The precision with which single cells of Dictyostelium discoideum can locate a source of cyclic AMP

Chaos, Solitons & Fractals 50 (2013) 3–12

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

The precision with which single cells of Dictyostelium discoideum can locate a source of cyclic AMP Abha Chopra a, Vidyanand Nanjundiah b,⇑ a b

Institute for Immunology and Infectious Diseases (IIID), Murdoch University, Perth, WA 6150, Australia Department of Molecular Reproduction and Genetics, Indian Institute of Science, Bangalore 560 012, India

a r t i c l e

i n f o

Article history: Available online 15 February 2013

a b s t r a c t When stimulated by a point source of cyclic AMP, a starved amoeba of Dictyostelium discoideum responds by putting out a hollow balloon-like membrane extension followed by a pseudopod. The effect of the stimulus is to influence the position where either of these protrusions is made on the cell rather than to cause them to be made. Because the pseudopod forms perpendicular to the cell surface, its location is a measure of the precision with which the cell can locate the cAMP source. Cells beyond 1 h of starvation respond non-randomly with a precision that improves steadily thereafter. A cell that is starved for 1–2 h can locate the source accurately 43% of the time; and if starved for 6–7 h, 87% of the time. The response always has a high scatter; population-level heterogeneity reflects stochasticity in single cell behaviour. From the angular distribution of the response its maximum information content is estimated to be 2–3 bits. In summary, we quantitatively demonstrate the stochastic nature of the directional response and the increase in its accuracy over time. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The ability to sense the direction of external chemical stimuli must be common to all microorganisms if not all cells. It is used by polymorphonuclear leucocytes to detect the site of tissue inflammation [55], yeast cells to respond to mating type pheromone [36] and, in the case most deeply investigated, by cellular slime mould amoebae to aggregate by chemotaxis to an attractant released by themselves [29,3,14]. A decisive step in the study of directional sensing in the cellular slime mould Dictyostelium discoideum was the identification of cyclic AMP (cAMP) as the aggregation pheromone [30]. Since then acrasins – Shaffer’s [43] evocative name for the chemo-attractant – have been identified in many other cellular slime mould species [45,12,49]. The chemotactic response of D. discoideum cells to cAMP improves with the duration of starvation [43,6,40], the

⇑ Corresponding author. E-mail addresses: [email protected] (A. Chopra), [email protected]. ernet.in (V. Nanjundiah). 0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.01.007

reason being the progressive differentiation of a sensory and motor mechanism [47]. There is a substantial literature on the molecular details of sensing, signal transduction and movement that lie behind the chemotaxis of D. discoideum to cAMP. The binding of extracellular cAMP to cell surface receptors initiates a complex series of responses; cytosolic components of the cAMP-sensing apparatus are rapidly mobilised at the site of stimulation [41] and a combination of local and global events leads to directed cell movement (reviewed in [25,27,47]. Behavioural observations have tried to correlate spatiotemporal features of an externally supplied cAMP signal with directed cell movements, mostly on cells that are very sensitive to the stimulus on account of being starved for some hours (e.g. [46,16,17,51,8,9,50,52]). Generally what has been monitored is the ability of cAMP to attract individual cells over long distances or to elicit a population-level response. Mato et al. [33] measured chemotaxis in small populations of confined cells to cAMP provided at different distances and concluded that the spatial gradient of cAMP across the cell was the relevant signal. Temporal gradients have also been implicated [16,51,50]. It has been suggested

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that an advancing pseudopod could act as a scanner: a spatial gradient across the cell could be inferred after integrating the inputs from temporal gradients sensed by a number of pseudopods around the cell [19,15]. Whatever the actual inputs, the upshot is that an amoeba can home in on an external source of cAMP and move towards it. However, we lack quantitative information regarding an important aspect of the accuracy with which an amoeba can sense a source of cAMP, namely how it builds up over time. The present work attempts to fill this gap. We focus on the first visible sign of the direction in which an isolated cell will move, not bulk cell movement, and interpret it as the cell’s guess of the location of the signal. The cells used range in stage of development from freshly starved to just before aggregation. In other words we do not restrict ourselves to cells that are aggregation-competent – and therefore highly polarised and possibly in phases of an oscillatory cycle that vary in sensitivity [18]. Finally, the cells belong to the wild-type, not one of the commonly used axenic derivatives that, being mutants, may differ in aspects of behaviour from the wild-type (for example, based on their comparison of chemoattractant cAMP wave propagation in the multicellular stage, Dormann and Weijer [13] conjecture that there may be ‘‘inherent, as yet uncharacterised differences’’ between wild type and axenic strains). Specifically, we address three questions. (a) How accurately can an amoeba sense the direction of a nearby localised source of cAMP? (b) Does the accuracy change with developmental age, meaning time following starvation? (c) Does the response of the population mirror the response of the individual cell?

