computer evidence concerning the chemotactic … · fig. 1. pulsatile aggregation that is largely...

J. Cell Sci. 25, 191-204 (1977) 191

Printed in Great Britain

COMPUTER EVIDENCE CONCERNING THE

CHEMOTACTIC SIGNAL IN DICTYOSTELIUM

DISCOIDEUM

HANNA PARNASDepartment of Developmental Biology, Hebrew University, Jerusalem, Israel

AND LEE A. SEGELDepartment of Applied Mathematics, The Weizmann Institute of Science,Rehovot, Israel, andDepartment of Mathematical Sciences, Rensselaer Polytechnic Institute,Troy,N.Y. 12181, U.S.A.

SUMMARY

Observed pulsatile aggregation of cellular slime mould amoebae is simulated on a computer.One spatial dimension is considered. In the simulation, attractant is rapidly secreted by the cells,after a delay period, when a superthreshold attractant concentration is sensed. Cells arerefractory to further signals after secretion. Once secreted, the attractant diffuses and ishydrolysed.

Movement results if a cell's extending pseudopods sense a supercritical increase of attractant;if increases are sensed on both sides, a sufficiently large difference can also initiate movement.The movement continues for a period independently of further signals, but then can be reversedby an attractant increase (at the back of the cell) that surpasses a high threshold. After 100s,motion stops and the threshold for movement reverts to normal.

With the above rules, and with parameter values taken, as far as possible, from the literature,the simulation provides the observed pattern of aggregation. Outward moving waves ofattractant and organized inward pulsatile ' steps' of cell movement surround a cell that secretesautonomously every few minutes. Other rules fail to give this picture, or give it only for arelatively narrow range of parameter values. It appears that of the various possible signals forchemotaxis, the most likely to be used by the amoebae is a temporal increase of attractant assensed by extending pseudopods. Nonetheless, we cannot rule out the' classical' hypothesis thatcells directly sense concentration differences.

INTRODUCTION

The cellular slime mould Dictyostelium discoideum can perform the aggregativemovements in its life cycle by means of a striking pattern of concentric waves that moveout from the centre of an aggregation territory. Each wave corresponds to a number ofcells that typically move markedly inwards for something less than 2 min, and thenmake no further significant progress for several more minutes. This patterned move-ment provides an easily observed response to coordinating signals, and thus has beenused by a number of investigators as a model for studying development. (For anintroduction to the extensive literature see, for example, the recent review of Gerisch& Malchow, 1976.)

13-2

192 H. Parnas and L. A. Segel

The basis for pattern in Dictyostelium cells lies in their secretion of cyclic AMP andtheir chemotactic response to this chemical. It appears that some of the cells secretecAMP autonomously every few minutes, while other relay cells signal only whensubjected to a suitable stimulus. Indeed, Gingle & Robertson (1976) conclude that' almost all cells are capable of relaying a signal before there are any autonomoussignals to relay'.

It is the way that chemical signals guide cell movement that will be the focusof attention here. Perhaps the most natural assumption is that the cells compare theattractant concentrations around them, and move toward relatively high concentrations.As will be discussed further below, some version of this suggestion has underlainseveral past discussions of the nature of the chemotactic signal. But Gerisch & Hess(1974) suggested that it is increases in cAMP concentration with time that stimulatechemotaxis. Their conclusions were based on observations of spikes of decreasedoptical density that appeared in cell suspensions of Dictyostelium. Pulses of cAMPmodified the spikes but continuous applications had little effect until cAMP flow rateswere so large that the spikes were extinguished.

Although the observations of Gerisch & Hess (1974) are consistent with the suggestionthat it is a positive temporal derivative of cAMP concentration that underlies thechemotactic signal, the matter cannot be considered closed. It is not certain whatresponse of the cells brings about the changed optical density. Furthermore, cells insuspension might exhibit rather different behaviour from that of cells on a substrateundergoing normal aggregative movements.

We have employed a computer simulation of aggregation to study possible types ofchemotactic response. Provided that various delays and refractory periods are incor-porated into the computer model, we show that response to a positive temporal deriva-tive of cAMP concentration can be the central part of a set of rules for individual cellbehaviour, which is in quantitative accord with present experimental knowledge andwhich provides aggregation patterns of the type observed. We also report observationson aggregative behaviour when it is assumed that individual cells compare cAMPconcentrations around their boundary in order to ' decide' on chemotactic movement.We find that cells with such a comparison ability can aggregate nicely, but that properbehaviour is more sensitive to the choice of parameters.

