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Elementary behavioural rules as a foundation for morphogenesis
J. theor. BioL (1975) 54, 23-34
Elementary Behavioural Rules as a Foundation for Morphogenesis HER~C,A~I~ B. LOCK
Laboratoire de Botanique Analytique et Structuralisme Vdggtal (C.N.R.S.-E.R. 16), Facultd des Sciences et Techniques de Saint-Jdr6me, F 13013 Marseille, France (Received 11 November 1974, and in revised form 20 March 1975) Two properties of a propagative deterministic context-free Lindenmayer system permit the determination of values which can be used in tests for histogenetic and morphogenetic hypotheses, just as segregation proportions after hybridization can be confronted with values from the Mendelian model. The procedure used is explained by means of an example of filamentous growth of a green alga, Chaetomorpha linum KOtz. (Cladophorales), and of the branching system of another green alga, Codium fragile (Sur.) Hariot (Siphonales). 1. Introduction Some 100 years ago, Sachs compared the length variations of parts of a growing organ, which he delimited by inkmarks, and defined the "great period of growth". Subsequently, morphogenetic research was largely concerned with cell-multiplication and cell-growth and their relationship to organ-growth and form-evolution. Notions like "mean cell length" and "critical cell length" etc. were introduced in botanical terminology and largely discussed. Two main ideas repeatedly appear: (1) the idea of considering cell-dimensional variations as being random, and therefore to explain organ-growth by statements which are based on averaged cell counts and measurements, and (2) the idea of considering--qualitatively--only those cells which are characterized by a specific position in the growing tissue, as for instance at the rear of a meristem (apical cells) or at the beginning of the outgrowth of a new organ (initial cells) or cells which are recognized as mother-cells of specific cellular configurations, such as mother-cells of stomatal apparatus. Only since the forties, has a more advanced approach to an explanation of organ-growth based on cellular behaviour arisen. Goodwin & Stepka (1954) introduced the notion of "elemental rate of cell-production", Erickson & 23
24
H.B.
LOCK
Sachs (1956) that of "elemental growth rate", Hejnowicz (1959) bases his arguments on an abstract "cell density". Dormer (1972) stated that "such an analysis represents an impressive approach to finality". Indeed, the taking into account of an elemental behaviour, mostly that of cells, in the context of overall-length variation, is new. In the works mentioned, and in others too, growth or division rates of elements are always evaluated from an analysis of the overall kinetics and represented as a function of the position of the elements in the growing entity. Averages of single cell measurements are no longer used. Single cells are supposed to grow in accordance with their pqsition in the whole meristem. However, two questions remain unanswered: Is it correct to accept a priori that differences in the behaviour of cells which are in a similar position are only due to random variations ? Is it right to assume that daughter-cells will behave in the same way? We feel the necessity to look for an approach in which the behaviour of ceils is not considered in the light of the organgrowth, but as a result of rules acting independently in each cell, thus deliberately accepting the possibility of a differential behaviour of daughterceils. Positional relationships should only be taken into account as relative positions between single cells. Overall-growth will, in that way, be explained as the resglt of the differential growth of the whole set of cells. But growth and division-rates are not the only attributes of meristematic cells. Differential cellular behaviour may lead to cell differentiation, which, when looked at from an integrated viewpoint, manifests itself as a histological pattern formation. In a series of earlier publications, we emphasized the close relationship between cell multiplication and histological pattern formation of differentiated cells. The ordered distribution of stomatal apparatus on the internodal epidermis of Tradescantia fluminensis Veil. appeared to be a direct consequence of the modalities of cell multiplication in the apex. The particular behaviour of the mother-cells of stomatal apparatus seemed to be less connected with the special disposition of their internal organization, than with the rules governing the evolution of the whole epidermal cell society. The statistical methods of investigation we used at that time, did not, however result in hypotheses about a probably future evolution of single protodermic cells and their derivates (Ltick, 1959, 1960a, b, 1962; Ltick & Liick, 1961). In other words, we have to look for a model which is capable of simulating cellular behaviour in such a way that good fitting of observed data will still be provided on a level higher than the elemental integration level, and which would enable predictions on the elemental level also. Such a model should be based on very simple assumptions, in order to be applicable in a very general way. It is assumed:
ELEMENTARY RULES FOR MORPHOGENESIS
25
(1) that elements are bifurcating, i.e. on the cellular level, dividing; in branching systems, ramificating, (2) that the behaviour of the two daughters can be different, and (3) that only one rule governs the behaviour of every element which may be found in different states; this, in order to reflect the genetical homogeneity of all cells of a plant. But there is another point which should be considered. Heidenhain (1932) emphasized that the development of leaf form and of animal organizations, such as the branching patterns of the alveoli in the human lung, have an underlying recursive structure. To take these findings into account in the model-construction would reinforce its fitting-power. Propagative deterministic context-free Lindenmayer systems (PDOL systems) (Lindenmayer, 1968, 1971, 1975) correspond to these outlines. In the following pages it will be shown that some properties of PDOL systems may be used for the description of developing tissues in a way which permits the use of quantifiable hypothetical values, exactly as classical genetics used the theoretical segregation proportions supplied by the Mendelian model. Two applications will be discussed, the first, on the cellular level for filamental growth and the second, on a higher organization level for a branching system.
