Branching and Self-Organization in Marine Modular Colonial Organisms:
An Ecological Approach.
Juan A. Sánchez (1), Howard R. Lasker (1),
J. Dario Sánchez (2) and Michael J. Woldenberg (3).
(1) Department of Biological Sciences, 109 Cooke Hall, University at Buffalo (The State
University of New York), Buffalo, NY 14260, USA. [email protected]; [email protected]
phone: 716-645-2881; fax: 717-645-2975
(2) Departamento de Matematicas, Universidad Nacional de Colombia,
Bogotá, Colombia. [email protected]
phone 011-571-2692265; fax 011-571-2697853
(3) Department of Geography, University at Buffalo,
(The State University of New York), NY 14260, USA.
(716) 645-2722 x29
[email protected]
Key words: branching, modular organisms, self-organization, gorgonian coral, colony
development, branch interference.
Running head: Branching in modular organisms.
1
Abstract
Regardless of the relevance and universality of branching pattern in modular colonial
organisms, there is no clear explanation about its development nor what makes these organisms
to preserve shape during growth. Modular organisms such as gorgonian corals branch subapically and depicting hierarchical mother-daughter relationships among branches. Mother
branch size frequency distribution followed a scaling power-law dependence suggesting selforganized criticality. Shape is preserved in these tree-like networks by maintaining a constant
ratio between total branches and mother branches (c). It is assumed that c, an integer in terms of
number of branches, is dynamically maintained by the production of mother branches (from an
old daughter branch) when the number of total branches is off the neighborhood of c. Using that
simple rule, we modeled both the intrinsic process of branching along with the global-ecological
effects by adapting a discrete logistic equation. It exhibited a very predictable trajectory ending
without fixed points or stable equilibrium, converging to a fixed number of branches. The
branching trajectory was sigmoid with a rapid exponential phase that ended in a short asymptotic
period, which has been observed by a number of empirical studies on hydroids and octocorals.
Different colony architectures may have very different qualitative behaviors depending on the
relationship between branching (r) and c. The inclusion of another parameter accounting for
branch interference and allometry, had endorsement from size and resource capture constraints.
The complex and dynamic nature of branching is still modulated criticality by the interaction
between r and c but global ecological constraints prevail throughout colony development.
Introduction
The study or branching and tree-like networks concerns a great number of disciplines as
well as both physical and biological systems. Branching networks are present from the tiniest
2
vessels in the human body or insect wing to the continental size of the Amazon or Mississippi
basins. Many marine modular colonial organisms, such as sponges, corals and hydroids among
others, are also branching structures themselves (i.e. Waller and Steingraeber, 1985). The great
importance of the study of branching for these organisms is that their main interaction with the
environment is through branching. Life-history and fitness in sessile modular organisms are
related to the colony shape and size (i.e. Hughes, 1983; Hughes et al., 1992). Despite the
relevance and universality of the problem of branching, there is no clear explanation about how
is branching dynamically achieved during development nor what makes these organisms to
preserve form or to stop growing (see reviews in Buss, 2001 and Lasker and Sánchez, 2002).
Some of them are indeed classified as having indeterminate growth (i.e. plastic attenuating
growth: Sebens, 1987). This study, based on empirical observations, aims to explain the limits of
branching in modular colonial organisms according to a few parameters that interact dynamically
and how this system self-organizes to preserve shape throughout development.
Although being among the most primitive and basal metazoans, modular colonial
organisms such as cnidarians contain complex developmental machinery such as homeotic genes
(i.e. Finnerty and Martindale, 1997). Genes, such as Cnox-2 for instance, seem to be iteratively
expressed throughout development of hydroid colonies, providing a simple basis for an intrinsic
mechanism of growth but no clear distinction for producing a new branch or extending an old
one has been found (Cartwright et al., 1999; Cartwright and Buss, 2000; Cartwright, pers. com.).
Colonial growth seems to be partially controlled and shaped by environmental stimuli (Buss,
2001). Branching in these organisms, consequently, is not just a developmental matter but an
ecological process too. Computer models demonstrate that with a few rules and environmental
input is possible to mimic colony growth of modular colonial organisms with remarkable realism
3
(see review in Kaandorp and Klueber, 2001). It is therefore reasonable to think that branching in
a modular colonial organism be a combination of both internal and external factors that shape
dynamically the system like in any other ecological process (e.g. logistic population growth,
host-parasite, predator-prey interactions, etc.). A challenging goal for a model of branching in
modular organisms is to include the interplay between an intrinsic growth process and global
ecological effects.
The study of branching networks, in contrast, has usually involved the study of ordering,
which depicts hierarchical relationships among branches as different orders. Branching in noncolonial systems and rivers has had considerable theoretical attention. Ordering systems are
classified into two groups: centrifugal and centripetal. Centrifugal, is when orders increase in the
same direction as a growing tree. Keill (1613-1719) developed elaborated centrifugal scaling
laws based on measurements of casts of the arterial system assuming dichotomous branching of
the arteries (Woldenberg, 1997). Although centrifugal approaches were very realistic on the
hierarchy of branches and nodes, it was replaced almost completely by centripetal schemes.
Horton (1945) proposes a centripetal ordering scheme assigning orders from the periphery
towards the trunk, where unbranched tributaries are assigned order one and excessive side
branches have to be ignored or treat like a constant. When two branches of order one meet, they
create a branch of order two and so on. Strahler’s (1952) modification of Horton’s method is the
most widely used ordering system in the study of river and biological tree-like networks. Both
systems are well suited to creating empirical geometric series or number of branches, mean
branch length, mean basin area, etc. for asymmetrical trees. A similar centripetal system allows
more orders than Strahler or Horton systems and the geometric series are not as generalized
(Horsfield, 1981). In spite of the dominance of the Strahler order systems for rivers and
4
biological systems (i.e. Woldenberg et al., 1993), there is some discontent with the fact that the
Horton-Strahler ordering system does not change when lower order branches join the higher
order branch.
These generational systems are well suited for modeling the growth of symmetrically
bifurcating networks. Unfortunately, few, if any of these systems occur in nature (but see review
in Kaandorp and Klueber, 2001). Instead, naturally occurring trees have asymmetrical branching.
