table 2 in the present paper), and with the already estimated actual duration of a timestep (Table 1), each division state of the octets, as well as the location of their mother cells with respect to the root tip, can be precisely determined at different times during root growth (Fig. 2). The different pathways achieve the observed cellular configurations in different periods of time. In P , $ about 63 h are required ; the PE pathways require about 51 h (Table 4). It follows that octets analysed at 600 µm in younger roots (! 63 h-old and ! 51 h-old for P and PE , " $ respectively) are all descended from pre-formed ancestral

Region A

P3

b

cells, whereas in older roots they must be descended from post-formed ancestors. The lifespans of pre-formed quartets and doublets have also been estimated (Table 4). Reading the basi-apical orientation of cells within a packet from left to right, the structuring of two theoretical cell packets up to the octet stage for pathways P and PE is " $ shown in Fig. 3. The structure of a quartet of cells descending from region A and behaving according to P , $ conforms to what we refer to as ‘ A-type ’ cell behaviour and has, on average, a basal pair of equal-sized cells associated with an apical pair in which the basal cell is the larger. Its structure appears late (" 63 h) in root development (Table 3). By contrast, quartets of cells descending from region B, and following pathway PE (conforming to ‘ B-type ’ cell " behaviour) show longer apical cells in both pairs of cousin cells. The patterns are repeated at the octet stage and they appear earlier in root development (15–34 h) (Table 3). Interpreting the data from the cell packets observed at 600 µm in terms of these A- or B-type cell behaviours (Fig. 2), it follows that most pre-formed octets have B-type cell behaviour, whereas post-formed octets have either an Atype or a mixed A- and B-type cell behaviour. Extrapolating back in time, prior to germination, the initial doublets or quartets of the pre-formed packets mainly behave as A-type cells. However, octets in dormant radicles show evidence of having had mixed behaviour, as though A- and B-type cells co-existed. These features point to a change, at the resumption of root growth, of the behaviour of all pre-formed cells. They take on a B-type of behaviour, the exception being cells above the QC which conform exclusively to A-type cell behaviour. Later, in the zone distal to the pre-formed packets, the cells tend once more to take the characteristics of A-type cells. This is also the case at the proximal end of the meristem, at 1000 µm, except for one pair of sisters within the octets (see Table 3). These findings can be summarized as follows. In the embryo, the cells have A-type behaviour and adhere to P $ but, proximal to 600 µm from the tip, there is also some B-type cell behaviour in accordance with P . Upon ger## mination, the zone behind the tip in which cells have the Atype of behaviour becomes very short and, up to at least 600 µm from the tip, there is B-type cell behaviour

Region B

a

381

PE 1

b

a

F. 3. The structured development of two theoretical cell packets up to the octet stage following (A) pathway P and (B) pathway PE . Differences $ " between cell lengths of sister cells were assessed for statistical significance. The longer cell of each pair is fully shaded ; in the case of near-equality, both cells are lightly shaded. b, a : the basal and apical ends of the packet.

382

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment

T     5. Mean octet length in relation to their location within the root and to the length (age) of the root Mean length of octet Location 600 µm from tip

Above QC 1000 µm

Root length (mm)

µm³s.e.

Dormant 10 30 60 80 105 120–130 70 105

79±8³1±5 81±9³1±9 79±8³1±6 75±1³1±5 72±3³1±6 81±9³1±8 72±3³1±4 99±9³3±2 97±9³3±5

Loge µm ³s.e.

n

4±36³0±02 4±38³0±02 4±36³0±02 4±30³0±02 4±26³0±02 4±38³0±02 4±26³0±02 4±56³0±02 4±53³0±03

84 100 100 100 100 100 100 88 100

everywhere. Later, the zone with A-type behaviour enlarges and, in roots of 80 mm length (52 h after germination), it has extended to a distance of 600 µm from the tip. Later still, it may spread along the entire meristem, with A-type cells gradually displacing all the B-type cells. In other words, cellular behaviour within the meristem changes when growth resumes and returns to the pattern of behaviour which had previously acted in the embryo. However, it is also possible that the B region is progressively displaced away from the tip of the root due to an extension of the meristem itself. The main characteristics of A- and B-region cells are that, in the A-type packets, the basal part of the packet grows and its cells multiply faster than those of the apical part. By contrast, in the B-type packets, any cell located apically within a pre- or post-formed packet tends to enter division sooner (irrespective of cell generation). The A- or B-type of cell behaviour does not seem to be related to the length of the octets (Table 5). Their lengths are greatest just above the QC and at the proximal end of the meristem, but are least in the centre of the meristem. The change from A-type to B-type cellular behaviour is a change only in the polarity of the octets with respect to the location of the faster-growing sister cells. The location of shorter cells and increased proliferation rate would correspond to a junction between the A and B zones.