2. Chaetomorpha--Analysisof Differential Cell Production In order to understand better the composition and the kinetics of the components of a given tissue we felt the necessity to analyse not only an instantaneous situation but to look for a plant material which would also permit the detection of both the genealogical relationship of all its cells and the time-course of their history. Observations should thus be recorded in terms of very general attributes, degree of parental relationship between single cells in the context of their positional relationship with regard to each other. Chaetomorpha linum Ktitz. is a green alga (Cladophorales) which is perfectly suited to the analysis of the timed division hierarchy of its ceils (Liick, 1973). The cells of this alga are lined up on unbranched filaments, a fact which makes the description of their relative position on the whole filament easy. Each cell contains 100-150 nuclei (coenocytic organization). Each cell divides. Cell divisions are asynchronous. The length of a cell enables a careful observer to find again in daily observations each cell of a filament, or eventually the two resulting daughter-cells, and so to construct a precise timed division hierarchy. The overall-growth of a filament can be, from observation to observation, from day to day, described by the states of all
26
H . B . LOCK
cells building it up. The developmental states of each cell can be quantified with the help of the attribute "probability of cell division in the next night". At the beginning of a cellular life cycle this probability will be very small, the cell first has to grow in length, but from day to day its value will become greater until the cell will finally divide into two daughter-cells. According to the polarity axis of the whole filament the apical daughter-cell will have a life cycle of m days, and the basal one of n days, with m > n, i.e. daughtercells in basal position have a shorter life cycle than those in complementary apical position. The mean values (m, n) describe the whole evolution of an observed filament starting with the very first mother-cell which had been experimentally isolated earlier. A careful observation of membrane structures (thickness, outlines etc.) reveals differences which emphasize the existence of boundaries between cells, those on one side of the boundary having closer genealogical relations with each other than with neighbouring cells on the other side. A real typology concerning the composition of "cell families" could be established. Definition: A cell family is a set of coexisting cells which are the totality of derivates of a single mother-cell. The genealogical relationship between all these cells is closer than to any other complementary cell of the tissue to which it belongs. A subfamily is a single or repeated bipartioning of a cellular family. It can be considered as a cell family (Fig. 1).
'( <
a0 4
><
O'l
)-
t =0
,
I= I
]~G. I. Typical cell family (a°) of Cl~etomorpha linum (a~ and a]: subfamilies of bipaztitiordng degree I = I; ao2, ~ , and a~: subfamilies of bipartitioning de~'e~ l = 2).
The existence of circadian rhythms of cell division corresponding to the discreteness of PDOL systems, the existence of overall rules governing the timing of the division of each cell (of types of cell families) during the iliamental growth, corresponding to the production rules of those mathematical constructions, as well as the possibility to observe cell family formation all make PDOL systems a good tool of adjustment, establishing hypotheses on elemental, cellular behaviour and on differential growth phenomena of parts of the entire filament studied.
ELEMENTARY
RULES
FOR
27
MORPHOGENESIS
Let S, = S l - , , + S , - .
(l)
be a locally catenative PDOL system where i refers to the power of transformation and the parameters m and n to the maximal number of developmental states of the apical and basal daughter-cell during their fife cycles, the length of which being m and n days. The production rules of such a system could be summarized in the following kinematic diagram
,
J
a 0 -4 a I -4 a 2 -4
...
-4 am_ n-~
. ..
--~ a m _ I
t
7
being developmental states during a life cycle of a cell. (The transition am-t -4 a 0 am_ ~ may be related to the "quantal division" of Holtzer, after Brachet, 1974.) The two terms S t _ m and S/_. of the recursive rule (I) correspond to two cellular subfamilies issuing from a bipartitioning of degree l = I. Two properties of such a simple PDOL system could be used for morphogenetical investigation:
n o , a l , a2 . . . . .
a, . . . . . .
a,.
(a) At a given instant the number of cells ui in the whole developing cell family simulated by equation (1) will be estimated by the sum of the cell number of the two subfamilies of the bipartitioning of first degree (l = 1). u~ = Ui_m + Ui_ n. (2 For a higher degree of bipartitioning (this always in a time invariable situation) the number of subfamilies will become 2 t. So, for the second bipartitioning degree the entire family will be composed of four subfamilies with the following cell numbers Ui =
Ui_2m-~l, li_m_nJVbtl_n_m-~Ui_2n
.
(3)
In this case three classes of subfamilies a~ are sufficient to describe the whole family a °, the upper index corresponding to the bipartitioaing degree l, the lower on to k e (0, 1, . . . . l) aO
,.~2,~ 2,.w2 n 2
=
(4)
.OU, l~ 1.2 .
The frequencies of a~ classes are given by binomial coefficients ,,
= k ! ( l - k) !
(5)
In the special case of l = 2 it will be Class
ao2 a 2 a22
Frequency
1
2
1
28
H . B . LOCK
The number of cells in each of the akz classes for a given bipartitioning degree is uk = Ui-t~-k)m-kn with k ~ (0, 1, 2, . . . . l). (6) According to equations (5) and (6) the cell number of the whole evoluting descendance a ° will finally become |li ~
E
k=O
(') k
Ui-(l-k)ra-kn "
(7)
(b) The ratio Ul-,/Uf-m, in the biologically relevant space, stabilizes rather quickly. (It may be noted that this ratio is for m = 2 and n = 1 the well known prol~ortio divina, derived from the standard Fibonacci series.) The calculated theoretical limit values (see Appendix A)can be related to the ratio of cell numbers (if they are not too small) of two observed subfamilies. An estimation of the nearest mean (m, n) values will become possible, even if the evolution of the cell family had not been followed. (c) Application. Suppose that an observed cell family is composed of 45 cells. An analysis of membrane structures reveals that the two subfamilies of the first bipartitioning degree have 19 and 26 cells respectively and those of the second degree 8 and 11 respectively and I0 and 16 respectively. The adjustment of a PDOL system to these data make it possible to test hypotheses about the ontogeny of this cell group: the ratio of the observed cell numbers of the two subfamilies is -
26 19 -
=
1.37.