2. Methods Cells of D. discoideum NC4 (a gift from Th.M. Konijn) were handled using standard protocols that included regular sub-cloning [39]. Exponentially growing amoebae were

washed free of bacterial nutrient (E. coli) from 2% SM-Difco agar plates, suspended in ice-cold sodium–potassium phosphate buffer at pH 6.1 and spread evenly at a density of 5.105/cm2 on 5 cm ‘control’ Petri dishes containing 3 ml of agar made up in the same buffer. After allowing a few minutes for cells to settle, excess buffer was decanted. Plates were incubated in the laboratory at 22 °C, this being taken as the starting time T0 (Tn stands for n hrs after T0). At various times thereafter a few cells were scraped off a control plate with a wire loop and, after brief agitation in ice-cold buffer, transferred to an ‘experimental’ plate which was similar to the control plate except that it had a much thinner layer of agar (about 1 mm as against 1.6 mm). Buffer was occasionally added to an experimental plate that appeared to be at risk of drying. The cell density on an experimental plate was 103/cm2 and typically 1–2 cells were visible a field of view. Visible aggregation could be seen on control plates at T7–T8. In contrast, aggregation never took place on experimental plates. Amoebae up to age T6–T7 were monitored individually on experimental plates for their responses to externally provided cAMP. Glass capillaries were pulled to a taper of 5° and inner diameter of slightly less than 1 lm at the tip and filled with cAMP (made up in phosphate buffer) at the required concentration or with buffer alone (‘sham stimulus’). A capillary was moved to a distance of about 5 lm from a cell on an experimental plate with the help of a Zeiss micromanipulator and gently lowered so as to just touch the agar. This provides a point source of cAMP and high spatial and (initially) temporal gradients at the near surface of the cell, both ideal conditions for a cell to sense direction (for a schematic sketch see Fig. 1). Observations were made with a Zeiss Ergaval microscope using a 32 objective and 16 eyepieces. Distances and directions were estimated with the help of a grid attached to the eyepiece. The visual field was centred on the cell under observation and divided into eight equal sectors of 45°. The stimulus originated

Fig. 1. Schematic of the experimental arrangement.

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from the mid-point of any sector chosen arbitrarily (i.e., with no reference to the shape of the cell at the time). However, the observations are reported as if the stimulus always originated from due East in a conventional sense; correspondingly the response is taken to lie at the middle of the appropriate sector. The response index was defined as cosh, where h is the angle between the directions of stimulus and response (doing so implicitly clubs responses to the right and the left of the source). Cells were not especially elongated during our observations. Apart from transient membrane protrusions, the shape of a cell was circular to oval. If oval, a long axis and a short axis perpendicular to it were apparent. In such cases we took care to make sure that the stimulus was oriented randomly with respect to the intrinsic axes; the direction of pseudopod extension was uncorrelated with them. Membrane extension and pseudopod extension were the two responses monitored. We recorded the time after the capillary tip touched the agar at which the responses were seen and their direction relative to the location of the capillary tip. For the direction of either response to be reproducible, i.e., distinguishable from the response to a sham stimulus, the capillary had to be left in place for 15 s at least; responses did not improve significantly with longer stimuli. After the responses were noted, or after waiting 1 min if there was no visible alteration in cell morphology, which happened rarely, the capillary was lifted and moved close to a second, distant, cell. A capillary was used for no more than 15 min before being replaced by a newly filled one. Usually a cell was tested just once, and when the same cell was subjected to repeated stimuli we waited for de-adaptation to take place. Data were analysed using standard statistics [24] or statistical procedures appropriate for orientation data (see Appendix in [42]). Statistical analysis of circular data and their graphical representation was made possible thanks to an evaluation version of the Oriana software kindly provided by Kovach Computing Services, Wales.