Another major problem of Horton-Strahler ordering has been excessive side branching found in
nature (e.g. pinnate branching networks: Fig. 1). Tokunaga (1978) proposed an elegant way to
assign order to asymmetrical trees with many side branches. Using this modification of the
Horton-Strahler ordering systems, statistical symmetry has been recently found between
geological and geophysical branching patterns (Turcotte and Newman, 1996) as well as between
living and non-living systems (Pelletier and Turcotte, 2000; Kaandorp and Klueber, 2001). Some
modular colonial organisms such as gorgonian coral trees have been analyzed using the
traditional Horton-Strahler ordering (Brazeau and Lasker, 1988; Mitchell et al., 1993) whereas
some other could not due to excessive side branching (Sanchez et al., submitted). Using the
Tokunaga ordering scheme, a common feature of rivers and leaves’ networks was the nearly
perfect linearity in the number of branches (N) per order (i) when using semi-log scale (Pelletier
αi
and Turcotte, 2000), N ∝10 , where α ≈ -0.65. The mean branch length (L) in terms of order in
βi
rivers and leaves is also a linear function using semi-log scale, L ∝10 , where β ≈ 0.34.
However, other modular organisms such as gorgonian corals do not exactly follow the same
pattern. For the “simple” pseudo-dichotomous species (e.g. candelabrums Fig. 2 B) there was not
apparent change among or within orders in the semi-logarithmic plot, which suggests that all
orders have approximately the same branching properties (e.g. growth rates). On the other hand,
5
“complex” pinnate species (e.g. feathers Fig. 1) exhibit a great deal of within- and among-order
variation that could be due to the differential growth rates observed between main and side
branches (i.e. Lasker et al., in prep.). Nevertheless, L(i) for corals was very different from which
βi
was found in rivers and leaves (L ∝10 ). Therefore, the behavior of number of branches and
mean branch length per order are not universally similar in all branching patterns of modular
organisms such as gorgonian corals (see Sanchez et al., submitted). This could be due to the
different nature of the marine animal networks or simply because they do not really bifurcate
during branching, which seems to be the case at least for gorgonian corals.
In this paper, we provide new theoretical ground on the ordering and dynamical behavior
of branching and colony development. We aimed to explain the problem of branching with selforganization and ecological grounds. Particular goals of the study were (1) to examine the
branching and ordering process in modular organisms using gorgonian corals as model system,
(2) to identify critic parameters for the branching dynamics and (3) to propose a model
explaining such dynamics. Particularly for such model we deduced analytically an expression
combining a difference logistic equation (i.e. May, 1976; Case, 2000) with the process of
branching and ordering observed empirically. Finally, the convergence of the discrete succession
of branching was examined to see if the model provides an explanation for determinate growth
and when/how a colony stops branching. This model does not aim to explain both mechanistic
and external controls on branching but to help understanding the numerical behavior of
branching as a discrete process. We wanted to show that although the internal controls face the
changing environment of the colony forming a dynamic ecological interaction colony form is
preserve until the end of development.
6
Colonial growth in marine modular invertebrates
Multiple observations on colony growth/development from several marine modular
organisms have a common feature: growth decreases with size/age. For instance, complete
observations on the colony growth kinetics of the branching hydroid Campanularia flexuosa
have shown that there is a decline in growth as size increases (Stebbing, 1981). It is worth noting
that those hydroid colonies after stopping growth keep desorbing hydrants without becoming
senescent. Even more interesting is the numerical behavior of the growth rate of C. flexuosa
though time, which exhibited cyclic non-linear behavior always decreasing after a high peak (i.e.
Stebbing, 1981, Figs. 11-12). Colony growth in gorgonian corals, our model system, has also
been documented to stop/decrease in a determinate fashion when the colonies are reaching
certain size. For instance, growth of the Pacific gorgonian coral Muricea californica decreases as
function of height (Grigg, 1974). Sea whip gorgonians Leptogorgia spp. from the Gulf of
Mexico also exhibit reduction of growth rate with height (Mitchell et al., 1993). Colonies of the
Mediterranean gorgonian coral Paramuricea clavata exhibit also size-specific negative growth
(Coma et al., 1998). Similarly, Lasker et al. (in prep.) showed in an extensive survey of the
Caribbean gorgonian Pseudopterogorgia elisabethae how colonies decrease and finally stop
growing near certain size. Growth from the Mediterranean sea-fan Eunicella clavolinii adjusts to
sigmoid growth models (Velimirov, 1875). Other octocoral such as the deep-sea soft octocoral
Anthomastus ritteri presents a sigmoid Gompertz growth trajectory ending asymptotically at the
biggest/oldest colonies (Cordes et al., 2001). Consequently, it is reasonable to think that if
colonies are decreasing growth when reaching certain size, there could be such a fixed number
indicating the maximum capacity of branches before complete interference. These observations
7
also suggested a sigmoid- or logistic-like growth trajectory, which must be met by the
predictions of a realistic model of branching.
Branching in modular organisms: a self-organized process
Modular colonial organisms such as gorgonian corals branch sub-apically and depicting
hierarchical mother-daughter relationships among branches, which could be observed on young
colonies of Pseudopterogorgia bipinnata (Fig. 1 A-B; see also Lasker and Sánchez, in press).
This pattern also occurs among species with sub-apical growth (production of branches below
the apex), which are the case for most branching cnidarians as well as most plants (see review in
Prusinkiewicz, 1998) and fungi (i.e. Watters et al., 2000). Mother branches resemble a pine
branch with many pinnules and are self-similar (e.g. first mother main stem, secondary mother
branch, tertiary mother branch, etc.). Daughter branches, the pinnules, have determinate growth
in short periods of time (Lasker et al., in prep.). The pattern defining the mother-daughter
ordering scheme is far from having fixed branching ratios. It has indeed a parabolic-like behavior
in terms of branches per order and a logistic- or sigmoid-like appearance when accumulating the
number of branches per order. However, any of the colonies follow the same developmental
trajectory and it is reasonable to think that there is no such deterministic rule as the case of
centripetal of centrifugal ordering and bifurcation schemes. Hundreds of colonies of the
gorgonian coral Pseudopterogorgia elisabethae were examined during two years showing that
any daughter branch can turn in a mother branch but its exact location under natural conditions is
uncertain (unpublished). Similarly, every colony branch distribution (e.g. locations of mother
branches, number of daughter branches, etc.) has its unique identity like a fingerprint but they
preserve similar species-specific form. These observations suggested us to think a different