This value corresponds most closely to the theoretical limit of a series with the parameters m = 5 and n = 3, the limit of this series being t'42. It may be tested if the values of the two parameters are good estimates of mean values of rn and n which summarize the timing of the divisions which led to the observed histological situation. For each of the four subfamilies of the bipartitioning degree 1 = 2 [the concatenation of them being given by equation (4) ] a theoretical cell frequency u~ may be calculated according to equation (6) and (7), taking u~ = 45. These values can be compared, by means of the X2 test, with the observed frequencies shown in Table 1. The statistical test shows that the parameter estimation by means of an adjusted PDOL system fits the real observed situation quite well. (This result has been confirmed by a direct analysis of the real timed division hierarchy.)
29
E L E M E N T A R Y R U L E S FOR M O R P H O G E N E S I S TABLE i
Comparison of values using g 2 test 1= 2 Theoretical cell frequency Observed cell frequency
ao2
Subfamilies a~ a~
a]
7-7 8
10.9 11
15.5 16
10.9 10
Total 45 45
;~2¢n)-- 0'108, P > 0.99. At the same time the estimated parameters give a quantified expression of the morphological organziation of the cell family considered. Small values of m and n signify a high overall growth rate, which corresponds to the expression "vitality". A great difference between the absolute values of the two parameters reflects a strong asymmetry of the general morphogenetic growth pattern, and highly divergent differential growth rates (in the sense of Avery, 1933).
3. Codium--Analysis of a Natural Branching System On another, non-cellular, organization-level an adjustment of PDOL systems to observed data and the application of successive bipartitioning as a help to mobilize hidden information, is of heuristic value as well. This can be demonstrated with another green-alga, Codium fragile (Sur.) Hariot (Siphonales) (Liick & LiJck, 1973). The cells of Codium show, just as those of Chaetomorpha, coenocytic organization, but no cell divisions intervene. They grow out to long, independent tubes which agglomerate into cylinders of 5-90 mm in length. The attached basal cylinder will bifurcate on the top (first branching degree). The two outgrowing branches will bifurcate later on (second branching degree). A strictly binary branching system develops and floats in the sea in all directions. In order to analyse the overall organization of such a thallus we assimilate the number of distal cylinders to the number of cells in a cell family. The successive bipartitioning of the entire number of distal cylinders into subsets (subfamilies) could be done according to the branching degree of the thallus. Such subfamilies are of different size. But there is a difficulty which was not encountered in the case of the filaments of Chaetomorpha, where, for each cell division, the position of the two daughter-cells could always be qualified as basal or apical. In Codium a positional order between two homologous segments of the branching system
30
n.B.
LOCK
Ca) Number of dislol cylinderz
Branching degree
32
I=0 I=1
14 6 2
18 8
4
4
8 4
4
1=2
I0 4
5
5
11 22 22 2 2 2 2 2 2 2 3 2
1=5 1=4
• (b) Observedclasses
I
INumber of distal cylindersI
I ~ ..."~I .. '
~? 2
-
3
,/
J
°~I
0
° I
,2
To,o,l 2
~ ~
I l~o,a, 14~.~ ~o
16 2
1I,~
/ Frequencyof observed / ] classes I
(c) FIG. 2, (a) Branching system of
Codiumfragile (young
individual). (b) N u m b e r o f distal
cylinders at different branching degrees (1 ---- 0 to l = 4). (c) Class frequency between subsets o f a P D O L concatenation (a~. . . . . al) a n d observed subfamilies (ha~. . . . . b~), branching degree l = 4.
ELEMENTARY
RULES
FOR MORPHOGENESIS
31
is not evident. No devices for the transcription of the three-dimensional appearance into a two-dimensional plan can be given. It seems that the application of the locally concatenative rules of P D O L systems would not be possible. Nevertheless, a positional constraint could be found: two homologous segments issuing from a bifurcation in the whole branching system at the level of branching degree l determine two subsets of distal cylinders with unequal numbers. Each of these two subsets may be bipartitioned as well, referring to the branching degree 1+ 1. It can be shown that between the four subsets of the level 1+ 1, that one which has the greatest cylinder frequency, belongs, between the two subsets of the level 1, to the subset which also has the greatest cylinder frequency. This property corresponds exactly to the properties of the mentioned P D O L systems. The proof can easily be deduced from equations (4) and (6). A branching hierarchy can, on this basis, be established without any ambiguity. The two homologous segments of each bifurcation should only be ordered in such a way that the segment to which the higher cylinder frequency corresponds always appears on the same side. For a given bipartitioning degree the resulting subsets can be assimilated to the different a[ classes of a P D O L system which produces, by means of its locally catenative rules, the necessary positional attributes. Once again, a highly complex morphological situation will be reduced to a table of correspondences of frequencies, such as appears--this time in the form of class-frequencies--in Fig. 2. In this form statistical decision aids can be adopted. Morphological complexity can, by help of a propagative deterministic context-free Lindenmayer system, be analysed from the view-point of an experimental biologist; classical statistical models proved to be unable to respond to structural recaarsive rules of morphogenetic processes.