SETTING THE MARGIN OF THE MERISTEM The pathway PE satisfactorily describes the pattern of " cellular development within a generalized cortical cell file of a growing root. However, since the underlying division process produces cells at an exponential rate, it is necessary to introduce limits to the operation of this bootstrap system within the file. There is, therefore, a demand not only for the eventual arrest of proliferative activity at some point along the file, but also a means of introducing new cells into a file. A meristem may be said to be established when a proliferative region is delimited by proximal and distal margins. An oriented root meristem exists during the embryonic phase of radicle development, is absent during dormancy, but is re-established upon germination, eventu-

ally attaining and then maintaining a constant length. The proximal margin of the cortical meristem is about 1000 µm from the dome of the root ; a zone just behind the dome, on the basiscopic surface of the quiescent centre, may be taken as the distal margin of the meristem. The proximal margin defines the point at which cells leave the meristematic phase whereafter they continue to elongate for a certain time. The distal margin defines where new cells are injected into preexisting files. Except for the stem cells, cells in a lineage do not divide indefinitely, but cease mitotic activity after a definite number of generations. At the molecular level, recent evidence suggests a possible mechanism for regulating such a phenomenon. In Šitro, cell populations divide with concomitant loss of telomeric DNA sequences during each successive cell division cycle (Hastie et al., 1990 ; Harley, 1991) and it is thought that the loss of these sequences provides a counter of the number of divisions accomplished by cells within a lineage. The proximal limit of the meristematic zone of the root could be regulated with an analogous device which might involve the inactivation of cyclin gene expression (see Doerner et al., 1996 ; Zhang, Letham and John, 1996). Another related possibility is that cell division potential is limited as a result of some influence from the mature portion of the root (Barlow, 1996). Suppose that after a number (λ) of cell generations with respect to a given mother cell (λ ¯ 0), cells within a file stop dividing (Lu$ ck and Lu$ ck, 1978). A counter (k) of successive cell generations must therefore supplement the bootstrap formalism of the cellular system. Thus, cells enter the elongation zone when k ¯ λ. The final number of cells which have ceased division activity, and which have all descended from a common ancestor cell, is equal to 2λ.

An elementary sigmoidal growth curŠe From an automata-theoretical point of view, the following two rules apply for the exponential development of a cell file : (1) cell state transition rules such as cn,p U cn+ ,p, and (2) " diŠision rules such as cmx,p U b ,q a ,q, where, with respect to " " c, the cell : c, p, q ` ²a, b´, with a and b being apical or basal sister cells located in a file, and p and q determining the apical or basal character of the mother cell ; with respect to mx, the lifespan of a cell : mx ` ²mbb, mab, mba, maa´ denotes the four lifespans differentiated within a quartet of four cousin cells, the left-hand subscript denoting the b or a character of the cell and the right-hand subscript denoting the character of the mother cell ; and with respect to n, the cell states (timesteps) traversed during a lifespan : n ` ²1, 2, … , mx− , mx´ " denotes each of the possible cell states according to the derivation graph of Lu$ ck et al. (1994 b) and Table 2 in the present paper. As an example, in pathway PE , where (mbb, " mab, mba, maa) equals (10,9,9,8), a cell, c, will possess the characteristics indicated in Table 6. Terminal states in the cellular state alphabet Σ lead to cessation of growth following the cessation of cell divisions. A maximum number, λ, of successive divisions is permitted in a packet, where k (the division counter) indicates the number of divisions accomplished. Accordingly, k ` ²0, 1, … , λ´.

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment T     6. Characteristics of a cell, c, within a bootstrap system that generates pathway PE " c bn,b an,b bn,a an,a

Orientation

Mother cell

b a b a

b b a a

Lifespan (mx) Cell states (n) 10 9 9 8

(1, … ,10) (2, … ,10) (1, … ,9) (2, … ,9)

a and b refer to a cell’s apical or basal location within a packet. n indicates the state or current timestep of a cell within the interdivisional period.

For cell proliferation to cease, the cell production rules are extended to (1) cell state transition rules : cn,p,k U cn+ ,p,k ; " (2) diŠision rules such as the following : cmx,p,k− U b ,q,k a ,q,k ; " " " and cmx,p,λ− U b ,q,λ, a ,q,λ ; and (3) terminal state specification, " " " such that : cmx,p,λ U cmx,p,λ denoting that once a cell reaches the last development state mx in the cycle and in the final generation where k ¯ λ, the cell will neither divide nor show any further change. For a two-parameter system based on two lifespans (mb, ma), i.e. mbb ¯ mba and mab ¯ maa, and which conforms to a simple Fibonacci series, the growth function h(i) is composed of an exponential part f(i) from which the non-produced cellular descendants of terminal state cells g(i) are subtracted. The frequency of occurrence of terminal state cells is given by binomial coefficients (Lu$ ck, 1975 ; Lu$ ck and Lu$ ck, 1978). Terminal state cells t(i) persist and have to be included in the function. The function is thus : h(i) ¯ f(i)®g(i)­t(i),

and

i & 0;

f(i) ¯ f(i®mb)­f(i®ma), the exponential Fibonacci growth series ; and

01

λ λ g(i) ¯ 3 f [i®(λm­kd)], k k=! the non-produced packets of cells in terminal state. The significance of m, λ and k are as mentioned above. d ¯ mb®ma, and m is the lifespan of the first cell. Then, ! λ λ f [i®(λm­kd )®1]. t(i) ¯ g(i)® 3 k k=! The first indication of a deviation from exponential development, i.e. when the first cell acquires a terminal state, appears at time i ¯ m ­λm®1, with m ¯ min ²mb, ! ma´. Complete cessation of division occurs at i ¯ m ­λm, ! when m ¯ max ²mb, ma´.