3. Results 3.1. Cell behaviour in the absence of external stimulation In addition to the more familiar filopods, lamellipods and pseudopods, D. discoideum amoebae spontaneously extend local arc-like glassy protrusions that lack organelles, named blebs [31,54]. We too find that amoebae at all stages display continuous activity at the cell surface. Every 30 s or so a cell puts out a hollow-looking, symmetrical, balloon-like protrusion that we call a membrane extension (because the kinetics of what we see and bleb formation are not identical we prefer the neutral term membrane extension). A membrane extension can originate from any point. It either retracts very soon, within a few seconds, or, rarely, circulates like a wave around the cell’s periphery until it gets resorbed elsewhere. In the latter case the balloon becomes flattened and elongated in the same direction that it circulates. Approximately 10 s after the initiation of a membrane extension there is a slower

and narrower outgrowth of the surface that soon fills up with cytoplasm and can be recognised as a typical pseudopod. The mean frequency of pseudopod formation is 2/ min, much higher than the 0.58 ± 0.39/min reported by Varnum-Finney et al. [50] for aggregation-competent A3 cells. Often, but not always, the pseudopod arises at the same location on the cell as the membrane extension. As far as can be judged both emerge perpendicular to the cell surface [47]. 3.2. Response to stimulus 3.2.1. Preliminary observations Working with T5–T6 cells, 106 M cAMP in the capillary was just sufficient to elicit a directional response that was discernibly different from the response to a sham stimulus. The variability in the response decreased as the cAMP concentration was raised and remained at the same level until it reached 103 M. At that point the response was occasionally poorer than with 3.106 M cAMP. This may have been because the response pathway was getting saturated or because with 103 M cAMP at the source, the (relative) spatial gradient across the cell, i.e., the difference in concentrations divided by the mean, was too shallow to be detected. We settled on 105 M as the standard cAMP concentration in the capillary. Only the first membrane and pseudopod extensions following the application of the stimulus were monitored, not later events such as pseudopod splitting or translocation of the whole cell [48]. 3.2.2. Detailed observations When a capillary containing cAMP is brought into contact with the agar near a cell, two early responses are visible. First, initiated within 20 s, there is a single membrane extension similar to that seen in unstimulated cells. Rarely, up to three or four extensions erupt simultaneously on a cell. Usually this happens when the cells are freshly starved and exhibit the most vigorous membrane activity from among all developmental stages. Relatively advanced cells (aged beyond T6) tend to be elongated and in some cases functionally polarised: a cell can respond when stimulated from one direction but not another. At times, especially when a cell presents a broad front to the stimulus, multiple membrane extensions are observed from the longer side. The second response is

Table 1 Times of initiation of membrane and pseudopod extension at different hours of development. Each set of values (mean ± s.d.) is based on at least 44 observations; the distribution of pseudopod responses is shown in Fig. 2. The times measured correspond to the interval between touching the capillary to the agar and the onset of the first visible response of each type. Developmental stage (h)

Membrane extension (s)

Pseudopodl extension (s)

T0–T1 T1–T2 T2–T3 T3–T4 T4–T5 T5–T6 T6–T7

16.93 ± 8.08 12.09 ± 5.49 17.04 ± 9.56 16.63 ± 8.10 22.56 ± 13.25 19.23 ± 11.67 20.25 ± 8.75

35.72 ± 12.59 26.07 ± 7.37 32.00 ± 13.67 30.07 ± 10.02 39.00 ± 19.58 36.12 ± 17.05 34.50 ± 13.67

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T0-T1 cells (n=45) 90º

T1-T2 cells (n=44) 90º

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T2-T3 cells (n=45) 90º

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T3-T4 cells (n=45) 90º

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T4-T5 cells (n=46) 90º

T5-T6 cells (n=45) 90º

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T6-T7 cells (n=45) 90º

T5-T6 cells: sham stimulus (n=45) 90º

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Fig. 2. The angular distribution of pseudopod extensions to 105 M cAMP diffusing from a capillary tip that touches the agar 5 lm from a cell and from the 0° direction at the indicated number of hours after starvation. The area of a rectangle stands for the proportion of cells of that age cohort that responded within the corresponding 45° sector. The long radial line indicates the mean direction and the outer arc delimits the 95% confidence interval about the mean. The standard deviations calculated from circular statistics are 137.7° (T0–T1), 78.5° (T1–T2), 54.0° (T2–T3), 61.6° (T3–T4), 46.1° (T4–T5), 35.5° (T5–T6), 28.3° (T6–T7) and 178.8° (T5–T6 cells, sham stimulus).