REFERENCES AVERY,G. S., JR (1933). Am. 3. Bot. 20, 565. BRACHET,J. (1974). Coll. Morphologie, Soe. Bot. p. 3. France: Orsay.
DORMER,K. J. (1972). Shoot Organization in Vascular Plants. London: Chapman & Hall. ERICKSON,R. O. d~.SACHS,K. B. (1956). Proc. Am. phil. Soc. 100, 487. GOODWlN,R. H. & STEPKA,W. (1954). Am. J. Bot. 32, 36. HEIDENHAIN,M. (1932). Die Spaltungsgesetze der Bli~tter. Jena: Fischer-Verlag. HEJNOWICZ,Z. (1959). Physiologia PI. 12, 124. LINDENMAYER,A. (1968). J. theor. Biol. 18, 280. LINDENMAYER,A. (1971). 3". theor. Biol. 30, 455. LINDEI'~AYER, A. (1975). In Developmental Systems and Languages (G. T. Herman & G. Rozenberg, Amsterdam: North-Holland). LOCK,H. B. (1959). Naturalia monspel., Sdr. Bot. 10, 33. LOCK,H. B. (1960a). Naturalia monspeL, S~r. Bot. 11,~9.
32
H.B.
LOCK
LOCK, H. B. (1960b). Bet. dt. bot. Gea. 73, 57. LOCK, H. B. (1962). Annls ScL nat. (Bot.) 3, 1. LOCK, H. B. (1973). Acta hot. neerl. 22, 251. LOCK, H. B. & LOCK,J. (1961). Naturalia monspel., Sdr. Bot. 13, 15. LOCK, J. & LOCK, H. B. (1973). Revue Cyt. Biol. vdg. 37, 118.
1.....
u,,_ 1
urn=
u i _ . < 106 a n d u i _ m ~ 106 i f n > m .
u t - , t> 106 a n d u l - m >/ 106 i f n = m,
2,...ifn=m;
u._ a = l,u.=2
....
urn_, = 1, Ur. = 2 . . . .
= I .... u.=
1. . . . .
u . = 1. . . . .
< 106 a n d u~_ m < 106;
1, u I = 1 . . . . .
1 =
u~_, >i 106 a n d ui_m < 106 i f n < m,
ui_ .
uo=
l,u
with
E x t r a c t e d f r o m a six-figure table, v a l u e s r o u n d e d off (i> 5 . 1 0 - 4 ) .
and
i, s u c h t h a t
i - I, s u c h t h a t
and
uo =
Uo = 1, u l = 1. . . . .
V a l u e s o f the r a t i o Zli_n/Ui_ m o f the series u i = u ~ - m + u ~ - , ,
Appendix A
ifnm,
LO LO
rn
o o r~ z
0
o
,-1 >.
z
2
1"618 1.000 0.755 0"618 0"529
0'466 0-418 0"380 0-350 0-325
0-303 0-285 0.270 0"255 0"242
1
1'000 0'618 0'466 0"380 0"325
0"285 0.255 0"232 0"213 0-197
0"184 0"173 0-163 0'155 0"147
6 7 8 9 10
11 12 13 14 15
n
0-404 0"380 0.362 0'342 0.325
0.618 0"554 0-505 0-466 0-430
2"148 1-325 1"000 0"819 0.701
3
0.499 0"466 0-433 0.418 0-391
0"755 0.680 0'618 0-563 0"529
2.630 1"618 1"221 1"000 0"858
4
0-565 0"544 0"511 0"498 0'466
0'890 0"798 0"717 0"673 0-618
3"080 1"891 1"426 1"165 1.000
5
0-673 0'618 0"565 0"554 0.529
1.000 0-918 0"819 0.755 0-701
3.506 2"148 1-618 1"325 1"123
6
0-718 0.685 0.673 0-618 0-565
1-090 1"000 0-943 0-828 0-796
3"915 2'393 1"805 1"471 1-253
7
0"811 0"755 0-715 0"680 0"673
1"221 1"061 1.000 0"957 0-858
4-310 2"630 1"979 1"618 1"395
8
m
0-890 0"819 0'796 0'718 0"701
1"325 1"208 1'045 1.000 0'962
4"691 2"860 2.148 1"776 1'485
9
0"963 0-890 0-824 0"798 0"755
1"426 1"256 1'165 1'040 1.000
5"064 3"080 2-324 1"891 1"618
10
1-000 0.964 0'890 0-828 0.812
1"487 1"393 1"233 1-124 1-038
5'430 3"298 2.477 2"005 1"770
11
1"038 1"000 0-964 0-918 0.858
1"618 1.460 1"325 1"221 1-123
5'786 3"506 2'630 2"148 1"840
12
1"123 1'038 1"000 0"964 0"943
1.770 1"487 1.400 1"256 1"213
6.127 3.705 2"761 2"311 1"956
13
1"208 1"090 1"038 1.000 0"964
1"805 1"618 1"471 1"393 1-253
6"462 3-915 2"922 2"393 2.007
14
1"232 1.165 1"061 1"038 1.000
1"891 1.769 1-487 1.426 1-325
6-790 4"129 3"080 2"557 2"148
15
t-, ~: (3
Elementary Behavioural Rules as a Foundation for Morphogenesis HER~C,A~I~ B. LOCK
Laboratoire de Botanique Analytique et Structuralisme Vdggtal (C.N.R.S.-E.R. 16), Facultd des Sciences et Techniques de Saint-Jdr6me, F 13013 Marseille, France (Received 11 November 1974, and in revised form 20 March 1975) Two properties of a propagative deterministic context-free Lindenmayer system permit the determination of values which can be used in tests for histogenetic and morphogenetic hypotheses, just as segregation proportions after hybridization can be confronted with values from the Mendelian model. The procedure used is explained by means of an example of filamentous growth of a green alga, Chaetomorpha linum KOtz. (Cladophorales), and of the branching system of another green alga, Codium fragile (Sur.) Hariot (Siphonales). 1. Introduction Some 100 years ago, Sachs compared the length variations of parts of a growing organ, which he delimited by inkmarks, and defined the "great period of growth". Subsequently, morphogenetic research was largely concerned with cell-multiplication and cell-growth and their relationship to organ-growth and form-evolution. Notions like "mean cell length" and "critical cell length" etc. were introduced in botanical terminology and largely discussed. Two main ideas repeatedly appear: (1) the idea of considering cell-dimensional variations as being random, and therefore to explain organ-growth by statements which are based on averaged cell counts and measurements, and (2) the idea of considering--qualitatively--only those cells which are characterized by a specific position in the growing tissue, as for instance at the rear of a meristem (apical cells) or at the beginning of the outgrowth of a new organ (initial cells) or cells which are recognized as mother-cells of specific cellular configurations, such as mother-cells of stomatal apparatus. Only since the forties, has a more advanced approach to an explanation of organ-growth based on cellular behaviour arisen. Goodwin & Stepka (1954) introduced the notion of "elemental rate of cell-production", Erickson & 23