01

Frequencies and locations of terminal-state cells Considering now the bootstrap L-systems which act over quartets or octets of lifespans, their sigmoidal growth functions are of the same structure as for doublets, but have more sophisticated expressions. In these more refined systems, a and b cells may have different periods of lengthening, and so the time of appearance of their terminal states has to be given separately in each case. This is

383

illustrated in the simulations of Fig. 4 B where, at each step, the frequencies fra and frb of new terminal-state a and b cells, respectively, are shown separately. The frequencies clearly follow binomial coefficients. However, as already mentioned, to conform with observations from the maize root cortex, this system PE , has to be initiated by a pair of " cells with lifespans (9,6) instead of (9,8), whereupon the binomial coefficients become hidden and the two classes of terminal cells, a and b, appear temporally displaced, or staggered (compare Fig. 4 C with 4 B). The sigmoidal growth curve obtained from the system P (Fig. 4 A) is even $ more complicated. Here there are clearly different frequencies of a and b cells ; the growth curve therefore cannot be formulated in conformity with a simple Fibonacci series. The frequencies of a and b cells are obtained from the binomial coefficients. For increasing values of λ, the frequencies are summed cumulatively at successive timesteps. In the case of PE , starting with initial lifespans (9,8) " and where λ ¯ 3, the binomial coefficients are 1, 2 and 1 ; these are the frequencies of terminal a cells at timesteps i ¯ 23, 25, and 27, while for the b cells these frequencies occur at timesteps i ¯ 24, 26, and 28 (i.e. the frequencies of the b cells are displaced by one timestep relative to those of the a cells). When the values of two such series, temporally displaced by one timestep, are summed, the result is the binomial coefficients for λ ¯ 4, i.e. 1, 3, 3 and 1 (these being the frequencies of the terminal a cells at timesteps i ¯ 31, 33, 35, and 37, and of the terminal b cells at timesteps i ¯ 32, 34, 36, and 38) ; and so on for each successive value of λ. With the initial lifespans (9,6), the value fra ¯ 2, at i ¯ 25 (or frb ¯ 2 at i ¯ 26) is replaced by two values of fra, now at times i ¯ 23 and 25 (or frb ¯ 1 at i ¯ 24 and i ¯ 26). When frequencies of terminal state a and b cells differ, as in system P , the addition of frequencies with increasing λ values is $ cross-wise, i.e. the values of frequencies fra are added to those of frb. For the determination of new coefficients, the temporal staggering of the two series can become quite complex. In fact, in all bootstrap systems the addition of frequencies is cross-wise, but this is not explicit when the frequencies of a and b cells are identical. Up to the inflection point, the growth curves of the cell populations are identical to exponential curves. The sigmoidal growth curves of Fig. 4 are asymmetric, especially for PE , where the divisions in each cell generation occur in " a basipetal wave. The inflection point is situated in the upper part of the curve, at i ¯ 45 (Fig. 4 B and C). With regard to cell numbers, the cell population changes abruptly from exponential increase to non-growth. This is not the case for P , for here some divisions still occur after the first $ appearance of terminal cell states at i ¯ 20 (Fig. 4 A). After cessation of growth, there are some more steps before all cells reach a terminal state, at i ¯ 30 and i ¯ 59 for P and $ PE , respectively. " THE COMPLETE DEVELOPMENT OF A CORTICAL CELL FILE Having established the behaviour of a cell packet as it completes its allotted number (λ) of cell divisions, the corresponding bootstrap systems and growth functions can

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment

384 A

5

10

4

8

3

6

2

4

1

2 5

Log2 number of cells ( , )

0 6

10

15

20

25

30

35

40

45

50

55

0

B

5

10

4

8

3

6

2

4

1

2 5

0 6

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15

20

25

30

35

40

45

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55

0

C

5

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8

3

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2

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1 0

Number of t-state cells (fra or frb) at each timestep

6

2 5

10

15

20

25

30 35 Timestep (i )