Our simulation incorporates parameter values from the literature, and the conclu-sions we draw are partially dependent on the accuracy of those values. The dependenceis only partial, because our results frequently will not be appreciably altered by achange in a factor of two or five, say, in a certain parameter value. This is because oftenonly ratios of parameters are of significance, and a second parameter - whose value isknown only roughly - can be altered to give the same ratio. Each new accuratedetermination of a parameter is likely to restrict the qualitative cellular behaviour thatis in accord with experiment. It is this ability to derive qualitative significance fromquantitative results that is perhaps the central feature of simulations such as ours.Also of interest are graphs of attractant waves and attractant variation at a particularlocation, for these can hardly be obtained in any other way than by computercomputation.

Simulation of amoeba aggregation 193

DESCRIPTION OF THE BASIC MODEL

We now describe our basic simulation. Most of its assumptions correspond togenerally accepted components of cellular behaviour. Its remaining features will bedefended and/or will be compared in their effect with other reasonable alternatives.

All cells are assumed to be located along the x-axis between x = o and x = L.The assumption of one-dimensionality, reasonable except near centres, permits theestablishment of qualitative conclusions without undue effort (Fig. 1).

o 0 0 0 0 0 o o

o 0 0 0 0 0 o o

o 0 0 0 0 0 o o

- * — 0 0 0 a — e 0 e ^ - x

Fig. 1. Pulsatile aggregation that is largely in the direction of the arrow can bemodelled by considering the cells (dots) along the .Y-axis. Concentric circles of cellsmoving toward a centre can be well modelled by a one-dimensional array, as here,provided the radii of the circles are large compared to a typical distance between cells.Our one-dimensional analysis makes the further approximation of replacing thevarying concentration along any line x = constant by its average value.

Attractant concentrations C are assigned on grid points a distance Ax apart. Theconcentrations on these points change over a time interval (t, t + At) according to theequation

C(x, t + At)- C(x, t) = D\f£Q [C(x + Ax, t) - zC(x, t) + C(x - Ax, t)] + Os + Qu.

(iff)

Here the first terms on the right side provide a discrete version of diffusion withdiffusion constant D. The Q terms account for the creation and destruction ofattractant.

The term Q$ provides the cell secretions, either the periodic secretions of auton-omous oscillators or the triggered secretions of relay cells. Typically one cell (whichusually becomes the aggregation centre) is programmed to signal autonomously, witha fixed period P s of a few minutes between signals. This cell is located in the middleof the line; when cells reach it they become 'adhesive' and their motion is stopped.

Other cells secrete only after receiving an appropriate signal and then passingthrough an assigned delay period Sr. The signal for this relay is a suitable super-threshold concentration Cr. For both autonomous and relay cells the total amount ofa single secretion is taken to be a constant^ and the secretion takes place over a periodTs. After secretion the relay cells are refractory for a time Tr to further stimulation.


We assume a refractory period of Tr = 7 min, so that cells relay every other auton-omous signal of period Ps = 5 min. A more refined treatment of refractoriness isdiscussed by Parnas & Segel (A computer simulation of aggregation in the cellularslime mould Dictyostelium discoideum (in preparation)).

Data on the amount of a given secretion are converted to concentration by assumingthat the secretion is instantly spread through a spatial region centred on the secretionpoint, having thickness 1 /*m (the assumed vertical extent of the fluid layer in whichthe amoebae move) and area (Ax)2. That is, with our value of Ax = 20 fim, a secretionof q molecules at a point is responsible for a local increase of concentration amountingto

q molecules q ..mole . ,.2 =

3 x 1 o (10)6 x io23 molecules/mole 400 (/«n)3 24 litre'

Michaelean hydrolysis of cAMP by enzyme (phosphodiesterase) bound to the cellsis modelled by the term Qu, where

(At)Vm,xC(x,t)n(x,t)Q^X' t] ~ C(x,t) + Km • ( 2 )

Here Vmax is the maximum rate of hydrolysis per cell and Km is the associatedMichaelis constant. The term M(X, t) in (2) denotes the number of cells that are' officially' located at point x at time t. Once a cell is officially located at some gridpoint x, its official location remains unchanged until its cumulative motion (kepttrack of by the computer) is sufficient to move it all the way to an adjacent grid point,either x — Ax or x + Ax. A source of error is introduced by this caricature of cell motionas well as by various other approximations such as the particular secretion rules that wehave adopted, the neglect of chemical diffusion into the agar on which the cells areusually placed, and the non-Michaelean nature of hydrolysis. Evidence indicating thatno serious flaws result thereby is discussed elsewhere (Parnas & Segel, in preparation).