24
H.B.
LOCK
Sachs (1956) that of "elemental growth rate", Hejnowicz (1959) bases his arguments on an abstract "cell density". Dormer (1972) stated that "such an analysis represents an impressive approach to finality". Indeed, the taking into account of an elemental behaviour, mostly that of cells, in the context of overall-length variation, is new. In the works mentioned, and in others too, growth or division rates of elements are always evaluated from an analysis of the overall kinetics and represented as a function of the position of the elements in the growing entity. Averages of single cell measurements are no longer used. Single cells are supposed to grow in accordance with their pqsition in the whole meristem. However, two questions remain unanswered: Is it correct to accept a priori that differences in the behaviour of cells which are in a similar position are only due to random variations ? Is it right to assume that daughter-cells will behave in the same way? We feel the necessity to look for an approach in which the behaviour of ceils is not considered in the light of the organgrowth, but as a result of rules acting independently in each cell, thus deliberately accepting the possibility of a differential behaviour of daughterceils. Positional relationships should only be taken into account as relative positions between single cells. Overall-growth will, in that way, be explained as the resglt of the differential growth of the whole set of cells. But growth and division-rates are not the only attributes of meristematic cells. Differential cellular behaviour may lead to cell differentiation, which, when looked at from an integrated viewpoint, manifests itself as a histological pattern formation. In a series of earlier publications, we emphasized the close relationship between cell multiplication and histological pattern formation of differentiated cells. The ordered distribution of stomatal apparatus on the internodal epidermis of Tradescantia fluminensis Veil. appeared to be a direct consequence of the modalities of cell multiplication in the apex. The particular behaviour of the mother-cells of stomatal apparatus seemed to be less connected with the special disposition of their internal organization, than with the rules governing the evolution of the whole epidermal cell society. The statistical methods of investigation we used at that time, did not, however result in hypotheses about a probably future evolution of single protodermic cells and their derivates (Ltick, 1959, 1960a, b, 1962; Ltick & Liick, 1961). In other words, we have to look for a model which is capable of simulating cellular behaviour in such a way that good fitting of observed data will still be provided on a level higher than the elemental integration level, and which would enable predictions on the elemental level also. Such a model should be based on very simple assumptions, in order to be applicable in a very general way. It is assumed:
ELEMENTARY RULES FOR MORPHOGENESIS
25
(1) that elements are bifurcating, i.e. on the cellular level, dividing; in branching systems, ramificating, (2) that the behaviour of the two daughters can be different, and (3) that only one rule governs the behaviour of every element which may be found in different states; this, in order to reflect the genetical homogeneity of all cells of a plant. But there is another point which should be considered. Heidenhain (1932) emphasized that the development of leaf form and of animal organizations, such as the branching patterns of the alveoli in the human lung, have an underlying recursive structure. To take these findings into account in the model-construction would reinforce its fitting-power. Propagative deterministic context-free Lindenmayer systems (PDOL systems) (Lindenmayer, 1968, 1971, 1975) correspond to these outlines. In the following pages it will be shown that some properties of PDOL systems may be used for the description of developing tissues in a way which permits the use of quantifiable hypothetical values, exactly as classical genetics used the theoretical segregation proportions supplied by the Mendelian model. Two applications will be discussed, the first, on the cellular level for filamental growth and the second, on a higher organization level for a branching system.