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extension to be a meaningful indicator of the direction of chemotaxis. Exceptionally, a pseudopod is retracted and a second one is extruded immediately in a fresh direction; when that happens, the cell moves in the new direction. These rare exceptions are seen either in control experiments with sham stimuli (i.e., when the capillary is filled with buffer) or when the cAMP concentration in the capillary is 106 M or lower. They are not included in the results presented. Intermittently we also examined the responses of aggregating cells and cells dissociated from aggregation streams or centres. Such cells tend to be strongly polarised with functional anterior and posterior ends (not shown). When stimulated close to the posterior, the cell forms a pseudopod at the anterior and performs a U-turn; polarity can be reversed by prolonged stimulation. Both behaviours are as Bonner [4] and Swanson and Taylor [46] reported. Fig. 3. The mean response index cosh for the experiments shown in Fig. 1 (ordinate) plotted as a function of the time of starvation in hr (abscissa). Mean times are shown, so that 0.5 stands for T0–T1 and so on. For standard deviations see Table 2. The smooth curve is a least-squared best fit to the negative exponential y = a⁄ (1.0exp (-bx)) flanked by 95% confidence intervals drawn with the help of free software (http:// zunzun.com//; a = 0.99, b = 0.29). The response index is defined as cosh where h is the angle made by a pseudopod with the direction of the cAMP stimulus. cosh would be 1 for a perfectly accurate response and 0 ± 0.76 for a uniformly distributed random response. When compared to a random response, all indices are statistically significant except for the one at T0–T1 (t-test for means, p > .025 for T1–T2, p < .001 for the others; n = 45 in each case).

the extension of a pseudopod and it is initiated about 30 s after the stimulus is applied. Both times show substantial variations about the respective means (Table 1). As with unstimulated cells, the pseudopod is almost always put out from same location as the initial membrane extension. If the capillary is withdrawn quickly (within 5 s) sometimes one can see a pseudopod extended without prior membrane extension. Futrelle et al. [17] observed, and other workers have confirmed, that cells contract immediately after cAMP stimulation. Under our conditions, as in the study by Varnum-Finney et al. [50], this ‘cringe’ response’ could be seen only rarely. The reasons are unclear. If the capillary is left in place it becomes obvious that the pseudopod extension presages a commitment on the part of the entire cell to move in the same direction. For this reason we consider the direction of pseudopod

3.3. Development of precision Neither the membrane nor the pseudopod extensions necessarily form at the site on the cell nearest to the capillary. The frequency with which a pseudopod is formed in one or the other direction (relative to that of the stimulus) is an indicator of the quality of the response; the intensity of the response appears to be invariant. When cells belonging to the same age cohort are stimulated, each of them just once, the direction of pseudopod extension varies from cell to cell, and the pattern of variation changes with the duration of starvation. This can be described in terms of the distribution, the response index or the proportion of responses within the correct sector. The circular histograms in Fig. 2 indicate both the spread in the directional response at various hours and the manner in which accuracy builds up after varying lengths of starvation. The Rayleigh test for circular data [42] indicates that except for cells at T0–T1 and the controls with a sham stimulus, all distributions are significantly different from that corresponding to an undirected response (mean vector r > 0.39, n > 40; p < 0.01). Responses become tightly bunched about the mean as development progresses, but there is a fair degree of scatter. The large standard deviations are due to a relatively small number of extreme events (Fig. 2, Table 1). Calculation of the mean response index, hcoshi, shows that the probability of an accurate

Table 2 The directional accuracy of pseudopod extension and its information content as a function of developmental age. The calculations are based on the data shown in Fig. 2 and controls with sham stimuli (n = 60 cases) with buffer in the capillary. As explained in the text, the information content, calculated is a measure of the gain in information relative to a completely random response. For a uniformly distributed random response, the numbers in columns 2–4 would be 0.125, 103.9° (s.d.) and 0 ± 0.71. The responses of T0–T1 cells and of T5–T6 cells to the sham stimulus (control, last line, which refers to T5–T6 cells) are statistically indistinguishable from it. Developmental stage (h)

Fraction of responses in forward sector

Response direction (mean ± s.d.)

Response index (cosh) (mean ± s.d.)