8
direction to understand branching. This system seems to grow in a more autonomous and
individualistic way.
Branching on modular organisms seems to be critically controlled by a parameter
indicating the ratio between total branches (N) and mother branches (S) or c =
N
S
(Fig. 2).
Colonies from two different gorgonian species (Gorgoniidae: Pseudopterogorgia bipinnata;
Plexauridae: Plexaura flexuosa) maintain a linear relationship between total branches and mother
braches, which slope is c (Fig. 2). Shape is preserved in these tree-like networks by maintaining
a constant ratio between total branches and mother branches (c). It is assumed that c, an integer
in terms of number of branches, is dynamically maintained by the production of mother branches
(from an old daughter branch) when the number of total branches is off the neighborhood of c. It
means that every time mother branches are producing daughter branches they are moving the
system off c, which eventually reaches c again after new growth fronts (e.g. mother branches) are
produced and so on. It is assumed that c is related to a branching threshold or an intrinsic
mechanism of branching, which is species- or colony form-specific.
To explain the branching process dynamically let us describe branching as a discrete
recurrent process. Dynamical models may not always be transmitted in terms of differential
equations for those which variables change discretely (Case, 2000). Studying slow-growing
modular organisms such as Pseudopterogorgia spp., there is no way to calculate an
instantaneous growth rate. This, indeed, should be calculated from discrete periods of time (i.e.
Goh and Chou, 1995; Coma et al., 1998; Lasker and Sánchez, in press). Finite discrete equations
also show the iterative nature of colony development where the growing variables are a function
of the previous conditions or X t+1 = f (X t) (i.e. Kaandorp, 1994). Therefore, let us consider a
population of branches N (daughter and mother branches) integrated as a colony from a
9
primordial mother branch S0=1. Under initial conditions, the number of branches per mother
branch increases r daughter branches after every iteration from t to t + 1 (branching rate). As the
number of daughter branches increases from iterations of t to t + 1, it reaches eventually c
branches (or is close to), that provokes the production of mother branches from an old daughter
branch. It indeed produces “grandchildren branches” respect to S0 and so on (Fig. 1 A-B).
Therefore the further production of mother branches or growth “avalanches” has this recurrence
form:
S t +1 = S t +
St r
c
 r
S t +1 = S t 1 + 
 c
or,
(1).
Expression (1) describes the process of branching producing self-similar mother branches
in a self-organized process. For instance, a hypothetical species with r = 12 and c = 20 produced
109 mother branches after 11 iterations and their size frequency distribution (assuming an
increment of r daughter branches per mother branch after every iteration) follows a power law
(Fig. 3 C). If branching is leaded by a critical state the size frequency distribution of branching
fronts size (daughter branches mother branch -1, D) must be dependant of a scaling power law of
the form D(n) ~ n -τ, where n is the frequency of D(n) daughter branches per mother branch and
-τ is a fractal exponent. Interestingly, if we compare the mother branch size frequency
distribution of different colonies from Pseudopterogorgia bipinnata and Plexaura flexuosa, it is
observed a very similar pattern as the observed iterating expression (1) and only noting
qualitative differences between species due to different fractal scaling powers (-1.6 and -2.4: Fig.
3 A and B). Consequently, there is an exponential decrease in branch size (or number of daughter
branches mother branch -1) as their frequency in the branching system increases.
10
These theoretical and empirical observations suggest the great importance of parameters
such as c for branching preserving shape. An idea to explain this finding is the behavior of some
dynamic systems that evolve spontaneously from apparently stationary stages without tuning of
parameters (self-organized criticality, see review in Bak, 1996). Such system evolution around a
parameter is due to the critical effect of such parameter as variables increase such as the case of
the slope in the sand pile/avalanche model (i.e. Carlson and Swindle, 1995). Branching and
colony development can be reduced as the production of new and overlapping growth fronts of
mother branches. Branching can be considered as a self-organized criticality phenomenon
oscillating as c is being approached/retreated. Since branching continues as well as the effect of c
in the process, these “avalanches” of mother branches will be more frequent and the colony will
keep branching at many mother-daughter hierarchies simultaneously like in a self-organized
criticality (e.g. Fig. 1 C). Besides to be an actual phenomenon, this process explains satisfactorily
empirical observations on branching for gorgonian corals and the c parameter reveals a
morphospace when compared with module size for the different colony architectures of 24
Caribbean octocoral species (Sanchez et al., in prep.). However, this “intrinsic” branching
mechanism does not seem to explain when this process should end. Expression (1) predicts an
exponential production of branches (Fig. 2 D), which is not observed empirically. The rapid
growth by means of new growth fronts of mother branches brings along both allometric and
branch interference constraints that could influence to resource capture and growth rates.
Consequently, it is reasonable to think that there is an independent effect from the “intrinsic
branching” process that prevents colonies to grow exponentially. The following model aims to
settle both the intrinsic process of branching with the global-ecological effects.
11
An ecological approach
This model shapes the intrinsic mechanism that produces and links mother and daughter
branches in a colony explained above (1) by including global ecological effects. The model
predicts a reduction of branching as the number of mother branches reaches a maximum number
or charge capacity (k). As we have seen previously, the two variables (mother and daughter
branches) are phenomena well distinguished during the growth of branching colonies. Although
there is a centrifugal ordering involved in the process of colony development, this model predicts
the production of branches only with the information of mother-daughter branch ratio, which is
independent of ordering. However, the exponential increment for the number of mother branches
observed in (1) does not count for crowdedness effects or the density-dependant/allometric
constraints for the rapidly increasing colony size (e.g. Fig. 3 D). Counting for likely branch
density-dependant constraints, a maximum number of mother branches due to branch
interference, or k, could shape dynamically branching in a logistic-like form. Here, it is assumed
that the natural biggest colonies of a population exhibiting low (or asymptotic) branching are in
the neighborhood of such number. Now the aim is to portray branching in a logistic-like dynamic
process but still dependant of mother branch production (as empirically observed). Under such
circumstances we can modify a population discrete logistic equation like this (2):
 S 
N t +1 = N t + S t r 1 − t 
k 