We next specify rules for movement. In this we follow the suggestion of Gerisch,Hiilser, Malchow & Wick (19756) that the signal depends on BCjdt as sensed byextending pseudopods. Thus a cell in official position x at time t — At will send outpseudopods that at time t will sense the concentrations Cv~ and CP

+ at the left andright sides of the cell, where

Cp(x, /) = C(x, t) + vvAt[C(x-Ax, t)-C(x, f)]/Ax, (3 a)

C£(x, t) = C(x, t) + vPAt[C(x + Ax, t)-C(x, t)]/Ax. (36)

These formulae are based on the notion that if vP is the speed of pseudopod movement,then a pseudopod will extend a distance vP At in time At. Thus (according to a linearinterpolation), this pseudopod will sense a change in concentration proportional to thefraction vPAt/Ax of the concentration gradient over the appropriate interval oflength Ax.

Estimates of the time derivatives sensed, at time t, by the left and right sides of thecell are thus given by

Cp(x,t)-C(x,t-At) Cp-(x,t)-C(x,t-At)JiOCl ,

At At


Note that at the left side of the cell the estimate of the derivative can be written, using(3 a), as

Cy{x, t) - C(x, t - At) C(x, t) - C{x, t-At) C(x, t) - C(x - Ax, t)At = A~t +V+Vp

This shows that (for unsteady signals) there are 2 contributions to the estimate, thefirst coming from the local temporal derivative and the second from the local spatialderivative. The same comment holds for the estimate of dCjdt made at the right sideof the cell. Thus, operationally, the role assumed for the extending pseudopod is toprovide a signal that is a linear combination of the local spatial and temporal derivatives.

If 8C/dt is positive on just one side of the cell, then the cell is hypothesized to movetoward this side with speed vc, provided that dCjdt > 0D, dD a constant. Motion atthis speed in the direction of the higher value of dCjdt will also commence if dCjdt ispositive on both sides provided that the difference between the 2 values is greaterthan 0D.

After moving steadily for a time Tu (typically of the order of 10 s) the cell ispresumed again to acquire the ability to respond to outside signals. If dC/dt on its'back' side is positive and becomes greater than on its 'front' by a difference largerthan 0'1)y O'J:i >̂ d^,, then it changes its direction. If not, it continues to move in itsoriginal direction.

Once the cell has moved in one direction for T^ seconds the threshold for movementreverts to the relatively low value dD. With every reversal of movement, just as with thecommencement of a new movement, the sequence of events begins anew with anabsolute refractory period. One can regard our rules as setting a total refractoryperiod for movement lasting Tjt time units; for a time TM refractoriness is absolute(infinite threshold for new motion signals) and after that is relative (a higher thresholdthan normal). It is likely that during the relative refractory period the true thresholdcontinuously decreases to the basic value, rather than jumping after 7 ^ seconds, butsuch a refinement is generally not expected to have a significant effect on the results.

DISCUSSION OF THE SIMULATION AND ITS RESULTS

Table 1 provides a summary listing of all the parameters used in the present simula-tion, a typical value of each parameter, and - if such is available - the source for thevalue used. Employing these parameter values, we typically begin a simulation runwith 40 cells evenly distributed, one on each of the 40 grid points. 'Reflexion'boundary conditions are used, so that the observed pattern should be regarded as onesegment of a pattern that is continually repeated in space. Under these circumstancesthe simulation provides a representation of aggregation that is in accord with experi-ment. Waves of attractant are relayed out from the centre (Fig. 2). Their width andspeed conform with the observations of Alcantara & Monk (1974). In agreement withwhat these authors observed in the early stages of aggregation, every 10 min the wavesinduce the cells to take 100-s 'steps' toward the centre (Fig. 3, solid line). Significantaggregation takes place on the time scale of 1 h (Fig. 4). Moreover, the cells will movesteadily up a fixed attractant gradient.