Information content (bits)

T0–T1 T1–T2 T2–T3 T3–T4 T4–T5 T5–T6 T6–T7 Control

0.22 0.43 0.60 0.67 0.74 0.70 0.87 0.23

2.00 ± 103.72° +7.16 ± 86.23° 3.00 ± 60.98° 6.00 ± 75.30° +1.96 ± 59.97° +14.00 ± 39.35° +5.00 ± 38.64° (s.d.) 109.9°

.01 ± .69 .38 ± .72 .63 ± .58 .55 ± .75 .71 ± .60 .78 ± .38 .86 ± .41 0 ± .77

0.30 0.65 0.98 1.23 1.56 1.66 2.20 0.13

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Cell 1 (n=10)

Cell 2 (n=10)

Cell 3 (n=11)

Cumulative (n=59)

Cell 4 (n=10)

Cell 5 (n=9)

Cell 6 (n=9)

Fig. 4. Trial-to-trial variability in the directional response of the same cell; the representation is as in Fig. 2 (also see Table 3). Cells differ widely but the cumulative distribution (centre) shows a highly accurate response at the population level (Rayleigh test, mean vector r > 0.3, n = 59, p < 0.01).

Table 3 Variation in the response of a cell between repeated trials. The cells in this series were all at stage T5–T6 and were stimulated by 105 M cAMP in the capillary. Six cells were tested independently every 5 min; see Fig. 4 for the distributions. The response index cosh is defined in Fig. 2 and the text. Cell No.

No. of trials

Response index cosh (mean ± s.d.)

1 2 3 4 5 6

10 10 11 10 9 9

0.36 ± 0.80 0.84 ± 0.32 0.77 ± 0.51 0.46 ± 0.74 0.33 ± 0.79 0.89 ± 0.33

response increases steadily until just before aggregation commences, at which time it becomes close to 1 (Fig. 3). For T0–T1 cells the value of hcoshi is statistically indistinguishable from that for a sham stimulus or a uniformly distributed response (t-test for means, p > 0.75 in both cases); it is significantly different at all later times (p < 0.025 for the T1–T2 data, p < 0.001 for the later times). The proportion of responses in the correct direction increases monotonically with developmental age (Table 2), but the mean response index decreases slightly as cells age from T2–T3 to T3–T4. It is not obvious what the underlying reason

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might be or what the decrease might imply, but a parallel increase in variance between the two sets of observations suggests that the decrease is real (Table 2). 3.4. Variability in the population and within a cell As we have seen, it appears that cells belonging to an age cohort do not respond in the same way. Is this because the population is heterogeneous or can the same cell exhibit time-dependent variations in the response? One has the impression that there is a small degree of persistent cell to cell variability, but it is insufficient to account for heterogeneity at the population level. Fig. 4 shows the example of six spatially well separated T5–T6 cells that were tested in sequence. An interval of 5 min separated successive stimulations of the same cell. This limited the number of trials that could be made with a cell that was notionally of the ‘same’ age. The trial to trial variation in response of cells that belonged to the same age cohort varied widely. Rayleigh’s test for circular data shows that cells No. 1, 4 and 5 display a response that is no better than random (mean vector r < 0.5, n = 9 or 10; p > 0.05) whereas Nos. 2, 3 and 6 respond with a high degree of accuracy (mean vector r > 0.65, n = 9, 10 or 11; p < 0.01). Still, the extent of variation from trial to trial (Table 3) is such that the single cell response and the population response are not significantly different (one-way ANOVA; between groups, mean squares = .61 with 5 d.f.; within groups, mean squares = .38 with 53 d.f.; F = 1.62, p = 0.17). Again we see that a cell can be accurate most of the time and yet do poorly ‘on average’, all on account of a few large-angle deviations from the correct direction (Fig. 4 and Table 3). 4. Discussion For the responses to be reliable (i.e., distinguishable from those to a sham stimulus) the cAMP stimulus had to last for a minimum of 15 s, as Futrelle et al. [17] found while studying chemotaxis in a population. Sometimes the membrane and pseudopod extensions occur separately, not as a pair, implying that the two are independent but correlated events. Also, they occur regularly whether or not a stimulus is presented to the cell. A possible interpretation is that the dynamics of membrane and pseudopod extensions is independent of the presence of an external stimulus, and that the stimulus only biases the probability that they occur nearest to it, i.e., are directed towards it. Meinhardt [35] proposed a stochastic model for direction-sensing by cells based on local self-activation and two antagonistic reactions, one acting rapidly and globally, the other slowly and locally. The model showed that in a cell that formed membrane extensions autonomously at random locations, an external stimulus could bias the probability that the extension occurred at the site closest to the stimulus. Formally, our findings are in accord with with Meinhardt’s and other models that explicitly consider stochasticity [1,11,32,41,48]. D. discoideum cells are spaced about 50 lm apart from each other at the threshold density required for aggregation [28]. A cell that receives a cAMP signal locates a source