2. Chaetomorpha--Analysisof Differential Cell Production In order to understand better the composition and the kinetics of the components of a given tissue we felt the necessity to analyse not only an instantaneous situation but to look for a plant material which would also permit the detection of both the genealogical relationship of all its cells and the time-course of their history. Observations should thus be recorded in terms of very general attributes, degree of parental relationship between single cells in the context of their positional relationship with regard to each other. Chaetomorpha linum Ktitz. is a green alga (Cladophorales) which is perfectly suited to the analysis of the timed division hierarchy of its ceils (Liick, 1973). The cells of this alga are lined up on unbranched filaments, a fact which makes the description of their relative position on the whole filament easy. Each cell contains 100-150 nuclei (coenocytic organization). Each cell divides. Cell divisions are asynchronous. The length of a cell enables a careful observer to find again in daily observations each cell of a filament, or eventually the two resulting daughter-cells, and so to construct a precise timed division hierarchy. The overall-growth of a filament can be, from observation to observation, from day to day, described by the states of all
26
H . B . LOCK
cells building it up. The developmental states of each cell can be quantified with the help of the attribute "probability of cell division in the next night". At the beginning of a cellular life cycle this probability will be very small, the cell first has to grow in length, but from day to day its value will become greater until the cell will finally divide into two daughter-cells. According to the polarity axis of the whole filament the apical daughter-cell will have a life cycle of m days, and the basal one of n days, with m > n, i.e. daughtercells in basal position have a shorter life cycle than those in complementary apical position. The mean values (m, n) describe the whole evolution of an observed filament starting with the very first mother-cell which had been experimentally isolated earlier. A careful observation of membrane structures (thickness, outlines etc.) reveals differences which emphasize the existence of boundaries between cells, those on one side of the boundary having closer genealogical relations with each other than with neighbouring cells on the other side. A real typology concerning the composition of "cell families" could be established. Definition: A cell family is a set of coexisting cells which are the totality of derivates of a single mother-cell. The genealogical relationship between all these cells is closer than to any other complementary cell of the tissue to which it belongs. A subfamily is a single or repeated bipartioning of a cellular family. It can be considered as a cell family (Fig. 1).
'( <
a0 4
><
O'l
)-
t =0
,
I= I
]~G. I. Typical cell family (a°) of Cl~etomorpha linum (a~ and a]: subfamilies of bipaztitiordng degree I = I; ao2, ~ , and a~: subfamilies of bipartitioning de~'e~ l = 2).
The existence of circadian rhythms of cell division corresponding to the discreteness of PDOL systems, the existence of overall rules governing the timing of the division of each cell (of types of cell families) during the iliamental growth, corresponding to the production rules of those mathematical constructions, as well as the possibility to observe cell family formation all make PDOL systems a good tool of adjustment, establishing hypotheses on elemental, cellular behaviour and on differential growth phenomena of parts of the entire filament studied.
ELEMENTARY
RULES
FOR
27
MORPHOGENESIS
Let S, = S l - , , + S , - .
(l)
be a locally catenative PDOL system where i refers to the power of transformation and the parameters m and n to the maximal number of developmental states of the apical and basal daughter-cell during their fife cycles, the length of which being m and n days. The production rules of such a system could be summarized in the following kinematic diagram
,
J
a 0 -4 a I -4 a 2 -4
...
-4 am_ n-~
. ..
--~ a m _ I
t
7
being developmental states during a life cycle of a cell. (The transition am-t -4 a 0 am_ ~ may be related to the "quantal division" of Holtzer, after Brachet, 1974.) The two terms S t _ m and S/_. of the recursive rule (I) correspond to two cellular subfamilies issuing from a bipartitioning of degree l = I. Two properties of such a simple PDOL system could be used for morphogenetical investigation:
n o , a l , a2 . . . . .
a, . . . . . .
a,.
(a) At a given instant the number of cells ui in the whole developing cell family simulated by equation (1) will be estimated by the sum of the cell number of the two subfamilies of the bipartitioning of first degree (l = 1). u~ = Ui_m + Ui_ n. (2 For a higher degree of bipartitioning (this always in a time invariable situation) the number of subfamilies will become 2 t. So, for the second bipartitioning degree the entire family will be composed of four subfamilies with the following cell numbers Ui =
Ui_2m-~l, li_m_nJVbtl_n_m-~Ui_2n
.
(3)
In this case three classes of subfamilies a~ are sufficient to describe the whole family a °, the upper index corresponding to the bipartitioaing degree l, the lower on to k e (0, 1, . . . . l) aO
,.~2,~ 2,.w2 n 2
=
(4)
.OU, l~ 1.2 .
The frequencies of a~ classes are given by binomial coefficients ,,
= k ! ( l - k) !
(5)
In the special case of l = 2 it will be Class
ao2 a 2 a22
Frequency
1
2
1
28
H . B . LOCK
The number of cells in each of the akz classes for a given bipartitioning degree is uk = Ui-t~-k)m-kn with k ~ (0, 1, 2, . . . . l). (6) According to equations (5) and (6) the cell number of the whole evoluting descendance a ° will finally become |li ~
E
k=O
(') k
Ui-(l-k)ra-kn "
(7)
(b) The ratio Ul-,/Uf-m, in the biologically relevant space, stabilizes rather quickly. (It may be noted that this ratio is for m = 2 and n = 1 the well known prol~ortio divina, derived from the standard Fibonacci series.) The calculated theoretical limit values (see Appendix A)can be related to the ratio of cell numbers (if they are not too small) of two observed subfamilies. An estimation of the nearest mean (m, n) values will become possible, even if the evolution of the cell family had not been followed. (c) Application. Suppose that an observed cell family is composed of 45 cells. An analysis of membrane structures reveals that the two subfamilies of the first bipartitioning degree have 19 and 26 cells respectively and those of the second degree 8 and 11 respectively and I0 and 16 respectively. The adjustment of a PDOL system to these data make it possible to test hypotheses about the ontogeny of this cell group: the ratio of the observed cell numbers of the two subfamilies is -
26 19 -
=
1.37.