We thus see that the various rules for attractant distribution and cell movement can

provide a 'correct' pattern of pulsatile aggregation. Our experience shows that it is

by no means easy to postulate a set of rules with this feature, so that we have one

piece of evidence that the temporal derivative of cAMP may be the signal for move-

Table 1. Parameters used, with typical values and literature references justifying the

choice of these values. In the absence of such references, a letter refers to a paragraph on

pages 196 to 199 of the text.

Asterisk indicates that preceding parentheses give a concentration, equivalent according toequation (16)

Notation Meaning

L

NAxAt

Ps

A

Ts

Cr

Sr

Tr

rM

T'

On

O'n

vc

VP

D^ max

Km

Length of interval in whichcells move

Number of cellsSpatial grid widthTemporal grid width

Periods of autonomous signalAmount of chemical per signal

pulse per amoeba (concentra-tion equivalent)

Secretion duration per pulseConcentration threshold forrelay

Relay delayRefractory period for relay

Duration of absolute movememrefractory period

Duration of relative move-ment refractory period

Primary 8C/dt threshold formovement signal

Secondary dC/dt threshold formovement signal

Speed of cell movementSpeed of pseudopod movement

Value

General

800 fim

40

20 /tmo-oi min

Signalling

5 minio7 molecules

(40 fiM)*

o-oi min3/tM

12 s7 min

Movement

t 12 s

100 s

0-002 /«vi/o-oi min

2 JUM/COI min

15 /tm/min75 /tm/min

Reference

—

a—

—

Gerisch & Hess (1974)Roos et al. (1975) and h

bc

Alcantara & Monk (1974)Durston (1974)

^ 12 s according toAlcantara & Monk (1974)

d

e

f

Alcantara & Monk (1974)g

Diffusion and hydrolysis of attractant (cAMP)

Attractant diffusion constantMaximum velocity ofhydrolysis

Michaelis constant forphosphodiesterase

2-4 x io~4 cm2/minio7 molecules/min/cell(40 /iM/min/cell)*0-5 JIM

Cohen & Robertson (1971a)h

Malchow & Gerisch (1974)

ment. To put this evidence into perspective, we shall now briefly discuss various

parameter choices that were not directly mandated by experimental observations.

(a) The number of cells was selected to give a cell density of 2-5 x io5/cm2 in

accord with typical experimental values. Parnas & Segel (in preparation) examine the


effects of using different numbers of cells, and of beginning with random distributionsof cells. These effects do not alter the conclusions of the present paper.

(b) We took the secretion duration Ts to be only o-oi min, so that the pulses aresecreted very rapidly. An alternative which is suggested by some of the availableevidence is that a secretion pulse lasts for a minute or two. This alternative is discussed

60 r

50

40

5 30

20

10

200 400

Distance, //m600 800

Fig. 2. Outward-moving waves of attractant. (Here and in the next 3 figures, the'standard' model is simulated, with parameters as in Table 1.) Cells signal in blocksof 3, essentially as observed by Alcantara & Monk (1974). Autonomous oscillator atcentre. Successive graphs depict the situation at the following times (in min). A, o-i;B, 0 3 ; C, 0 5 ; D, 0 7 ; E, 10 ; F, 1-5.

at length by Parnas & Segel (in preparation), who conclude that it is possible but lesslikely than short secretion.

(c) The threshold for relay Tr was chosen to give a critical density for relay ofabout 4X io4 cells/cm2 (Alcantara & Monk, 1974).

(d) A relative movement refractory period Tj, of 100 s was selected because it ledto the 100-s movement steps observed by Cohen & Robertson (1971 b) and Alcantara& Monk (1974).

(e) That the primary movement threshold 0D is low seems clear from such experi-


ments as those of Mato, Losada, Nanjundiah & Konijn (1975). The lower 0v is, theless the cells' need' short secretions to build sharp gradients. The particular value usedhere was selected so that chemotactic movement would be readily stimulated underthe conditions of our simulation.

Fig. 3. Solid line; the inward 'step' of a cell, according to the standard model.Dashed line: back and forth movement resulting from a model in which attractantconcentration C, not BCjdt, is taken as a signal for movement, and kinetic parametersare somewhat altered so that signal duration is prolonged by about 20 %.

20

10o

ttmtfm t tt tttftttttt200 400

t600 800 0

Distance, /<m

tt tt ft t t ft t t tt tt tt200 400

t600 800

Fig. 4. A, aggregation after 30 min. Initially, the cells were uniformly distributed, withan autonomous pulser at arrow. B, aggregation after 50 min.