(2).
Where N is the total number of branches in a colony (both daughter and mother) and S is the
number of mother branches. Even though expression (2) includes the intrinsic branching rate r it
does not show how the branching process occurs. Therefore, the model is a system that needs the
recurrence expressions (1) and (2) to fully depict the dynamic interaction of branching.
12
Results
According to the results from integrating expression (1) as shown in the Appendix A, we
have expression (3) for any value of t giving only the initial conditions S0.
 r
S t = S 0 1 + 
 c
t
(3).
This expression is also needed in order to find the analytic solution for all N t+1, our variable of
interest, in terms of St, the leading variable that behaves according to a mother-daughter
relationship. Now we can replace (3) to reduce (1) to initial conditions (4):
N t +1
t

 r

S
1
+

t
0
c
 r 

= N t + rS 0 1 +  1 −
k
 c 









(4).
Following the analytic solution for Nt according to initial conditions S0 (expressions 5-10,
Appendix B). Then we have production of branches Nt+1 in terms of parameters and initial
conditions (10):
t
2t

 r
 r 
1 − 1 +  
 1 − 1 + 
rS 0 
c

 c 
− S0
Nt = N0 +
k
2
r
k 
 r 
−
1 − 1 +  

c

 c  
There is also a special case when
Nt = N0 +
r
= 1 , or r = c and we can quite reduce the expression (11):
c
rS 0  S 0

1 − 2 2t − k 1 − 2 t 

k 3

(
) (
(10).
)
(11).
Our empirical observations showed us that all tree-like modular organisms such as gorgonian
corals start always with one single source branch (S0 = 1), which is the only branch too at initial
13
conditions (N0 = 1). Then we can have the model only in terms of parameters after the first
iteration N1 (12):
2

 r
 r 
 1 − 1 +  1 − 1 +  
r
 c −
 c ,
N1 = 1 + k
2
r
k
 r 
−
1 − 1 +  

c

 c  


 r
 1 − 1 + 

r
 c
N1 = 1 + k
− 1


r
k
−


c


N1 = 1 + r −
r
k
or,
(12).
This shows that only after the second iteration the organism start having daughter branches or
“branching” as the reason of the discrete branching rate r with a minor effect of the charge
capacity, because there is only one mother branch and low interference is expected. After the
second iteration the branching increases in a more complex way under the full scope of
parameters, when it is presumably reaching the neighborhood of c (13).
2
4

 r
 r 
1 − 1 +  
 1 − 1 + 
r
c
 c 

N 2 = N1 + k
−
2
r
k
 r 
−
1 − 1 +  

c

 c  
(13).
Fixed point, convergence, and qualitative behaviors. If expression (10) is the solution
for a system describing the quantitative behavior of branches during colony development, it is
expected a predictable outcome per species because it is a process to build a discrete structure
instead of a continuous stabilizing population. Therefore, the process should have a very stable
14
trajectory before to reach a fixed point otherwise should converge to a common numerical
neighborhood for each colony (or species) sharing the same branching parameters. Initially, if we
want to know a fixed point for Nt we should get:
lim N t = N * ,
t →∞
as:


t
2t 

rS
1−α
1−α
N t = N 0 + 0 k
− S0
,
r
k 
1−α t 
−
c


where α = 1 +
r
. Therefore, the lim N t exists only if
c
n →∞
t
r
r
r
 r
lim α = 0 ⇔ lim1 +  = 0 ⇔ 1 + < 1 ⇔ −1 < 1 + < 1 ⇔ −2 < < 0 .
t →∞
t →∞
c
c
c
 c
t
However, r and c are by definition positive parameters (number of branches). Therefore, r > 0
and c > 0, then
r
will never be lower than 0 and Nt does not have a fixed point. If there is not
c
fixed point for a branching colony, then, what is the fate of this dynamic system? We can study
the convergence of the iterative system from a different approach. For instance, since there is
proportionality between the variables (mother and total branches) or c (14):
St =
Nt
c
(14).
This expression could be also used to examine the convergence for all N through time. For
instance, if we substitute (14) in (1), we obtain (15):
2
 r N r
N t +1 = N t 1 +  − t 2
 c  kc
(15).
15
Let us suppose initially that the succession is convergent and then there is the lim N t = L . Thus,
n→∞
taking the limit at both sides of (15):
r
 r
lim N t +1 = lim N t 1 +  − 2 lim N t2 .
t →∞
t →∞
 c  kc t →∞
However, with {N t +1 }t∈Ν it is a sub-succession of {N t }t∈Ν , then,
lim N t +1 = lim N t = L , and, lim N t2 = L2 .
t →∞
t →∞
t →∞
Hence, we can calculate the value of L in the expression (16):
r
 r
L = L1 +  − 2 L2
 c  kc
(16).
Since N0 ≠ 0, then L ≠ 0 and we can simplify (16) obtaining:
1 = 1+
r
r
r L r
− 2 L ⇔ ⋅ = ⇔ L = kc
c kc
c kc c
This is the case where the succession {N t }t∈Ν is convergent to L = kc and we have two
possibilities (see appendix C for convergence criterion):
I)
2
 r N r
If N t +1 = N t 1 +  − t 2 > Nt in some interval (e.g. when the curve
 c  kc
r
 r
2
N t +1 = N t 1 +  − 2 N t is over the straight line Nt = Nt + 1) then the succession
 c  kc
{N t }t∈Ν
II)
increases as long as its values remain in that interval.
2
 r N r
If N t +1 = N t 1 +  − t 2 < Nt in an interval (e.g. when the curve
 c  kc
r
 r
2
N t +1 = N t 1 +  − 2 N t in under the straight line Nt = Nt + 1) then the succession
 c  kc
{N t }t∈Ν
decreases as longs as its values remain on that interval.
16
The convergence can also be corroborated by using the quantitative solution of
expression (10) and hypothetical values of parameters. Overall there is initially a steady
increment in the number of new branches Nt produced between discrete periods of time that
turned abruptly after reaching maximum production per period of time (Fig. 4 A). The turning
point was usually above the line Nt = Nt + 1 but it crosses the trajectory at the turning point when
the value or r was much smaller than c (Fig. 4 A). The cumulative number of branches time
series, indicator of colony size, had a sigmoid trajectory with a short period of low growth
followed by exponential increment and ending after a short asymptotic period (Fig. 4 B).
Hypothetical species also showed how the interplay between parameters r and c resulted
in modified qualitative behaviors of the branching trajectory (Fig. 4). There are two main
qualitative behaviors. One is when r << c corresponding to feather-like pinnate colonies such as
Pseudopterogorgia bipinnata, which branches produce many daughter branches before than one
of these start producing grandchildren branches (Fig. 5 A). The other outcome was those species
with very similar values for r and c, where almost every branch will for sure have daughter
branches (Fig. 5 B). These are pseudo-dichotomous or candelabrum-like colonies, such as
Plexaura flexuosa, that present many overlapping generations of mother branches. The model
has a simplified version due to r ≅ c (expression 11: Fig. 3 B). Even though the fate of branching
seemed very deterministic, Rekin diagrams of these two outcomes show how the pinnate
colonies might keep growing in a cycle of period 10 around the turning point (Fig. 5 A). The
absence of a fixed point at the intercept with the line Nt = Nt + 1 suggest that this is a very unlikely
event but qualitatively possible. Nonetheless, it is clear that the case of pseudo-dichotomous
colonies is more predictable than pinnate ones because the values of Nt + 1 under the turning point
are still in the neighborhood of the maximum branch production in the latter (Fig. 5 A). This
17
kind of behavior might be also observed if external perturbation affects the critical parameter c
(e.g. grazing, disturbance, etc.).
Discussion
Branching in modular colonial organisms can be understood as self-organized process
that produces fast pulses of branching. This process is shaped by the changing environment of
the growing colony (e.g. branch interference and global colony size). Like in many ecological
systems, branching may have size- and density-dependant constraints explaining reduction of
growth. Colonial growth, though shaped logistically, converges very predictably to their
expected maximum number of branches, when branching stops completely by its own branch
interference. It was theoretically demonstrated that there is no fixed point during the process of
branching and that the recurrence succession converges to kc, which is nothing but the maximum
number of total branches. Consequently, the branching of modular colonial organisms under the
scheme exposed above follow a very predictable outcome. This is a dramatic difference with a
logistic population model where there may be stable points and/or periodic cycles and the
population perishes (i.e. May, 1976). The branching trajectory was sigmoid with a rapid
exponential phase that ended in a short asymptotic period. The new branches produced during
each discrete period of time was nearly symmetric above and below the line Nt = Nt + 1 indicating
a half-life turning point where growth starts to decrease. Nonetheless, the different colony
architectures may have very different qualitative behaviors depending on the relationship
between growth and production of new growth fronts as new mother branches. The model
predictions adjusted to what were observed by several studies on growth of marine modular
invertebrates, which were logistic- or sigmoid-like growth trajectories.
18
Branching and self-organized criticality. The model presented here showed a process
where a simple modular system can drive itself without the need of fine tuning of any parameter.
This self-organized process allowed to branch through time preserving colony shape. Although
this kind of behavior has been identified mostly on physical systems (e.g. sandpiles, avalanches,
forest fire, etc.), there is an increasing number of applications for biological systems including
branching process. Self-organized criticality (SOC) has been found in the “avalanches” of alveoli
activation during lung inflammation, which is a self-similar branching structure (see review in
Csahok, 2000). Branched polymer growth has been explained by a SOC state of a regulating rule
for the aggregation of monomers (Andrade et al., 1997). Even the punctuated equilibrium
evolution model, leading to evolutionary branching, can be explained as a SOC phenomenon
(see review in Bak and Paczuski, 1995). It was surprised to find that a model as SOC explains
more adequately the dynamic process of branching in modular colonial organisms than
traditional approaches such as ordering.
Here was exposed by the first time how the form and development of branching modular
organisms can be partially explained by SOC. It was clear how the scaling of mother branches
size (both empirical and theoretical) followed power law frequency dependence. It was identified
a critical parameter (c) that keeps the system in a spontaneous dynamics whereas preserving
form. Should mother branches produce more daughter branches, or, daughter branches produce
more mother branches? This is the critical state that keeps the colony actively growing. But,
what biological mechanism could provide such self-organized state? Buss (2001) model of
hydroid colony growth by intussusceptions is conceptually a SOC, where colonies keep adding
modules responding to certain threshold of internal fluid tension. A similar approach has been
proposed using redox control for the same kind of hydroids (Blackstone, 1999), which also
19
suggest the presence of thresholds triggering colonial expansion. Although we do not know the
interplay between resource transport and developmental genes expression in gorgonian corals,
the critical state c in a colony could indicate certain differential between resource availability and
surplus that could trigger new growth fronts. It is, as a result, testable to find a physiological or
proteonomical correspondence to the pulses of growth observed in gorgonian corals.
Crowdedness and branch interference. Size increment brings along a series of
constraints that affect the colony design and module interference (see review in Lasker and
Sánchez, in press). Space is the primary limiting constraint for clonal sessile taxa (Jackson,
1977). For instance, the design of branching colonies of modular organisms such as cheilostome
bryozoans have been shaped both to prevent dragging and breakage, and, to minimize
crowdedness and module interference (Cheetham and Thomsen, 1981; Cheetham, 1986). During
the growth of byozoans, branches initially divergence and progressively converge beginning to
interfere with their functions, which seems to limit the maximum colony size (Cheetham and
Hayek, 1983). Stebbing (1981) suggested that Campanularia flexuosa stops growing when the
spaces for asexual production of zooids are completely filled or in close proximity to other
zooids. In gorgonian corals, for instance, experimental evidence shows that crowding among
branches impedes the capture of resources at the shaded branches (self-shading: Kim and Lasker,
1997). There are indeed allometric constraints for resource capture during modular growth
because internal modules begin to be resource-depleted by the expansion of new exterior
modules (Kim and Lasker, 1998). Some octocoral colonies (soft coral Sarcophyton) start
growing and calcifying from the base up to the branches (Tentori E., Central Queensland Univ.,
personal communication). Both soft and gorgonian corals have direct connections from every
module down to the colony base (Bayer, 1961). In the case of gorgonian corals it can be
20
plausible that the growth and extension of their internal axial channels (solenia) be linked to the
production of new branches (unpublished). If resources are being depleted as the colony
“colonizes” its periphery, there should be such maximum extension point (e.g. k) when no
surplus is then provided to the base and growth and branching would stop.
The inclusion of a new parameter controlling branching in modular organisms such as k,
or maximum branch capacity, had supports from size and interference constraints, which are
evident phenomena during colony growth. The complex nature of branching is still modulated
criticality by the interaction between r and c but global ecological constraints prevail throughout
colony development. Nevertheless, direct empirical observations to fulfill a complete ecological
theory of branching are needed. Some biological other aspects of modular colonial organisms
such as reproduction and regeneration were not cover in this model and could have and
important effect on branching. Straightforward parameters have been identified and a number of
model organisms seem appropriate for such tests. Future observations on colony growth could
greatly increase the knowledge in this field by including branching parameters instead of
height/width or other indirect measurements of modularity.
Acknowledgments
J. A. Sánchez acknowledges Fulbright-Laspau-COLCIENCIAS for a doctoral scholarship
and great support during 1998-2002. The Complex Systems Summer School at the Central
European University, Budapest, Hungary, the Santa Fe Institute (New Mexico, USA), G. Yan
(University at Buffalo, SUNY), and A. Cheetham (Smithsonian Institution) gave to J.A. S. new
insights and good discussion in the studying of branching. The National Center for Ecological
Analysis and Synthesis (NCEAS), University of California, Santa Barbara, workshop “Modeling
of growth and form in sessile marine organisms” 1999 (J. Kaandorp and J.E. Kubler) provided
21
great feedback and discussions. The Bahamas Field Station (1999-2000), Gerace Research
Center-College of the Bahamas, San Salvador, Bahamas, provided field facilities for observing
gorgonian corals. Comments and discussions from C. Mitchell, D. J. Taylor, S.D. Cairns, M.A.
Coffroth and G. Yan (University at Buffalo, SUNY) greatly helped during early stages of the
study.
APENDIX A
Mother branches Sn for all t giving only initial conditions S0.
 r
S t +1 = S t 1 + 
 c
(2).
If we keep iterating equation (2) by discrete periods back in time to get the initial conditions S0
we got:
 r
S t = S t −1 1 + 
 c
 r
S t −1 = S t − 2 1 + 
 c
“
“
 r
S 2 = S 1 1 + 
 c
 r
S 1 = S 0 1 +  .
 c
And by multiplying them member to member,
 r
S t +1 S t S t −1 ...S 2 S1 = S t S t −1 S t −2 ...S1 S 0 1 + 
 c
t +1
.
22
We have a simplified form of (2) for all S given the initial conditions S0 iterated t+1 or t times
(3):
S t +1
 r
= S 0 1 + 
 c
 r
S t = S 0 1 + 
 c
t +1
or,
t
(3).
APPENDIX B
Analytical solution for Nt according to initial conditions N0 and S0.
We have this system of discrete difference equations:
 r
S t +1 = S t 1 + 
 c
(1).
 S 
N t +1 = N t + S t r 1 − t 
k 

