Polar Organisms With Apolar Individual Cells

POLAR ORGANISMS WITH APOLAR INDIVIDUAL CELLS

G. T. HERMAN State University of New York at Buffalo, Amherst, N.Y., USA

1. Introduction

This paper is intended to demonstrate how a particular automatontheoretical model for cellular interactions in development, the one originally proposed by LINDENMAYER (1968), can be used to answer a particular problem in biology. The problem that we shall be concerned with is whether polarity of individual cells is a necessary condition for the regulatory polar development of organisms. We quote SINNOT (1960) to explain the notion of polarity in biology. "A notable feature of these bodily forms of plants (and animals) is the presence in them of an axis which establishes a longitudinal dimension for organ or organism. Along this axis, and symmetrically with reference to it, the lateral structures develop. The two ends or poles of the axis are usually different both as to structure and physiological activity" . "This characteristic orientation of organisms, which is typically bipolar and axiate, is termed polarity". "Polarity is simply the specific orientation of activity in space. It refers to the fact that a given biological event, such as the transfer of material through an organ or the plane in which a cell divides, is not a random process but tends to be oriented in a given direction. If this were not so, an organism would grow into a spherical mass of cells, like tissue in a shaken culture". An example given by SINNOT (1960) refers to some original experiments of Yachting. "YaCHTING (1878), cut twigs of willow and kept them under moist conditions. Some he left in their normal, upright orientation and others were inverted. Regardless of orientation, however, roots tended to be regenerated more vigorously from the morphologically basal end and

666

G. T. HERMAN

shoots from buds at the original apical end. This is the classical example of polarity. If such a shoot were cut into two or more parts transversely, each part regenerated roots and shoots in the same polar fashion. Even very short pieces of stem showed this polar character. Yachting removed a ring of bark in the middle of a shoot and confirmed earlier observations that roots were formed above the ring and shoots below, just as if the stem had been cut in two. From these and similar experiments he concluded that polarity was a fixed and irreversible characteristic of the plant axis and that probably the individual cells of which the axis was formed themselves possessed a polar character". Our major concern will be, whether such a conclusion about the polar character of the individual cells is justified on the basis of the experiments. In other words, we shall investigate whether the kind of global behavior observed in the experiments can be achieved with individual cells which are apolar. What we shall describe now is a summary of HERMAN (1971, 1972). The interested reader will find a more detailed discussion in those papers. For the individual cell, polarity can have two kinds of manifestation: 1. External polarity, e.g., transportation of certain substances may take place in one direction and not in the other. 2. Internal polarity, e.g., unequal division, the daughter cells of a cell may have different characters. In this paper, we shall incorporate the notions of external and internal polarity into the automaton-theoretical model of Lindenmayer, and then investigate whether the kind of behavior, which led people to believe that there is polarization of individual cells, can be achieved with cells which are apolar. Our major result will be that complicated polar regulatory behavior can be achieved with apolar individual cells, and so the argument, which concludes the polarity of individual cells from certain kind of polar behavior of the organism as a whole, is false. This is not to say that we claim that cellular polarity never plays a role in achieving global polarity of the organism. This mayor may not be the case. What we shall show is that one cannot argue that cellular polarity is necessarily present from observations of the kind described by Sinnot. 2. Definition of Lindenmayer models If G is a nonempty finite set, G* denotes the set of all strings of elements of G. G* includes the empty string, which is denoted bye.

POLAR ORGANISMS

667

Given a string p E G*, pR denotes the string which we obtain by writing elements of p in reverse order. A Lindenmayer model L is a quadruple
<{O, I}, 1, 0, {1}), where 0 is defined by 0(1,0,1) = 00, 0(1,0,0)=0, Md, c ; b) = 1,

in all other cases.

is a Lindenmayer model with two states 0 and 1, where if a cell is in state o it divides into two cells of state 0 if both its neighbors are in state 1, and it changes into a cell in state 1 if its left neighbor is in state O. Under all other conditions cells do not change state. The standard environmental input is 1, i.e., we assume that unless otherwise stated the leftmost cell behaves as if its left neighbor was a cell in state 1 and the rightmost cell behaves as if its right neighbor was a cell in state 1. A Lindenmayer model is said to be propagating if and only if e is not oed, b, c) for any d, b, C E G. So in a propagating Lindenmayer model cells cannot simply disappear. The L in Example 1 is a propagating Lindenmayer model. A Lindenmayer model L =
=

o(d,b,f)

for all d, b, c, f E G. In a unidirectional Lindenmayer model a cell is only influenced by its left neighbor. Thus passage of information in the filament is allowed only from left to right. (The sense of the word 'information' in th~ context is explained in the first footnote of HERMAN, 1970.) The Lindenmayer model of Example 1 is not unidirectional, since 0(1 , 0, 1)

=1=

0(1, 0, 0).