(/) The secondary movement threshold O'-Q was taken to be sufficiently large so thatback and forth movement would be avoided. With lower values of 6'^, such movementcan be prevented with longer absolute refractory periods for movement.

(g) The speed of pseudopod movement was estimated from a knowledge of thespeed of cell movement. Some authors have observed filopods extending from thecells; these could doubtless move faster than pseudopods. At any rate, the effectivepseudopod speed can be raised or lowered without strongly affecting our results,provided that the movement thresholds are correspondingly increased or decreased.

(h) Malchow, Nagele, Schwarz & Gerisch (1972) report a value of Vmax of 1-2 x io8

molecules of cAMP hydrolysed by phosphodiesterase (PD) per cell per min. This wasthe peak value measured in cell suspensions contained in small glass tubes rotating at30 rev/min. Something less than this value is appropriate to our purpose because(i) not all PD will be fully exposed in cells on a Petri dish; (ii) we are interested in the


onset of aggregation, when maximum PD activity has not yet been attained; (iii) therotation will increase the apparent F m a x through stirring (a membrane-boundenzyme does not display the F m a x of the enzyme in solution, but rather an effectivevalue that is sensitive to the degree of mixing activity near the membrane); (iv) theexperiments were not run at the optimal conditions at which F m a x was measured.As a rough estimate, we assume that these factors lower F m a x to io7. Keeping in mindthat the time-scale of aggregation is minutes, we find this value of Fmax in satisfactorycorrespondence with our assumption of io7 molecules per pulse (taken from theestimate 'at least 6x io 6 ' for the number of molecules per pulse, found in Roos,Nanjundiah, Malchow & Gerisch (1975)). Note that, by (1 b), a pulse of io6 moleculescorresponds to a local concentration increase of 40 /tM.

We turn now to a reconsideration of the qualitative features of the basic simulationmodel.

A priori, cAMP concentration C itself, rather than its temporal derivative dCjdt, isperhaps the more 'natural' signal for movement. Thus 30 years ago Bonner (1947)already suggested that chemotactic Dictyostelium cells can somehow measure spatialgradients along their lengths. More recently, cell behaviour in the face of cAMPpulses has become an object of study. In this connexion, Cohen & Robertson (1971 b)postulated that the first side of the cell to receive a superthreshold concentrationtransforms into a pseudopod-bearing leading edge. In accordance with their observa-tions, they assumed that motion persists for some time (100 s) after cessation of thesignal. Polarity was regarded as lasting for many pulse periods. Cohen & Robertson'scalculations showed that the centrifugal attractant waves provided superthresholdconcentrations for only o-2 s, so that their postulated mechanism provided a way toobtain the observed 100-s inward steps.

A difficulty with the views of Cohen & Robertson (1971&) is that, as we haveremarked, it now appears that cells can lose their polarity between pulses and that theycan be stimulated to form a new leading edge within a few seconds. If C is to be thesignal, the latter observation, especially, seems to require the hypothesis that cells havethe ability to compare values of C and to move toward concentrations of sufficientlyhigh relative magnitude. Considerations favouring the existence of such a comparisonability were recently advanced by Mato et al. (1975).

What can simulations tell us concerning the possibility that movement is generatedby comparisons of attractant concentrations C? First of all, if the secretion rules areretained, then the waves of Fig. 2 correspond at a given cell location to the oscillatingconcentrations shown in Fig. 5. It thus seems that the cells cannot respond instan-taneously to the environment (whether C or dC/dt is measured), for such responsewould bring about a back and forth movement that is not observed. Some kind ofrefractoriness to a change in direction of movement must thus be introduced. Toprovide agreement with the observations mentioned above, absolute refractorinesscannot persist more than a few seconds, although relative refractoriness can last sometime.

We studied rules for movement involving concentration comparisons that againemploy the concentrations (equations 3 a, b) sensed by extending pseudopods. If

2 0 0 H. Parnas and L. A. Segel

a primary superthreshold concentration is sensed on just one side of the cell, the cellmoves towards this side. If both sides are above threshold, the cell will move - in thedirection of the higher concentration - only if the difference is also superthreshold.Rules for reconsideration, with a higher (secondary) threshold, and reversion afterTj|- seconds to the primary threshold are as before. The primary and secondarythresholds for the movement signal were 0-002 /MVI and 2 //.M respectively. Theremaining parameters were as in Table i.

z

60

50

40

30

20

10

1-0Time, min

2 0

Fig. 5. Variation of attractant concentration with time at a given cell location. Thefirst peak is the chemotactic signal that arrives from the left. The second peak comesfrom the cell itself and its nearest neighbour to the right. The next neighbour signalsalmost immediately. A somewhat diffuse peak follows as a result of signals from thenext block of 3 cells. (Here Cr = 1 /«vi - but C3 = 3 fiM gives almost the same graph.)