(2).
Replacing (1) into (2)
N t +1
t

 r

S 0 1 + 
t
c
 r 
= N t + rS 0 1 +  1 − 
k
 c 









(4).
For better manipulation of (4) we can define α (5) as:
α = 1+
r
c
(5).
Then our recurrence equation (4) gets the form (6):
 S αt 
N t +1 = N t + rS 0α t 1 − 0 
k 

N t +1 − N t =
(
S0r t
α k − α 2t S 0
k
or:
)
(6).
23
This is a telescopic formula given the first values or t, starting from t=0 and then t=1, t=2…etc.,
and it has the following solution:
t = 0, → N 1 − N 0 =
rS 0
(k − S 0 )
k
t = 1, → N 2 − N 1 =
rS 0
(αk − α 2 S 0 )
k
t = 2, → N 3 − N 2 =
rS 0 2
(α k − α 4 S 0 )
k
“
“
t + 1, → N t +1 − N t =
rS 0 t
(α k − α 2t S 0 ) .
k
Adding and equaling member to member it is obtained (7):
N t +1 − N 0 =
[(
)
(
rS 0
k 1 + α + ... + α t − S 0 1 + α 2 + ... + α 2t
k
)]
(7).
The summation of 1+α+ α2+…+ αt is a geometric progression and it may have a value Bt:
Β t = 1 + α + α 2 + ... + α t
(8).
Then if we multiply Bt by α we have:
αΒ t = 1 + α + α 2 + ... + α t + α t +1
So now,
Β t − αΒ t = 1 − α t +1
or:
Β t (1 − α ) = 1 − α t +1 .
If
24
α = 1+
r
≠ 0 , then
c
Β t = 1 + α + α 2 + ... + α t =
1 − α t +1
1−α
Likewise:
1 + α + α + ... + α
2
2t
1 − α 2t + 2
=
.
1−α 2
Therefore:
N t +1 = N 0 +
rS 0
k
 1 − α t +1
1 − α 2(t +1) 
k
S
−
0


1−α 2 
 1−α
(9).
Using the original parameters r and c according to (5), then we have production of branches Nt+1
in terms of parameters and initial conditions (10):
N t +1
t +1
2 (t +1)


 r
 r
−
+
−
+
1 1


 1 1

rS 0 
c
c



= N0 +
− S0
k
2
r
k 
 r 
−
−
1

1 +  
c
 c  

2t
t

 r 
 r
−
+
−
+
1
1
1
1



 

rS 0 
c
c 


− S0
Nt = N0 +
k
2
r
k 
 r 
−
−
+
1
1


 
c
 c  

or,
(10).
APPENDIX C
Convergence criterion.
In summary the convergence criterion can be portrayed as follows: Being L = kc, obtained above,
r
 r
the fixed point of the function f ( x) = x1 +  − 2 x 2 is the interval I, then
 c  kc
25
If f’(L) > 1, then the succession {N t }t∈Ν does not converge to L = kc, excepting the
(i)
case where the succession has a constant value L, it means that the succession would
have been reduced to {N0, N1,…, Nt, L, L, L,…}.
If 0< f’(L) < 1, then the succession {N t }t∈Ν converges to the limit L = kc, in a
(ii)
monotonous way in the neighborhood of the fixed point L = kc.
Additionally:
1) We say that {N t }t∈Ν converges to L in the neighborhood of L if {N t }t∈Ν tends to L = kc
when N0 belongs to the neighborhood of L.
r
 r
2) If , f ( x) = x1 +  − 2 x 2 , then
 c  kc
(x − kc )1 + r  −
(
r
x 2 − k 2c 2
2
f ( x) − f (ck )
 c  kc
f ' (kc) = lim
= lim
x → kc
x → kc
x − kc
x − kc
 r 