A Lindenmayer model L =
668

G.T. HERMAN

Thus, in an informationless Lindenmayer model the future of each cell is determined by the state of the cell only, and it is not influenced by the states of its neighbors. Note that every informationless Lindenmayer model is a unidirectional Lindenmayer model. As can be seen above, the notions of nondisappearance of cells, of unidirectional information flow and of autonomous (mosaic) cellular development can be introduced into the theory of Lindenmayer models by putting certain restrictions on the function b. The two notions given in the introduction for the non polarity of individual cells can be incorporated in a similar fashion. A Lindenmayer model L = G, g, (l, F) is said to be externally symmetric if and only if (l has the property that

<

(l(d,b,c)

=

(l(c,b,d)

for all d, b, c E G. L is said to be internally symmetric if and only if the property that

(l

has

(l(d, b, c) = [(led, b , C)]R

for all d, b, c E G. L is said to be symmetric if and only if it is both externally symmetric and internally symmetric. The L of Example 1 is not externally symmetric since (l(1, 0, 0) :F (l(0, 0,1). However, it is clearly internally symmetric. The central purpose of this paper will be the investigation of the properties of externally symmetric, internally symmetric and symmetric Lindenmayer models. If L = G, g, b, F) is a Lindenmayer model, any element p of G* is said to be a filament of L. A filament is said to be dead if and only if one of its symbols is in F (see HERMAN, 1969). For any filament p we define J.(p), the consecutive filament as follows. If p is of the form a1 02 ... -ak(k ~ 2), then

<

q

= J.(p)

if and only if q

= q1q2 ...

qk>

where qj q1

qk

=

(l(aj_l,

OJ,

aI+ 1 )

= (l(g, a 1 , a2 )

=

,

for 2

~ i ~

k-l,

,

(l(ak-l, ak, g).

If P = e, then J.(p) = e, and if pEG, then J.(p) = (l(g, p, g).

POLAR ORGANISMS

669

For all nonnegative integers n we define )."(P), the n'th consecutive filament of p, by ).0(P) = p,

)."+1(P) = ).()."(p)). Thus, in Example 1, if p

= 0, then

).°(0) = 0, ).1(0) = ),(0) = <5(1,0,1) = 00,

).2(0) = 01, ).3(0) = 001, etc.

3. The problem of universal computing ability A general developmental model for organisms should be powerful enough so that one can incorporate into it a number of phenomena. It can be argued (see, e.g., HERMAN, 1969) that one of these phenomena is the ability to implement in the model an arbitrary algorithm. An algorithm is, to quote MINSKY (1967), "a set of rules, which tell us from moment to moment, precisely how to behave". However, an algorithm cannot be divorced from the environment in which it operates. For example, the description of a chess game which appears in a newspaper is a perfectly clear algorithm for a chess player who wants to reproduce that game. However, if we want a computer to implement this algorithm, we must translate rules like P /K2-K4 into rules which the computer is designed to obey. Furthermore, since the computer cannot move pieces on a chessboard, it must somehow have a way of representing the positions of the pieces on the board inside its memory. In short, what we can do is to 'simulate' the chess player by a computer. In this paper, we shall assume that the general notion of simulation requires no further explanation. A short exposition of the general notion can be found in Section 1 of HERMAN (1969). Using the notion of simulation, the problem of universal computing ability can be stated as follows. Given any algorithm, can a (type X) Lindenmayer model always be found which simulates it? The difficulty with the above formulation is that the notion of an algorithm is somewhat vague, and so a precise mathematical analysis of the problem is impossible. This difficulty is avoided by introducing the concept of Turing machines. Turing machines are abstract computing machines about which