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of angular size 10 lm/50 lm or nearly 12°. In turn the local source moves towards some other cell that has stimulated it. Thus the direction from which the signal appears to come may not always be the best direction for a cell to aim at. It makes sense for a responding cell to incorporate some degree of variation while sending out a pseudopod, in other words to orient its movement within a range about a mean direction. A cell which adopts this strategy should exhibit a decreasing variance in its orientation response as time progresses and the location of the source becomes less uncertain. This is in accord with our observations except for the puzzling anomaly between T2–T3 and T3–T4 (Table 2). We have monitored only the first step of a chemotactic response, not taxis per se. Further, we cannot comment on whether the direction of membrane and pseudopod extensions can be ascribed to the concentration of cAMP, its spatial or temporal gradients or a combination of all three. Still, since the set-up we use is essentially of Futrelle et al. [17], we can estimate cAMP profiles in the same way and compare our numbers with older estimates. cAMP begins to diffuse on and within the agar as soon as the capillary tip touches the surface. With a capillary tip diameter of 1 lm, taper of 5° and diffusion coefficient of 0.97  105 cm2/s [10], the cAMP level at a cell’s near edge, 5 lm from the capillary tip, rises to a peak in about (5 lm)2/(4  0.97  105 cm2/s) or 4 ms. It falls very slowly, over a time scale 1 h or more. This means that for all practical purposes the cAMP level at a given distance goes through a very short and rapid rising phase after which it is constant in time as long as the tip is in contact with the agar. Spatially, the concentration decreases inversely with distance from the tip. If the capillary contains 105 M cAMP and the contour of the flattened cell on agar is a circle of diameter 10 lm, we can set lower limits on the relevant cAMP concentrations: 0.4 nM (at the peak) at the near edge of the cell; a difference of 0.27 nM between the front and back of the cell; and 0.18 nM when averaged over the volume of the cell. At steady-state the absolute spatial gradient of cAMP across the cell is 0.013 nM/lm and the relative gradient is about 72%. Bonner [3] (and Savage; 1947) estimated that a relative acrasin gradient of 3% could be sensed. Mato et al. [33] give 4.3 nM as the peak concentration averaged over the cell, 0.0036 nM/lm as the minimum spatial gradient required for a positive chemotactic response, and 0.9% as the relative gradient (all pertaining to a population of cells that show a threshold response, i.e., respond 50% of the time). Van Haastert [48] reports reliable chemotaxis at a mean concentration of 650 nM, an absolute spatial gradient of 0.5 nM/lm and relative gradient of 0.7%. In our study the mean temporal gradient in concentration experienced by the near side of the cell during the rising phase of stimulus is 0.4 nM in 4 ms or 100 nM/s. Varnum-Finney et al. [50] found reliable pseudopod extensions in response to a temporal gradient of 5.73 nM/min (our estimate from their data). The conclusion is that both the relative spatial gradient of cAMP across the cell and the temporal gradient used by us are rather high in comparison to those used in earlier studies. In contrast the absolute concentration and spatial gradient are much smaller. It may be that the cell can make