40

45

50

55

0

F. 4. Simulation of the sigmoidal pattern of growth in cell numbers in an idealized cortical cell file of a Zea mays root, with cell division ceasing after λ ¯ 6 generations. A, Growth curve for a cell file following pathway P . For each developmental step i (where i ¯ 1 , … , 29), (E) indicates the $ total number of cells generated and (D) indicates the number of cells that have ceased division and which have reached the terminal state t. Note that the numbers of both types of cells are presented as a doubling series (see the left-hand axis) ; i.e. cell numbers have been converted to their log value. The t-state cells may be either a or b cells and their respective frequencies, fra and frb, are indicated by unfilled and filled bars at each # timestep i. Their numbers (untransformed values) are indicated on the right-hand axis. The inflection point in the growth curve occurs at i ¯ 20 ; at i ¯ 30 all cells have reached the terminal state. B and C, Growth curves for pathway PE over i ¯ 59 timesteps. In B, which applies to PE , the " initial pair of lifespans is (9,8). At i ¯ 47 the inflection point occurs ; at i ¯ 59 all cells are in the terminal state. Symbols as for A. In C, illustrating PE also, the initial pair of lifespans is (9,6) in order to conform with the observed pattern of cells within the root cortex. The inflection point is " at i ¯ 45, and at i ¯ 59 all cells are in the terminal t-state. Symbols as for A. In B and C, unfilled bars precede filled bars because, according to the parameters of PE, the appearance of a and b t-state cells alternates in time, a preceding b, whereas in A, a and b t-state cells occur simultaneously in a time-step.

now be incorporated into a model that simulates the development of an entire cortical cell file. At the distal end of the cell file, there is a small zone of relatively slowly-growing founder cells. These cells can only have notional lifespans because of the existence of unseen radial and periclinal cell divisions. The founder zone juxtaposes with the exponentially and rapidly growing meristematic region. Beyond the proximal margin of the meristem, there is an elongation zone in which cells progressively stop their divisions. Their additional, but finite, amount of elongation will be considered in another paper, but has been briefly described by Lu$ ck, Barlow and Lu$ ck (1996). Each cortical cell file can be considered to be derived from a founder cell which is renewed periodically by one of the daughters of a stem cell (Barlow, 1997). Descendants of

the founder cell cease to divide after λ generations and the founder itself does not divide during this period. This situation is achieved in a bootstrap system of four parameters (i.e. with a quartet of cells (bb,ab,ba,aa) which obeys the following division rules : a\,a, U b\,a, a\,a, " " " b\,a,k U b\,b,k+ a\,b,k+ " " a\,b,k U b\,a,k+ a\,a,k+ " " b\,b,k U b\,b,k+ a\,b,k+ . " " The cell states need not be considered (their non-descript state is indicated by the left-hand subscript \), the middle (a or b) and right-hand (1, k or k­1) subscripts denote, respectively, the a or b location and the cell generation numbers of the mother cells. As a result, an oriented lineage

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment Basi-apical direction

0

a.,a,1

st

1

b.,a,1 a.,a,1

2

b.,b,2 a.,b,2 b.,a,1 a.,a,1

st ... .. .. .

3

b.,b,3 a.,b,3 b.,a,3 a.,a,3 b.,b,2 a.,b,2 b.,a,1 a.,a,1

4

t t t t b.,b,3 a.,b,3 b.,a,3 a.,a,3 b.,b,2 a.,b,2 b.,a,1 a.,a,1 4

5

t

t t t t b.,b,3 a.,b,3 b.,a,3 a.,a,3 b.,b,2 a.,b,2 b.,a,1 a.,a,1

6

t8 t t t t b.,b,3 a.,b,3 b.,a,3 a.,a,3 b.,b,2 a.,b,2 b.,a,1 a.,a,1

Number of cells ( , )

st

220 200 180 160 140 120 100 80 60 40 20 0

A

8 6 4 2 0 20

220 200 180 160 140 120 100 80 60 40 20 0

. . .

···

···

···

···

···

···

···

a.,a,1 . . .

etc F. 5. A refined sequence of cell productions within a cell file that incorporates a founder cell (a\,a, ) at the apical, or distal, end of the " file, and terminal cells, t, at the proximal end. After a period of one lifespan (a cell generation), new cells arise within the file in accordance with the division rules described in the text. In this instance, there are λ ¯ 3 generations before the terminal state is reached.

appears, always commencing with a\,a, cells. After λ " generations, cells acquire a terminal proliferative condition, as shown in Fig. 5 where, as an example, λ ¯ 3. The spatiotemporal structure of the meristem is maintained with respect to the disposition of cell generations along the file. At first there are two cells in generation one (one of them being the stem cell), then two cells in generation two, four cells in generation three, and at each subsequent division step four new terminal state cells, t, are added. It becomes immediately apparent that the total cell number will increase linearly, in spite of the presence of the eight meristematic cells which proliferate exponentially. Taking now lifespan values instead of generation times for the four parameters of P (Fig. 3 A), or those of PE (Fig. " $ 3 B), and redefining them in the way described above, the simulations of Fig. 6 show the structuring of the meristem, the progressive formation of the elongation zone represented by non-dividing cells, and the linear growth of the root. It is noteworthy that the terminal cell states appear, not together, but in a dispersed order along the file. The different zones have precise courses of extension which vary with a periodicity governed by the interdivisional duration of the stem cell. Thus, at every maath step, a set of 2λ−" cells that have reached their final length is added to the mature part of the root. In the example shown in Fig. 6 B, this occurs at i ¯ 58, 64, 72, 80 etc. with packets of 32 cells leaving the meristem at each of these timesteps. The result is that the