670

G.T. HERMAN

it has been claimed that given any algorithm there exists a Turing machine which simulates it. This claim is sometimes referred to as Turing's thesis (see, e.g., MINSKY, 1967, p. 108). In view of this, the precise version of the problem of universal computing ability reads as follows. Given any Turing machine, can a (type X) Lindenmayer model always be found which simulates it? We shall assume that the reader is acquainted with the notion of a Turing machine (see, e.g., HERMAN, 1969, 1971; MINSKY, 1967). The following sequence of theorems, whose proofs can be found in HERMAN (1971) gives a detailed analysis of the computing ability of the symmetric Lindenmayer model and some of its restricted versions. In particular, we have that symmetric Lindenmayer models and internally symmetric unidirectional Lindenmayer models have a universal computing ability, but externally symmetric unidirectional Lindenmayer models do not. THEOREM 1. For every Turing machine T there exists a symmetric propagating Lindenmayer model L which simulates T. THEOREM 2. A Lindenmayer model is informationless if and only both unidirectional and externally symmetric.

if

it is

THEOREM 3. There are Turing machines which cannot be simulated by an externally symmetric unidirectional Lindenmayer model. THEOREM 4. For every unidirectional Lindenmayer model there-exists an internally symmetric unidirectional Lindenmayer model which simulates it. THEOREM 5. For every Turing machine T, there exists an internally symmetric unidirectional Lindenmayer model L which simulates T. These theorems show that polarity of the individual cells is not a necessary property for implementing arbitrary algorithms in a developmental model. This in itself is a strong indication that much can be achieved in development without the notion of polarity. In the next section, we show that a problem of regulation can also be solved without the use of polarity in individual cells. 4. The French flag problem

This problem was originated by WOLPERT (1968). A thorough discussion of it from an automaton-theoretical point of view has been given by ARBIB (1969, 1972). The statement of the problem is somewhat imprecise. This

POLAR ORGANISMS

671

may have been done on purpose; after all, it has been designed to reflect a general biological phenomenon and not to provide mathematicians with a pretty, but particular, problem. The imprecision of the statement of the problem provides us with a certain amount of freedom in how to translate it into the terminology of Lindenmayer models. We shall, therefore, start with a discussion of the reasons behind the formulation we are going to give. Roughly speaking, the problem is to create a mechanism by which a filament of initially identical cells turns into a French flag (one third red, one third white, one third blue) and restores this pattern in spite of large external disturbances, e.g., the breaking of the filament into two or more parts. First of all, we assume that initially we have an array of cells in identical states, which gets disturbed at one end. It is not an unusual assumption in embryonic morphogenesis (see, e.g., ROSEN, 1970) that at some stage the organism consists of an arrangement of identical cells in unstable equilibrium, and pattern development will be the result of a small disturbance (which in nature will inevitably arise sooner or later), which will push the organism out of this equilibrium and thus lead to pattern formation. The orientation (polarity) of the final pattern will depend on the position (and nature) of the initial disturbance. (An interesting discussion of the dependence of the outcome on the initial disturbance has been given by ROSEN (1971). WEBSTER (1971) presents overwhelming evidence to show ,that in hydra "cell division and growth processes do not playa major role in morphogenesis" .) In our formulation of the French flag problem, we shall require that the region which turns into red is at the end, where the disturbance occurs. We shall assume that after this initial disturbance the only other disturbances which interrupt the normal development of the filament come about as a result of the removal of parts of the filament. (How this assumption can be weakened is discussed in HERMAN, 1972). What happens if the filament is broken into two or more pieces? In that case, we would like each of the pieces (provided they are at least 3 cells long) to turn into a French flag. Furthermore, the polarity of each of the parts must be the same as that of the original, i.e., if the red region was to be formed on the left end of the original, then the red region of each of the parts must be formed at the left end of the part (see WEBSTER, 1971; WOLPERT, 1968). Clearly, such a requirement is impossible to achieve without further restrictions. If a region from the middle of a filament is removed before

672

o. T.