r
r
r
r
= lim  1 +  − 2 ( x + ck ) = 1 + − 2 (2ck ) = 1 −
x → kc
c kc
c
  c  kc

)
r
r
Therefore f ' (kc) = 1 − . Since r and c are positive numbers 1 − may not be higher than 1.
c
c
Thus, 1 −
r
r
< 1 and because 0 < f ’(kc) < 1 and therefore 0 < 1 − < 1, then we should have
c
c
that r < c when there is a neighborhood for kc of
(kc − δ , kc + δ ) = {x ∈ ℜ / kc − δ < x < ck + δ },
such as
kc < f ( x) < x when x ∈ (kc, kc + δ )
kc > f ( x) > x when x ∈ (kc − δ , kc)
consequently, if N n ∈ (kc − δ , kc + δ ) then,
26
kc < N n +1 < N n when N n ∈ (kc, kc + δ )
kc > N n +1 > N n when N n ∈ (kc − δ , kc)
Therefore, the succession {N t }t∈Ν increases (or decreases) and is delineated by kc if
N 0 ∈ (kc, kc + δ ) (or N 0 ∈ (kc − δ , kc) respectively), and due to succession theory, the
succession {N t }t∈Ν converges in the neighborhood of kc. However, kc is the only limit of the
succession {N t }t∈Ν in (kc − δ , kc + δ ) consequently {N t }t∈Ν converges to kc monotonously in the
neighborhood of kc.
Literature Cited
Andrade Jr., J.S., L.S. Lucena, A.M. Alencar and J.E. Freitas. 1997. Self-organization in growth
of branched polymers. Physica A 238: 163-171.
Bak, P. 1996. How nature works: the science of self-organized criticality. Copernicus, New
York.
Bak, P. and M. Paczuski. 1995. Complexity, contingency, and criticality. Proceedings of the
National Academy of Sciences USA 92: 6689-6696.
Bayer, F.M. 1961. The shallow water Octocorallia of the West Indian region. Studies of the
Fauna of Curaçao 12: 1-373.
Blackstone, N.W. 1999. Redox control in development and evolution: evidence from colonial
hydroids. Journal of Experimental Biology 202: 3541-3553.
Brazeau D. and Lasker H. 1988. Inter- and Intraspecific variation in gorgonian colony
morphology: quantifying branching patterns in arborescent animals. Coral Reefs 7: 139143.
27
Buss, L.W. 2001. Growth by intussusception in Hydractiniid hydroids. In: Evolutionary
Patterns: Growth, Form, and Tempo in the Fossil Record., J.B.C. Jackson, S. Ligand,
and F.K. McKinney (eds). Chicago: University of Chicago Press.
Carlson, J.M. and G.H. Swindle. 1995. Self-organized criticality: sandpiles, singularities, and
scaling. Proceedings of the National Academy of Sciences USA 92: 6712-6719.
Cartwright, P. and L.W. Buss. 1999. Colony integration and the expression of the Hox gene,
Cnox-2, in Hydractinia symbiologicarpus (Cnidaria: Hydrozoa). Journal of Experimental
Zoology 285: 57-62.
Cartwright, P., J. Bowsher and L.W. Buss. 1999. Expression of a Hox gene, Cnox-2, and the
division of labor in a colonial hydroid. Proceedings of the National Academy of Sciences
USA 96: 2183-2186.
Case, T.J. 2000. An illustraded guide of theoretical ecology. Oxford University Press, New York.
Cordes, E.E., Nybakken J.W. G. and VanDykhuizen. 2001. Reproduction and growth of
Anthomastus ritteri (Octocorallia : Alcyonacea) from Monterey Bay, California, USA.
Marine Biology 138 (3): 491-501.
Cheetham, A.H. 1986. Branching, biomechanics and bryozoan evolution. Proceeding of the
Royal Society of London B 228: 151-171.
Cheetham, A.H. and L.-A. C. Hayek. 1983. Geometric consequences of branching growth in
adeoniform Bryozoa. Paleobiology 9(3): 240-260.
Cheetham, A.H. and E. Thomsen. 1981. Functional morphology of arborescent animals: strength
and design of cheilostome bryozoan skeletons. Paleobiology 7(3): 355-383.
Coma, R., Ribes M., Zabala M. and J.-M. Gili. 1998. Growth in a modular colonial marine
invertebrate. Estuarine Coastal and Shelf Science. 47(4): 459-470.
28
Finnerty, J.R. and M.Q. Martindale. 1997. Homeoboxes in sea anemones (Cnidaria: Anthozoa): a
PCR-based survey of Nematostella vectensis and Metridium senile. Biological Bulletin
193: 62-76.
Grigg, R.W. 1974. Growth rings: annual periodicity in two gorgonian corals. Ecology 55: 876881.
Goh, N.K.C., and L.M. Chou. 1995. Growth of five species of gorgonians (sub-class
Octocorallia) in the sedimented waters of Singapore. Marine Ecology 16(4). 1995. 337346.
Horsfield, K. 1981. The science of branching systems. In: Scientific foundations of respiratory
medicine. J.G. Scadding and G. Cumming (eds). London: William Heinemann Medical
Books. pp. 45-54.
Horton, R.E. 1945. Erosional development of streams and their basins: hydrophysical approach
to quantitative morphology. Geological Society of America Bulletin 56: 275-370.
Hughes, R.N. 1983. Evolutionary ecology of colonial reef-organisms, with particular reference to
corals. Biological Journal of the Linnean Society 20: 39-58
Hughes, T.P, D. Ayre and J.H. Connell. 1992. The evolutionary ecology of corals. Trends in
Ecology Evolution 7(9): 292-295.
Jackson, J.B.C. 1977. Competition on marine hard substrata: The adaptive significance of
solitary and colonial strategies. American Naturalist 111: 743-767.
Kaandorp, J.A. 1994. Fractal modelling. Growth and form in biology. Springer-Verlag, Berlin.
Kaandorp, J.A. and J. Klueber. 2001. The Algorithmic Beauty of Seaweeds, Sponges and Corals.
Springer-Verlag, Amsterdam.
29
Kim K. and H.R. Lasker. 1997. Flow-mediated resource competition in the suspension feeding
gorgonian Plexaura homomalla (esper). Journal of Experimental Marine Biology and
Ecology 215(1):49-64,
Kim K. and H.R. Lasker. 1998. Allometry of resource capture in colonial cnidarians and
constraints on modular growth. Functional Ecology. 12(4): 646-654.
Lasker, H.R., M. Boller & J.A. Sánchez. (submitted). Determinate growth of a modular marine