30

35

40

45

B

8 6 4 2 0 50

12 7 t ···

25

55

60

65 75 70 Timestep (i)

Number of t-state cells (fra or frb) at each timestep

Generation

385

80

F. 6. Simulation of the linear growth in cell number in a cortical cell file of a Zea mays root following either pathway P (A) or pathway PE $ " (B), and using the refined division rules mentioned in the text (see also Fig. 5). Only values between timesteps i ¯ 16 and i ¯ 46 (A) and i ¯ 47 and i ¯ 84 (B) are shown. Until timesteps i ¯ 22 and i ¯ 45 in each case, the growth curves are identical to those shown in Fig. 4 A and C (except that here the cell numbers, indicated on the left-hand axis, have not been transformed to logarithmic values). Cells cease division after λ ¯ 6 cell generations, while periodically a new cell is injected into the meristematic cell file from the pool of stem cells. A, Growth curve applying to pathway P . Symbols as for Fig. 4. The inflection point in $ the curve occurs at timestep i ¯ 20 and linear growth then continues. There is a portion of the meristem where terminal, non-dividing cells (tstate) mingle with cells still committed to division. Whether they are b or a cells is indicated, as before. The numbers of these t-state cells is indicated on the right-hand axis. B, Growth curve applying to pathway PE . Similar explanations as for A. The inflection point in the curve "occurs at timestep i ¯ 49, whereupon linear growth commences.

root grows quasi-linearly, even in the presence of an exponentially expanding apical zone. Linear growth is highlighted by the curve representing the number of cells in a terminal state. If the generation times are considered, then each doubling of cell numbers along the file takes, on average, 17±4 h for cells in pathway P , and 16±7 h for cells in $ pathway PE . Therefore, when λ ¯ 6, cells take about 4±2 d " to traverse the entire length of the meristem by means of transverse divisions and elongation growth. All these features approximate to those known for the developing primary root in Zea mays. DISCUSSION An automata-theoretical model of cell file growth, which makes use of a bootstrap L-system algorithm, successfully simulates the development of the cellular packets which comprise the cortical portion of the maize root primary meristem. Inherent to the bootstrap system is a spatially ordered variability of cell lengths and cell lifespans

386

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment

(interdivisional periods). Thus, the observed range of cell lengths and the pattern of asynchronous cell divisions within packets of the cortex can be readily accommodated by a suitable division algorithm. Moreover, the division patterns which develop as a result of the bootstrap system provide an explanation for some of the variation of interdivisional periods within areas of histological uniformity, noted using another method of analysis (Barlow and Macdonald, 1973). The bootstrap system also predicts that, along a cell-file, there will be no definite border between the meristematic and non-meristematic zones. As was shown by simulation, the border will be imprecise and will fluctuate rhythmically. At a certain distance along a file, non-cycling cells will be interspersed with dividing, cycling cells before division finally ceases at a slightly greater distance from the tip. The mixture of non-cycling and cycling cells in the proximal portion of a meristematic cell file has been discussed in terms of ‘ division probability ’ (Erickson, 1961) or ‘ growth fraction ’ (Lo! pez-Sa! ez et al., 1983). The former was estimated by direct observation of files of cells in the epidermis of Phleum pratense roots : whether or not a cell would divide once it had reached a certain distance from the tip defined the division probability for cells at that location. In the case of growth fraction, whether or not a cell divided was predicted on the basis of a model in which the ability to divide was expressed over only a certain distance from the root apex and there was also a counter of the number of divisions permitted within any given cell file. The present bootstrap system, together with a division counter, allows both these situations to be reinterpreted. By so doing, the necessity of hypothesizing longitudinal gradients of divisionpromoting substances (Lo! pez-Sa! ez et al., 1983), a threshold concentration of which determines whether or not a cell will divide, is eliminated. Moreover, in the present system a growth fraction less than unity in proximal regions of the meristem is a natural outcome of the bootstrap system. Only a division counter is required ; there is no need for an arbitrary positional limit to mitosis. However, the terms ‘ division probability ’ and ‘ growth fraction ’ can still be usefully employed in the specific context of structured groups of cells, for instance, octets (or even quartets), to describe the proliferative potential of the packet after different numbers of cell generations have been accomplished (e.g. Fig. 3). The distribution of mitoses along a cell file is a function of the file’s location within the tissue. In the central cortex, for example, the distance from the root tip to the most proximal mitoses is less than in either the inner and outer cortex, or in the epidermis and pericycle (Luxova! , 1975). This could be because different bootstrap systems apply in the respective cell files. In fact, this seems quite likely since in region B of the cortex, although P was frequent, other ## pathways also existed. The minority pathways may have been representative of cell files which were infrequently included in the sample, perhaps because of their more peripheral location. Hence, detailed analysis of these other locations (e.g. inner and outer cortex) might provide evidence for an association with other pathways. These would have different growth functions and hence mitosis