HERMAN

there was time for the disturbance to propagate from the end to the region in question, it is impossible for the cells in that region to differentiate depending on the end which was disturbed. There is, however, no need to go to the other extreme and require the pattern to be well established before it can regulate. In fact, we can show that in our model, if the length of the original filament is x, any part which is removed at least x units of time after the initial disturbance is guaranteed to form a French flag with the correct orientation, provided it is not cut again before the French flag is formed. If there is a second cut, then the parts resulting from the cut will also turn into French flags with the correct orientation, and so on. We must also introduce a mechanism which starts regulation in a removed subfilament. It seems reasonable to assume that when a break in the filament is caused, the two previously contiguous cells which get separated are both aware that something unusual has happened. (Both ARBIB, 1969, 1972, and WOLPERT, 1968, allow some special properties for the cells at the breaking point.) In our model, all we shall assume is that a cell which suddenly finds itself an end cell will go through the same kind of change of state, which the original end cell goes through as a result of the initial disturbance. In particular, in accordance with our notion of lack of polarity in the individual cells, the change of state will not depend on whether the break took place on the left or on the right of the cell in question. In view of the above comments, we claim that a symmetric propagating Lindenmayer model L with the following properties is a satisfactory solution to the French flag problem. L = (G,(g,O,O), b,F). (A) G = MxNxN, where M is a set symbols which in particular contains the symbols g, r, IV, b, sand i, and N = {a, 1,2,3, 4}. I.e., the states of L are ordered triples, the first element of which is a symbol, the second and third elements of which are numbers. If the first component of a state is r, we shall assume that a cell in that state appears to be red, similarly, for IV and white, and for b and blue. (B)

b«s, 0,0),
= b«g, 0, 0), (s, 0,0), (s, 0,0») = b(s, 0,0), (s, 0,0), (g,O,O») = b«g, 0, 0), (s, 0,0), (g, 0, 0») = (s, 0,0). This implies that any filament in which all cells are in state (s, 0, 0) will be in equilibrium.

673

POLAR ORGANISMS

(C) For any filament p = ... such that x ;?; 3, a 1 = i, ak = s, for 2 :::;;; k :::;;; x, there exists a to such that, for all t;?; to, ).t(p) = (r , 0, m l> ... (r , 0, m u> ... ... ,

where 13u-xl :::;;; 2, 13v-xl :::;;; 2 and 13w-xl :::;;; 2. This says that any filament, disturbed from the equilibrium state at one end, will turn into a French flag, with red at the disturbed end. (Since the model is symmetric, the property analogous to (C) with the right-hand end initially disturbed will be automatically satisfied.) In fact, to can be chosen to be 5x/3. Before describing the fourth property which L satisfies, we introduce the concept of a legitimately altered subfilament, to allow us a precise mathematical description of what happens when the filament is repeatedly cut. The definition is recursive. (I.e., legitimately altered subfilaments are defined using filaments about which we already know that they are legitimately altered subfilaments.) Let q =

>...

be any filament of L which is such that q = )./(p) for some filament p and nonnegative integer t, such that, either (0() p = ... is of length x and t > x, or (fJ) p is a legitimately altered subfilament and t ;?; 0. Then any filament of one of the following three forms is a legitimately altered subfilament.


C I,

dl> ...
C"

dr>,

3 :::;;; r :::;;; x,

... , 1:::;;; s < s+2 :::;;; r:::;;; x, ... , 1:::;;; s < s+2:::;;; x.

Assuming that a cut changes the first component of the state of the cell on either side of it to i, the intended meaning of the definition above is the following. A filament is a legitimately altered subfilament, if it is at least of length 3, and it has been obtained by a series of cuts from an originally disturbed filament, the only restriction being that the first cut can only be performed after a time equal to the length of the filament.

674

G. T. HERMAN

Using this terminology, we can now state the final condition which L must satisfy. (D) If p is a legitimately altered subfilament of length x, then there exists a to such that, for all t ~ to, )..f(p)