This value corresponds most closely to the theoretical limit of a series with the parameters m = 5 and n = 3, the limit of this series being t'42. It may be tested if the values of the two parameters are good estimates of mean values of rn and n which summarize the timing of the divisions which led to the observed histological situation. For each of the four subfamilies of the bipartitioning degree 1 = 2 [the concatenation of them being given by equation (4) ] a theoretical cell frequency u~ may be calculated according to equation (6) and (7), taking u~ = 45. These values can be compared, by means of the X2 test, with the observed frequencies shown in Table 1. The statistical test shows that the parameter estimation by means of an adjusted PDOL system fits the real observed situation quite well. (This result has been confirmed by a direct analysis of the real timed division hierarchy.)
29
E L E M E N T A R Y R U L E S FOR M O R P H O G E N E S I S TABLE i
Comparison of values using g 2 test 1= 2 Theoretical cell frequency Observed cell frequency
ao2
Subfamilies a~ a~
a]
7-7 8
10.9 11
15.5 16
10.9 10
Total 45 45
;~2¢n)-- 0'108, P > 0.99. At the same time the estimated parameters give a quantified expression of the morphological organziation of the cell family considered. Small values of m and n signify a high overall growth rate, which corresponds to the expression "vitality". A great difference between the absolute values of the two parameters reflects a strong asymmetry of the general morphogenetic growth pattern, and highly divergent differential growth rates (in the sense of Avery, 1933).
3. Codium--Analysis of a Natural Branching System On another, non-cellular, organization-level an adjustment of PDOL systems to observed data and the application of successive bipartitioning as a help to mobilize hidden information, is of heuristic value as well. This can be demonstrated with another green-alga, Codium fragile (Sur.) Hariot (Siphonales) (Liick & LiJck, 1973). The cells of Codium show, just as those of Chaetomorpha, coenocytic organization, but no cell divisions intervene. They grow out to long, independent tubes which agglomerate into cylinders of 5-90 mm in length. The attached basal cylinder will bifurcate on the top (first branching degree). The two outgrowing branches will bifurcate later on (second branching degree). A strictly binary branching system develops and floats in the sea in all directions. In order to analyse the overall organization of such a thallus we assimilate the number of distal cylinders to the number of cells in a cell family. The successive bipartitioning of the entire number of distal cylinders into subsets (subfamilies) could be done according to the branching degree of the thallus. Such subfamilies are of different size. But there is a difficulty which was not encountered in the case of the filaments of Chaetomorpha, where, for each cell division, the position of the two daughter-cells could always be qualified as basal or apical. In Codium a positional order between two homologous segments of the branching system
30
n.B.
LOCK
Ca) Number of dislol cylinderz
Branching degree
32
I=0 I=1
14 6 2
18 8
4
4
8 4
4
1=2
I0 4
5
5
11 22 22 2 2 2 2 2 2 2 3 2
1=5 1=4
• (b) Observedclasses
I
INumber of distal cylindersI
I ~ ..."~I .. '
~? 2
-
3
,/
J
°~I
0
° I
,2
To,o,l 2
~ ~
I l~o,a, 14~.~ ~o
16 2
1I,~
/ Frequencyof observed / ] classes I
(c) FIG. 2, (a) Branching system of
Codiumfragile (young
individual). (b) N u m b e r o f distal
cylinders at different branching degrees (1 ---- 0 to l = 4). (c) Class frequency between subsets o f a P D O L concatenation (a~. . . . . al) a n d observed subfamilies (ha~. . . . . b~), branching degree l = 4.
ELEMENTARY
RULES
FOR MORPHOGENESIS
31
is not evident. No devices for the transcription of the three-dimensional appearance into a two-dimensional plan can be given. It seems that the application of the locally concatenative rules of P D O L systems would not be possible. Nevertheless, a positional constraint could be found: two homologous segments issuing from a bifurcation in the whole branching system at the level of branching degree l determine two subsets of distal cylinders with unequal numbers. Each of these two subsets may be bipartitioned as well, referring to the branching degree 1+ 1. It can be shown that between the four subsets of the level 1+ 1, that one which has the greatest cylinder frequency, belongs, between the two subsets of the level 1, to the subset which also has the greatest cylinder frequency. This property corresponds exactly to the properties of the mentioned P D O L systems. The proof can easily be deduced from equations (4) and (6). A branching hierarchy can, on this basis, be established without any ambiguity. The two homologous segments of each bifurcation should only be ordered in such a way that the segment to which the higher cylinder frequency corresponds always appears on the same side. For a given bipartitioning degree the resulting subsets can be assimilated to the different a[ classes of a P D O L system which produces, by means of its locally catenative rules, the necessary positional attributes. Once again, a highly complex morphological situation will be reduced to a table of correspondences of frequencies, such as appears--this time in the form of class-frequencies--in Fig. 2. In this form statistical decision aids can be adopted. Morphological complexity can, by help of a propagative deterministic context-free Lindenmayer system, be analysed from the view-point of an experimental biologist; classical statistical models proved to be unable to respond to structural recaarsive rules of morphogenetic processes.