With the rules given in the previous paragraph, we found that a relative refractoryperiod of 100 s is associated with acceptable aggregation pattern in cells that compareconcentrations at their 2 ends. Here, however, there is noticeable sensitivity of theresults to our assumptions. As it is, the chemotactic signal is reduced to a very lowlevel in about 90 s, so there will be no effect when the movement threshold reverts toits original small value in 100 s. But this picture could be readily changed by moderate


alterations in the chemical kinetic parameters or by a more realistic characterization ofthe relative refractory period by a threshold that dropped continuously from high tolow values. Any set of rules that permits the cAMP level to remain above threshold atthe end of the 100-s refractory period leads to an outward step following the inwardone (Fig. 3, dashed line) and thus to poor aggregation. The back of the wave provokesbackward movement. No such problem arises with a dCjdt signal for chemotaxis,because of our hypothesis that only positive values of dCjdt are significant.

The relatively long persistence (just mentioned) of attractant above reasonablethresholds for C is in sharp contrast to the brevity of the signal found by Cohen &Robertson (1971 b). These authors obtained a lower limit for the chemotactic threshold'by estimating the period when the concentration exceeds the signalling (i.e. relay)threshold'. For this period they obtained the extremely short value of 0-2 s, fromwhich they concluded that the chemotactic signal itself is probably impulsive.

As shown in Fig. 5 we find the cAMP concentration to persist above the thresholdfor signalling for about 10 s, and this value would of course be greater if secretion werenot assumed to be so rapid. (As discussed in some detail by Parnas & Segel (inpreparation), this difference between our results and those of Cohen & Robertson(19716) can be attributed to the fact that we employ a more precise computation,using accurate parameter estimates that were unavailable at the time Cohen & Robertsonwrote their paper.) Furthermore, our calculations show that superthreshold conditionsfor chemotaxis can easily persist for longer than the 100-s step duration. It must bekept in mind that a cell's signal for relay comes from somewhat distant cells andthus is relatively attenuated compared to the local concentration that is importantfor chemotaxis, a concentration to which the cell's own secretion is the dominantcontribution.

Since the chemotactic signal does not in fact appear to be impulsive, we must againface the 'back of the wave' problem that was explicitly recognized years ago byShaffer (1957). Shaffer (1975) again emphasizes this problem in his study of cAMPrelay, stressing that one must now contend with Alcantara & Monk's (1974) findingthat the refractory period for movement is 12 s or less.

Returning to the main line of argument, we reiterate that we have found likely theassumption that the central part of the signal for chemotactic movement is a positivevalue of dC/dt. This conclusion is based on attempts to simulate the observed pulsatileaggregation patterns in Dictyostelium. But Dictyostelium cells will also move steadilyup a fixed spatial gradient. To adapt our assumptions accordingly we incorporatedthe proposal of Gerisch et al. (1975 £), that continual monitoring of attractant con-centration by extending pseudopods adds a multiple of the local spatial gradientinto the local temporal gradient.

It might be asked, if the signal for movement is dC/dt, why was the signal for relaytaken to be a superthreshold value of C itself? To answer this we note that becausethere is no equivalent to the back of the wave problem for the relay signal, we are infact unable to draw conclusions about it. Nevertheless, our inferences concerning thesignal for chemotaxis are unaffected, since our results would be essentially the samewhatever the relay signal.


Another major qualitative feature of our model whose validity must be examinedmore closely is the assumed nature of movement. Our rules amount to the assumptionof a short absolute refractory period, within which motion in a given direction isinevitably maintained, followed by a lengthier relative refractory period wherein asufficiently intense signal will cause a change in the direction of the motion. Motionstops at the end of the relative refractory period until a new superthreshold signal isreceived.