= (r,0,m t>(r,0,m2> ... (r,O,m u>(w,O,n t>(w,O,n 2> ... (w, 0, nv> cb, 0, II> (b, 0,12) .. , (b, 0, Iw>,

where 13u-xl ~ 2, 13v-xl ~ 2 and 13w-xl ~ 2. This says that any legitimately altered subfilament will turn into a French flag with the same orientation as the original filament would have done if no cutting took place. (Note again that the symmetry of the model, and of the definition of a legitimately altered subfilament, guarantees that a property analogous to (D), with the right-hand end initially disturbed, will be automatically satisfied.) HERMAN (1972) gives a symmetric propagating Lindenmayer model L, which has the properties (A)-(D). Thus, regulatory polar behavior can be achieved with apolar individual cells. For lack of space we cannot reproduce here the details of the solution. 5. Conclusion We have offered two pieces of evidence of the power of symmetric Lindenmayer models: universal computing ability and solvability of the French flag problem. Other evidence can also be given, e.g., the solvability of the firing squad synchronization problem (see HERMAN. 1972). This demonstrates that an argument for the polarity of individual cells on the basis of regulatory polar behavior of the organism as a whole is unacceptable. In a larger context, we have demonstrated the applicability of cellular automata to an honest-to-goodness biological problem. This is not an isolated instance. In HERMAN (1972), the relevance of the solution to the French flag problem given there is discussed, considering such biological notions as mosaic development, positional information, gradients, etc. We wish to conclude with the words of LONGUET-HIGGINS (1969), the truth of which is further supported by this paper. "We are beginning to realize that the interest of an organism lies, not in what it is made of, but in how it works". "The most fruitful way of thinking about biological problems is in terms of design, construction and function, which are the concrete

POLAR ORGANISMS

675

problems of the engineer and the abstract logical problems of the automata theorist and computer scientist". "If-as I believe-physics and chemistry are conceptually inadequate as a theoretical framework for biology, it is because they lack the concept of function, and hence of organization ... This conceptual deficiency of physics and chemistry is not, however, shared by the engineering sciences; perhaps, therefore, we should give the engineers, and in particular the computer scientists, more of a say in the formulation of Theoretical Biology". References ARBrB, M. A, 1969, Self-reproducing automata-e-some implications for theoretical biology, in: Towards a Theoretical Biology, vol. 2, Sketches, ed, C. H. Waddington (Aldine, Chicago), pp. 204-216 ARlnB, M. A, 1972, Automata theory in the context of theoretical embryology, in: Foundations of Mathematical Biology, vol. 2, ed, R. Rosen (Academic Press, New York, N.Y.), pp. 141-215 HERMAN, G. T., 1969, The computing ability of a mathematiial model for filamentous organisms, Journal of Theoretical Biology, vol. 25, pp. 421-435 HERMAN, G. T., 1970, The role of environment in developmental models, Journal of Theoretical Biology, vol. 29, pp. 329-341 HERMAN, G. T., 1971, Models for cellular interactions in development without polarity of individual cells, Part I: General description and the problem of universal computing ability, International Journal of System Sciences, vol. 2, pp. 271-289 HERMAN, G. T., 1.972, Models for cellular interactions in development without polarity of individual cells, Part II: Problems of synchronization and regulation, International Journal of System Sciences, vol. 3, pp. 149-175 LrNDENMAYER, A, 1968, Mathematical models for cellular interactions in development, Part I: Filaments with one-sided input, Part II: Simple and branching filaments with two-sided inputs, Journal of Theoretical Biology, vo!' 18, pp. 280-299; pp, 300-315 LoNGUET-HrGGINs, C., 1969, What biology is about, in: Towards a Theoretical Biology, vol. 2, Sketches, ed. C. H. Waddington (Aldine, Chicago), pp. 227-232 MINSKY, M. I., 1967, Computation: Finite and infinite machines (prentice Hall, Englewood Cliffs, N. J.) ROSEN, R., 1970, Dynamical system theory in biology, Vol. 1: Stability theory and its applications (Wiley-Interscience, New York) ROSEN, R., 1971, Some comments on the concepts of regulation and positional information in morphogenesis, International Journal of System Sciences, vol. 2, pp, 325-335 SINNOT, E. W., 1960, Plant Morphogenesis (McGraw-Hili, New York) VOcHrING, H., 1878, Uber Organbildung und Pfianzenreicb (Cohen, Bonn) WEBSTER, G., 1971, Morphogenesis and pattern formation in hydroids, Biological Reviews, vol. 46, pp. 1-46 WOLPERT, L., 1968, The French jfag problem: a contribution to the discussion on pattern development and regulation, in: Towards a Theoretical Biology, Part I: Prolegomena, ed. C. H. Waddington (Edinburgh University Press, Edinburgh), pp. 125-133