organism.
May, R. 1976. Simple mathematical models with very complicated dynamics. Nature 261: 459467.
Lasker, H.R. and Sánchez (2002). Allometry and Astogeny of modular organisms. In: R.N.
Hughes (ed.), Progress in Asexual Propagation and Reproductive Strategies. John Wiley
(Oxford). (in press)
Mitchell, N.D., Dardeau M.R. and W.W. Schroeder. 1993. Colony morphology, age structure,
and relative growth of 2 gorgonian corals, Leptogorgia hebes (Verrill) and Leptogorgia
virgulata (Lamarck), from the northern Gulf of México. Coral Reefs 12 (2): 65-70.
Pelletier J.D. & Turcotte D.L. (2000) Shapes of river networks and leaves: are they statistically
similar ?, Philosophical Transactions of the Royal Society of London B 355: 307-311.
Prusinkiewicz, P. 1998. Modeling of spatial structure and development of plants: a review.
Scientia Horticulturae 74: 113-149.
Sebens, K. P. 1987. The ecology of indeterminate growth in animals. Annual Reviews in Ecology
and Systematics 18: 371-407.
Stenning, A.R.D. 1981. The kinetics of growth control in a colonial hydroid. Journal of the
Marine Biological Association U.K. 61: 35-63.
30
Strahler, A.N. 1952. Hyposometric (area-latitute) analysis of erosional topography. Geological
Society of America Bulletin 63: 1117-1142.
Tokunaga E. 1978. Consideration on the composition of drainage networks and their evolution.
Geography Reports Tokyo Metropolitan University 13: 1-27;
Turcotte D. L. and Newman W. I. 1996. Symmetries in geology and geophysics. Proceedings of
the National Academy of Sciences USA 93: 14295-14300.
Turcotte D. L., J.D. Pelletier and W.I. Newman. 1998. Networks with side branching in biology.
Journal of Theoretical Biology 193: 577-592.
Velimirov, B. 1975. Growth and age determination in the sea fan Eunicella clavolinii. Oecologia
19: 259-272.
Waller D.M. and D.A. Steningraeber. 1985. Branching and modular growth: Theoretical models
and empirical patterns. pp. 225-257. In: Jackson J.B.C., L.W. Buss and R.E. Cook (eds.).
Ecology and evolution of clonal organisms. Yale Univ. Press, New Haven.
Watters, M.K., C. Humphries, I. De Vries and A.J. Griffiths. 2000. A homeostatic set point for
branching in Neurospora crassa. Mycological Research 104(5): 557-563.
Woldenberg, M.J. 1997. James Keill (1708) and the morphometry of the microcosm. Pp. 243264. In: D.R. Stoddart (ed), Process and form in geomorphology. Routledge, London and
New York.
Woldenberg, M. J., M. P. O'Neill, L. J. Quackenbush, and R. J. Pentney. 1993. Models for
growth, decline and regrowth of the dendrites of rat Purkinje cells induced from
magnitude and link-length analysis. Journal of Theoretical Biology 162(4): 403-429.
31
Figure Legends
Figure 1. Photographs of living colonies of the gorgonian coral Pseudopterogorgia
bipinnata (San Salvador, Bahamas). A-B. Examples of two young colonies at December 1999 (t)
and July, 2000 (t + 1), the arrows in t depict the daughter branches that turned into a new mother
branch in t + 1. C. Adult colony showing multiple growing mother branches (grid 10 x 10 cm).
Figure 2. Plots of the number of total branches vs. mother branches colony -1 with mean
and 95% predictive intervals indicated. A. From 20 colonies of Pseudopterogorgia bipinnata (c
= 19; r2 = 0.69; P<0.05). B. From 9 colonies of Plexaura flexuosa (c = 5; r2 = 0.88; P<0.05) (San
Salvador, Bahamas). Both inset photos: San Salvador, Bahamas.
Figure 3. A-C. Size frequency distribution of number of daughter branches branch -1 in
log-log scale. A. From 176 mother branches of 11 photographed colonies of Pseudopterogorgia
bipinnata (r2 = 0.93; P<0.05). B. From 211 mother branches of 6 photographed colonies of
Plexaura flexuosa (r2 = 0.92; P<0.05) (San Salvador, Bahamas). C. From a hypothetical colony
iterating expression (1) eleven times, with r = 12 (assuming an extension of r at every mother
branch per iteration) and c = 20, 109 branches mother branches (r2 = 0.96, P<0.05). D.
Cumulative time series of mother branches S, data from C.
Figure 4. Results from the iteration of the model for a hypothetical species with k = 30
and c = 20. A. Different number of new branches for Nt and N t + 1 with different values of r (4, 8,
10, 12, and 20). B. Cumulative total number of branches N for different values of r (4, 8, 10, 12,
and 20) along 6-month periods.
Figure 5. Rekin diagrams showing two different quantities behaviors for the number of
new branches (net growth per iteration) for the map of all Nt and N t + 1 (k = 30). A. When r = 4
and c = 20,. B. When r = 20 and c = 20.
32
B
A
t
C
t+1
t
t+1
400
800
A
Total branches
Total branches
1000
600
400
200
0
B
300
200
100
0
0
5
10
15
20
Mother branches
25
30
0
10
20
30
40
50
60
Mother branches
70
80
A
B
100
Branch Frequency
Branch Frequency
100
10
10
1
1
1
C
10
100
Daughter Branches Branch
10
-1
-1
120
100
80
10
S
Branch Frequency
Daughter Branches Branch
D
100
100
60
40
20
0
1
10
100
Daughter Branches Branch
-1
0
2
4
6
t
8
10
12
600
r =20
300
200
100
5000
4000
B
r=
4
1
t+
400
r=
4
N t+1
N
=N
t
3000
2000
1000
r=
20
A
500
N (cumulative)
6000
0
0
0
200
400
Nt
600
0
5
10
15
t
20
25
30
500
A
=
800
1
B
700
Nt
600
300
N t+1
N t+1
400
c >> r
N t+
200
c≅r
N
=N
t
t+
1
500
400
300
200
100
100
0
0
100
200
Nt
300
400
500
2t
t

 r
 r 
1 − 1 +  
 1 − 1 + 
rS 0 
c
c 

− S0 
Nt = N 0 +
k
2
r

k 
r


−
1
1
−
+




c
 c 

0
0
100
200
300
400
500
600
700
800
Nt
Nt = N0 +
(
) (
)
rS 0  S 0
2t
t 
1
2
k
1
2
−
−
−

k  3