would terminate at different distances from the tip. An additional possibility for the variation in the longitudinal extent of mitosis is that some files have different values of λ, the number of division cycles. This is evident in the metaxylem cell files of the maize root where λ ¯ 4, and is particularly conspicuous in the root cap files where λ ¯ 2. These considerations raise two important questions about meristem structure generally ; (1) is there a relationship between division pathway P and the histological identity of a cell file ? and (2) is there a division counter and, if so, how is its value, λ, set in any given cell file ? With respect to the first question, specificity of packet pathway is assumed for the reason that, of the large number of pathways theoretically available, there is evidence of only a few actually being used, and these differ according to the location of the cell files within the root (Lu$ ck et al., 1994 a). Since the maize root (or any root) approximates to a bundle of concentrically arranged cell files, it is possible to envisage that different pathways pertain to different cell files according to their radial distance from the root axis (or root surface). The regular occurrence of specific pathways in certain locations is indisputable, for the patterns of cell division are demonstrable within the packets of every maize primary root (of cv. LG11) so far examined. Moreover, the pathways are actively maintained : experimental intervention can disturb them—methanol, for example, forces unequal divisions to become equal—but the normal, expected patterns of divisions later resume in packets when conditions ameliorate (Barlow, 1989). One key to understanding division patterns must reside in the cytoskeleton, an intracellular network of microtubules (MTs) and actin filaments, to mention only two of its readily observable components. The position of a prominent band of MTs in the cortical cytoplasm of preprophase cells has been conclusively shown to determine the points of attachment of the new cell wall at the following cytokinesis (Gunning, 1982 ; Wick, 1991). Thus, the immediate problem of cell division specification could be resolved by discovering the factors responsible for the siting of the preprophase band (PPB) of MTs. Interestingly, MT dynamics, that is, the equilibrium between tubulin monomers, dimers and MTs and their interactions with organizing centres sited on membranes in the cytoplasm, may also show automatonlike behaviour (Hotani et al., 1992). Thus, the cellular automata which appear to be responsible for the structure of the division pathways and the consequent patterning of cells within a file, may, in their turn, be driven by intracellular molecular automata that manifest as pathways of specific MT behaviours (Barlow and Parker, 1996). The bootstrap L-systems described here are D0L systems. In the context of the cells and their packets to which the bootstrap systems apply, this means that there are no informational inputs other than those intrinsic to the cells themselves. There is no scope for cellular interactions. However, within tissues, potentially informational signals are being continually imparted to the individual cells, and some of these signals can be transduced into processes which govern the siting of division walls. Relevant among these are stresses—which may extend across many cells within a tissue—and structural and chemical information

LuX ck et al.—Automata-theoretical Model of Meristem DeŠelopment provided by neighbouring cells and their walls. The importance of stresses for influencing the pattern of transverse division in the proximal meristem of a uniformly growing, straight root is unknown, but may have some significance in regions where cell files curve and come to a focus, as in the zone of formative divisions at the root tip. However, neighbouring cells certainly influence the siting of a new cell wall in a dividing cell, for it has been repeatedly observed that a new wall avoids attaching to a site on the parent cell wall that is opposite an already established attachment site in an adjacent cell. That is, there is a preference for three-way, rather than four-way, wall junctions. Moreover, any given cell file is surrounded, on average, by six neighbouring files, each one having transverse walls that could influence the resultant location of a new wall in the enclosed central file. The off-setting of walls has, in its turn, a basis in an avoidance mechanism exhibited by the PPBs which tend not to centre themselves over threeway junctions, perhaps as a result of strain patterns transmitted through pre-existing walls, or because such junctions represent ruptures in the cytoskeleton-cell-wall continuum. This aversion of a PPB for three-way junctions seems to act over distances of a fraction of 1 µm (Barlow, 1982 ; Gunning, 1982). Therefore, while it may seem reasonable at present that certain components of the cytoskeleton are crucial for division patterns, little is understood of the interactions between neighbouring cells, either laterally between files or longitudinally within files, which could influence the cytoskeleton in ways that could predict the outcome of the bootstrap algorithm. This problem is not confined to division patterns in plant tissues. Locke (1990), for example, has drawn attention to spatial patterns of cytoplasmic and nuclear organization which are inherited from one cell generation to another in animal cellular systems, and for which no satisfactory interpretation exists. It is possible that inheritance of cytoplasmic structure may occur within the root cell packets for, as Gime! nez-Abian et al. (1988) have shown, tetranucleate cells in the onion root meristem, formed by the suppression of cytokinesis, display predictable spatial patterns in the rate at which the individual nuclei complete G and enter prophase. It was suggested that these # rates are controlled by the amounts of cytoplasm immediately surrounding each nucleus. Recent research (briefly summarized by Rousch, 1995) indicates that at least some classes of asymmetric division are under genetic control. Thus, the switch from one division pathway to another (e.g. P giving way to P ) may be the result of some immediately ## $ post-germination genetic programme ceding to another. The automata-theoretical model of cell file development requires the introduction of a limit, λ, to the number of cell divisions accomplished in each file. The loss of telomeric DNA sequences has been proposed as a counter of divisions in human cells in culture (see earlier), though this mechanism has not been studied in cells of tissues growing in ŠiŠo, and not at all in plants. In the root meristem, as mentioned earlier, a molecular counter of this type would be required to operate differently in different tissues to account for the varying number of meristematic cells along specific files in which λ may vary from two to seven or eight. Other