As we have mentioned, there is clear evidence gleaned from observing the effects ofartificial applications of cAMP, that cells neither respond instantaneously to environ-mental conditions, nor move unalterably in the direction of a first chemotactic trigger.The existence of an absolute refractory period for chemotaxis thus seems wellsupported, as does persistence of undisturbed motion for ioo s (to match the observedioo-s 'steps'). A relative refractory period for chemotaxis is required, for if thethreshold quickly returns to normal then the simulation yields considerable back andforth movement. The reason is that, as can be seen from Fig. 5, cAMP concentrationsfluctuate considerably for a time much, longer than the absolute refractory period.Graphs of cAMP concentration at adjacent cell locations (not shown) demonstrateeven more clearly that signals for backward movement will certainly be received afterthe absolute refractory period, unless a rather high threshold is maintained for someadditional time.

Aggregating cells adhere when they contact each other. This has been taken intoaccount here only in the rule that a cell which joins an autonomous oscillator does notleave it. Together with our assumption of one-dimensionality, this means that ourmodel does not produce accurate results in the region of an aggregate centre. Moreover,a one-dimensional model does not permit the study of streams or of percolationphenomena. It should be clear that these limitations do not affect our main conclusions.Also immaterial for present purposes are inaccuracies in the details of secretion,diffusion, and hydrolysis. Only when one proceeds to a subcellular model, for examplewith hypotheses about mechanisms for pseudopod extension, is it necessary to beprecise about the local distribution of attractant.

To mention a final qualitative feature of our model, we have regarded phospho-diesterase as membrane-bound, because of evidence (summarized by Gerisch &Malchow, 1976) that an inhibitor renders relatively unimportant the phospho-diesterase in the medium. In any case, simulations show no large differencesbetween the 2 possibilities. This is to be expected, as the cells remain fairly evenlydistributed.

CONCLUSIONS

The elaborate spatio-temporal pattern of aggregation that can be exhibited byDictyostelium provides an excellent opportunity to learn something of the details ofindividual cell behaviour. As we have indicated, and as is shown further by Parnas &Segel (in preparation), many a priori reasonable possibilities for individual cellbehaviour do not provide overall patterns of the correct type. Indeed, when account istaken of the fact that the cells also move steadily up an attractant gradient, our simula-


tion reinforces the suggestion (Gerisch et al. 19756) that the cells respond to a positivetemporal derivative of attractant concentration sensed by extending pseudopods orfilopods. The possibility remains, however, that chemotaxis is mediated by comparisonof attractant concentration itself, not its derivative.

Receptor response to positive binding rates is, not unprecedented. A well documentedexample occurs in bacterial chemotaxis. (Experimental findings on this matter arediscussed within a theoretical framework by Segel, 1977.) In addition, thermotacticresponse to temporal derivatives has been found in the protozoa Paramecium (Tawada,Miyamoto & Oosawa, 1972) and Chlamydomonas (Majima & Oosawa, 1975). There isevidence that cellular response to gradients rather than concentration is found amongplants (Sachs, 1974). And in Dictyostelium, other processes - connected with differen-tiation itself - respond to pulsatile signals of cAMP far better than to steady onesGerisch, Fromm, Huesgen & Wick, 1975 a; Darmon, Brachet, & Pereira da Silva,

1975)-Even if one grants the conclusion that positive values of dC/dt are the central com-

ponent of the chemotactic signal, it of course still remains to discern the mechanismby which this signal operates. Part of an investigation of this problem may usefullyinvolve computer simulations that take the overall picture discussed here for grantedand concentrate on subcellular detail.

This work was supported by a grant from the Albert Alberman Research Fellowship Fund(to H. P.) and by the National Science Foundation Grant No. MCS 76-07429 (to L. A. S.). Wethank G. Gerisch, M. Sussman and A. Goldbeter for helpful discussions.

REFERENCES

ALCANTARA, F. & MONK, M. (1974). Signal propagation in the cellular slime mould Dictyo-stelium discoideum. J. gen. Microbiol. 85, 321-324.

BONNER, J. (1947). Evidence for the formation of cell aggregates by chemotaxis in the develop-ment of the slime mould Dictyostelium discoideum. J. exp. Zool. 106, 1-26.

COHEN, M. & ROBERTSON, A. (1971a). Wave propagation in the early stages of aggregation ofcellular slime molds. J. theor. Biol. 31, 101-118.

COHEN, M. & ROBERTSON, A. (19716). Chemotaxis and the early stages of aggregation incellular slime molds. J. theor. Biol. 31, 119-130.

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{Received 6 August 1976 - Revised 8 December 1976)

computer evidence concerning the chemotactic … · fig. 1. pulsatile aggregation that is largely...

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