REFERENCES AVERY,G. S., JR (1933). Am. 3. Bot. 20, 565. BRACHET,J. (1974). Coll. Morphologie, Soe. Bot. p. 3. France: Orsay.
DORMER,K. J. (1972). Shoot Organization in Vascular Plants. London: Chapman & Hall. ERICKSON,R. O. d~.SACHS,K. B. (1956). Proc. Am. phil. Soc. 100, 487. GOODWlN,R. H. & STEPKA,W. (1954). Am. J. Bot. 32, 36. HEIDENHAIN,M. (1932). Die Spaltungsgesetze der Bli~tter. Jena: Fischer-Verlag. HEJNOWICZ,Z. (1959). Physiologia PI. 12, 124. LINDENMAYER,A. (1968). J. theor. Biol. 18, 280. LINDENMAYER,A. (1971). 3". theor. Biol. 30, 455. LINDEI'~AYER, A. (1975). In Developmental Systems and Languages (G. T. Herman & G. Rozenberg, Amsterdam: North-Holland). LOCK,H. B. (1959). Naturalia monspel., Sdr. Bot. 10, 33. LOCK,H. B. (1960a). Naturalia monspeL, S~r. Bot. 11,~9.
32
H.B.
LOCK
LOCK, H. B. (1960b). Bet. dt. bot. Gea. 73, 57. LOCK, H. B. (1962). Annls ScL nat. (Bot.) 3, 1. LOCK, H. B. (1973). Acta hot. neerl. 22, 251. LOCK, H. B. & LOCK,J. (1961). Naturalia monspel., Sdr. Bot. 13, 15. LOCK, J. & LOCK, H. B. (1973). Revue Cyt. Biol. vdg. 37, 118.
1.....
u,,_ 1
urn=
u i _ . < 106 a n d u i _ m ~ 106 i f n > m .
u t - , t> 106 a n d u l - m >/ 106 i f n = m,
2,...ifn=m;
u._ a = l,u.=2
....
urn_, = 1, Ur. = 2 . . . .
= I .... u.=
1. . . . .
u . = 1. . . . .
< 106 a n d u~_ m < 106;
1, u I = 1 . . . . .
1 =
u~_, >i 106 a n d ui_m < 106 i f n < m,
ui_ .
uo=
l,u
with
E x t r a c t e d f r o m a six-figure table, v a l u e s r o u n d e d off (i> 5 . 1 0 - 4 ) .
and
i, s u c h t h a t
i - I, s u c h t h a t
and
uo =
Uo = 1, u l = 1. . . . .
V a l u e s o f the r a t i o Zli_n/Ui_ m o f the series u i = u ~ - m + u ~ - , ,
Appendix A
ifn
LO LO
rn
o o r~ z
0
o
,-1 >.
z
2
1"618 1.000 0.755 0"618 0"529
0'466 0-418 0"380 0-350 0-325
0-303 0-285 0.270 0"255 0"242
1
1'000 0'618 0'466 0"380 0"325
0"285 0.255 0"232 0"213 0-197
0"184 0"173 0-163 0'155 0"147
6 7 8 9 10
11 12 13 14 15
n
0-404 0"380 0.362 0'342 0.325
0.618 0"554 0-505 0-466 0-430
2"148 1-325 1"000 0"819 0.701
3
0.499 0"466 0-433 0.418 0-391
0"755 0.680 0'618 0-563 0"529
2.630 1"618 1"221 1"000 0"858
4
0-565 0"544 0"511 0"498 0'466
0'890 0"798 0"717 0"673 0-618
3"080 1"891 1"426 1"165 1.000
5
0-673 0'618 0"565 0"554 0.529
1.000 0-918 0"819 0.755 0-701
3.506 2"148 1-618 1"325 1"123
6
0-718 0.685 0.673 0-618 0-565
1-090 1"000 0-943 0-828 0-796
3"915 2'393 1"805 1"471 1-253
7
0"811 0"755 0-715 0"680 0"673
1"221 1"061 1.000 0"957 0-858
4-310 2"630 1"979 1"618 1"395
8
m
0-890 0"819 0'796 0'718 0"701
1"325 1"208 1'045 1.000 0'962
4"691 2"860 2.148 1"776 1'485
9
0"963 0-890 0-824 0"798 0"755
1"426 1"256 1'165 1'040 1.000
5"064 3"080 2-324 1"891 1"618
10
1-000 0.964 0'890 0-828 0.812
1"487 1"393 1"233 1-124 1-038
5'430 3"298 2.477 2"005 1"770
11
1"038 1"000 0-964 0-918 0.858
1"618 1.460 1"325 1"221 1-123
5'786 3"506 2'630 2"148 1"840
12
1"123 1'038 1"000 0"964 0"943
1.770 1"487 1.400 1"256 1"213
6.127 3.705 2"761 2"311 1"956
13
1"208 1"090 1"038 1.000 0"964
1"805 1"618 1"471 1"393 1-253
6"462 3-915 2"922 2"393 2.007
14
1"232 1.165 1"061 1"038 1.000
1"891 1.769 1-487 1.426 1-325
6-790 4"129 3"080 2"557 2"148
15
t-, ~: (3