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mechanisms of cell division counting are possible, such as the dilution of some cytoplasmic factor within a lineage or the decay of some periodic biochemical phenomenon (Norel and Agur, 1991). Dilution of a transcription factor for a cyclin gene is a possibility (see Doerner et al., 1996). Moreover, there must be means of switching on the division counter in the functional initial (e.g. Raff et al., 1988) since the counting of divisions would not be desired in the stem cell population. In the telomere model, switching on the counter would be effected through a suppression of telomerase (Counter et al., 1994), an enzyme whose activity normally maintains the telomeric sequences. A corollary to the division-counting model of file development is that once mitoses have terminated, rounds of DNA endoreduplication can still continue in elongating cells. Again, the extent to which this occurs varies depending on the location of the file in question. The switch from mitotic to endoreduplicative cycles is probably accompanied by a change in the pattern of cell-cycle-related gene expression (e.g. Orr-Weaver, 1994 ; Nagl, 1995). In plant cell files, where the changeover from one cycle to the other occurs without interruption (List, 1963 ; Barlow, 1985), the switch must also be coupled with the division counting mechanism. Possibly, there could be a separate mechanism which counts the number of DNA replication cycles, λS. Although it remains to be established, it may be that all files undergo the same λS, but the number of division cycles (λD) varies. It is, of course, possible that there is no counter, but only an ability to turn off divisions as a result of some signal from mature cells (Barlow, 1996) or a decay in cytokinin levels to one which is below the threshold for mitotic activation (Barlow, 1976 ; Zhang et al., 1996). Such a model implies that ‘ positional controls ’ regulate the activity of the meristematic population. At a more formal, molecular level such a control could embrace aspects of the ‘ entelechia ’ model of size regulation of mitotic compartments proposed by Garcia-Bellido and de Celis (1992). A remarkable finding to emerge in the course of the present study is that the pathway of cortical packet development changes during the course of root growth. Central cortical cells whose mother cells were initially located in region B, 250 µm or more from the root tip, undergo a set of markedly unequal divisions in the early stages of root growth (Barlow, 1987). These correspond to pathway P (Lu$ ck et al., 1994 a). Later, the packets resulting ## from this pathway are swept from the meristem by descendants of cells that were initially located closer to the tip. Eventually, all cells in the cortex, and in the meristem generally, are descended from functional initial cells located on the border of the quiescent centre. From time to time, these initials are replaced by other cells emerging from the quiescent centre and in the process they, too, assume the status of functional initials. Cells and packets which develop from these functional initials adopt a different pathway, P $ (or PE), that persists thereafter, even though the cells move through zones of the root where pathway P was previously ## found. Thus, the pathway P is a temporary one, dictated ## by the developmental state of region B, rather than being a pathway constantly adopted by cells in a particular location in the cortex. Moreover, as far as can be ascertained from

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examining cells in the dry radicle, it seems that pathway P $ is also followed by cortical cells during the later stage of embryonic root development. One problem, then, is to understand why pathway P is ## interpolated into the proximal portion of the cortex at germination, whereas in the same region at both earlier (embryonic) and later (in roots " 40 mm length) times file development adheres to a different pathway. This, in turn, raises the further question of whether any of the observed pathways have an adaptive role for growth and survival of the root. Certain arrangements of cell walls may have particular advantages for subsequent cell differentiation as, for example, in the case of the unequal divisions of epidermal cells which anticipate root hair initiation, or pericycle cells that initiate lateral root primordia (reviewed in Barlow, 1984, 1996). Particular cell wall patterns may also have a role in enabling the root to penetrate a physically resistant medium (e.g. surface soil layers) during germination. It is known, for example, that different regions along the length of the maize root have different patterns of flexural rigidity, as assessed from patterns of root bending in response to a longitudinally compressive force (Silk and Beusmans, 1988). However, the contribution of the spacing pattern of transverse and longitudinal walls to these properties, as well as its relationship with the generation of the osmotic forces needed to drive the early growth of the root, is unknown. As suggested earlier (Barlow, 1987), unequal divisions would result in a greater dispersion of cell cycle phases. This may have some adaptive value in circumstances, such as germination and early growth, where there is a transition from an environment on the soil surface, which could be more stressful to a dividing cell population, to one that is more constant underground. It is evidence that an apparently simple dividing system, such as the plant root meristem (which has often proved to be a paradigm for the observation of chromosome and cell division—cf. McLeish and Snoad, 1958) can present many problems of division control that are, at present, beyond the scope of a reductionist, molecular biological approach. It may be objected that an automata-theoretical approach to cell file development is an artificial solution to a problem in pattern formation that could be more usefully addressed in terms of biochemistry, molecular biology, etc. The subjects of these latter reductionist disciplines obey the laws of the molecular level, whereas cellular systems, such as we have described, obey the law of the cellular level. In this sense, the automaton approach to cells is no less reductionist, except that the level of organization to which the system is ‘ reduced ’ is that of the cell, not the molecule. It is, perhaps, more precise to consider it a ‘ structuralist ’ approach to development, in that it seeks to uncover the basic units of structure and the developmental ‘ rules ’ by which these structures are interrelated. Simple biochemical systems, either in ŠiŠo or in Šitro, have predictable outcomes and, in this respect, are deterministic and can be expressed in terms of equations and algorithms. Indeed, their predictability enables their simulation (e.g. Garfinkel and Hess, 1964), which in turn can lead to a deeper understanding of the interplay of the parameters of the system. The same is true of cells, either as individuals or as groups, and their

apparently complex behaviours can likewise be reduced to equations and algorithms ; this principle may even be advanced into the workings of complex ecosystems (Ulanowicz, 1983 ; Kolasa and Pickett, 1989). The bootstrap L-system is but one class of algorithms that encapsulates a dynamic aspect of cellular behaviour within a structuralist concept of biological organization. In time, much of this behaviour will be interpretable in terms of comparable dynamic subcellular processes. But, for the present, the automata-theoretical approach provides a coherent conceptual framework for the analysis of cellular behaviour, and can also lead to a deeper appreciation of the processes that organize plant organs. LITERATURE CITED Barlow PW. 1976. Towards an understanding of the behaviour of root meristems. Journal of Theoretical Biology 76 : 433–451. Barlow PW. 1982. Root development. In : Smith H, Grierson D, eds. The molecular biology of plant deŠelopment. Oxford : Blackwell Scientific Publications, 185–222. Barlow PW. 1984. Positional controls in root development. In : Barlow PW, Carr DJ, eds. Positional controls in plant deŠelopment. Cambridge : Cambridge University Press, 281–318. Barlow PW. 1985. The nuclear endoreduplication cycle in metaxylem cells of primary roots of Zea mays L. Annals of Botany 55 : 445–457. Barlow PW. 1987. Cellular packets, cell division and morphogenesis in the primary root meristem of Zea mays L. New Phytologist 105 : 27–56. Barlow PW. 1989. Experimental modification of cell division patterns in the root meristem of Zea mays. Annals of Botany 64 : 13–20. Barlow PW. 1991. From cell wall networks to algorithms : The simulation and cytology of cell division patterns in plants. Protoplasma 162 : 69–85. Barlow PW. 1996. Cellular patterning in root meristems : its origin and significance. In : Waisel Y, Eshel A, Kafkafi U, eds. Plant roots. The hidden half. 2nd edition. New York : Marcel Dekker, 77–109. Barlow PW. 1997. Stem cells and founder zones in plants, particularly their roots. In : Potten CS, ed. Stem cells. London : Academic Press, 29–57. Barlow PW, Macdonald PDM. 1973. An analysis of the mitotic cell cycle in the root meristem of Zea mays. Proceedings of the Royal Society, London, Series B 183 : 385–398. Barlow PW, Parker JS. 1996. Microtubular cytoskeleton and root morphogenesis. Plant and Soil 187 : 23–36. Clowes FAL. 1961 a. Duration of the mitotic cycle in a meristem. Journal of Experimental Botany 12 : 283–293. Clowes FAL. 1961 b. Apical meristems. Oxford : Blackwell Scientific Publications. Counter CM, Hirte HW, Bacchetti S, Harley CB. 1994. Telomerase activity in human ovarian carcinoma. Proceedings of the National Academy of Science USA 91 : 2900–2904. De Boer MJM, De Does M. 1990. The relationship between cell division pattern and global shape of young fern gametophytes. I. A model study. Botanical Gazette 151 : 423–434. Doerner P, Jørgensen J-E, You R, Stepphuhn J, Lamb C. 1996. Control of root growth and development by cyclin expression Nature 380 : 520–523. Erickson RO. 1961. Probability of division of cells in the epidermis of the Phleum root. American Journal of Botany 48 : 269–274. Garcia-Bellido A, de Celis JF. 1992. Developmental genetics of the venation pattern of Drosophila. Annual ReŠiew of Genetics 26 : 277–304. Garfinkel D, Hess B. 1964. Metabolic control mechanisms VII. A detailed computer model of the glycolytic pathway in ascites cells. Journal of Biological Chemistry 239 : 971–989. Gime! nez-Abia! n MI, de la Torre C, Canova! s JL, Gime! nez-Martı! n G. 1988. Modulation of S and G2 by the amount of cytoplasm surrounding each nucleus in onion multinucleate cells. European Journal of Cell Biology 47 : 404–407.

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