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use of a range of inputs for chemotaxis, and use the strength of one to compensate for the weakness of another. The circular plots (Fig. 2) show that even a cell that is starved for an hour or less can mount a vigorous directional response to a cAMP stimulus; but that response is not reliable, as Futrelle et al. [17] pointed out. Within a tolerance of 22.5° either way, a cell which has just been deprived of food can detect the location of a cAMP source with a probability of 22%. Statistically, this is no better than the 12.5% success rate of a randomly directed pseudopod (Table 2). A statistically meaningful directional response first appears between 1 and 2 h of development. Over time it increases in reliability; so does chemotaxis, and the increase appears to be gradual rather than abrupt [40]. The probability of correct detection increases by roughly fourfold to 87% after 6.5 h of starvation. The improvement of precision in response could be related to dynamic anisotropy in a cell as embodied in a shifting distribution of cAMP receptors on the cell surface, a varying spatial arrangement of cytoskeletal elements responsible for motility (reviewed in [47]), or something else. Anisotropy, or inherent asymmetry, must not only be behind the differences from cell to cell at any one time [35,41] but also the differences within a cell from ‘moment to moment’. The high temporal variability shown by single cells is noteworthy (Fig. 4). As we have seen; this makes single cells and populations comparable in terms of both the mean response and the scatter (Table 3). Samadani et al. [41] worked with latrunculin-treated A3 cells that had been starved for 5 h, all the while being stimulated periodically by cAMP, and came to a rather different conclusion. They found significant differences between cells with respect to the translocation of molecules of CRAC-GFP internally towards a topical cAMP stimulus; local CRAC accumulation was used as a surrogate for the accumulation of PIP3, which indicates the direction of movement. We can think of different explanations for why the two sets of findings do not concur. (i) Samadani et al. used A3, a mutant of D. discoideum selected for axenic growth, not the wild type (which requires bacterial food). (ii) They looked at a very early response to cAMP (intracellular PIP3 accumulation in 3.5 s) as against the much slower initiation (30 s) of actual movement under natural conditions that we monitored. (iii) The cells they studied were in effect isotropic because the actin cytoskeleton had been disrupted by latrunculin. We worked under natural environmental conditions, under which temporal variations in directional sensing can and do occur on account of the translocation of membrane patches. The variations could have masked average cell-to-cell differences which would be apparent in cells that had been rendered isotropic by latrunculin. (iv) Finally, the recurrent cAMP pulses that Samadani et al. applied over 5 h prior to testing for chemotaxis means that the post-starvation development of the cells had been accelerated [20,53]. It is possible that in terms of developmental stage the cells used by them were much older than the oldest cells used by us (T6–T7). It is known that in parallel with and following aggregation a functional distinction arises in D. discoideum between presumptive stalk and spore cells [5,26]. The distinction is reflected in traits that include chemotaxis [34] and

movement speed [22]; it may have played a role in their study. The directional response of a cell can be analysed in terms of its information content (see [2] for an information theory-based mathematical model of chemotaxis in D. discoideum). Given 8 equiangular sectors within which a cell can extend a pseudopod, the a priori probability that it will choose a given sector, 1/8, is the same for all sectors; our uncertainty with regard to what it will do is maximal. Once a pseudopod forms the direction is known and the uncertainty vanishes. The amount of decrease in uncertainty is a measure of the information content of the response [38]. If p(i) is the a priori probability of pseudopod extension into sector i, the Shannon entropy or information content in bits is defined as I = Rp(i) log2p(i) where the sum extends over all sectors. In the case just considered, initially p(i) = 1/8 for every sector and the answer is 3 bits. In the presence of an external cAMP source the response will value not be the same in all directions; p(i) will vary from one sector to another. Once the cell has responded to the stimulus, the directional information conveyed by it will be less than 3 bits by an amount that depends on the precision of the response. The difference in the two values is a measure of the additional information about the direction of the cAMP source that can be gained by an observer monitoring the amoeba. For example, suppose the cell’s response is invariably in the correct direction. Then p(i) = 1 for that sector and =0 for all other sectors and Rp(i) log2p(i) = 0. In this case the information gained by an observer, the difference in information content between a completely uncertain response and a completely certain response, is the theoretical maximum of 3 bits. What about actual cellular behaviour? Table 3 summarises the data. With very young cells the response is more or less undirected and the gain in information content is 0.3, a statistically insignificant value relative to zero. The gain in information content rises steadily with the duration of starvation. The maximum is reached when cells are starved for 6–7 h. T6–T7 cells display a reasonably sharply directed response with a concomitant gain in information content of 2.2 bits. In other words the maximum information regarding the location of the cAMP source that can be conveyed by means of extending a pseudopod by a preaggregation amoeba is about 2.2 bits. Obviously the directional information contained in the cAMP signal itself must be higher than this. The angular diameter of the capillary tip at the nearest point on the cell surface is about 1 lm/ 5lm or 12°; this sets a lower limit to the angular resolution of the source. As pointed out earlier, 12° is also the angle subtended by a 10 lm diameter amoeba at a distance of 50 lm. (Successive bisections of a 45° angle by eye showed that we could estimate the location of the source to 10°, not all that different a figure.) Let us assume, then, that the direction of the source is uncertain to within 12°. Further, as a measure of the uncertainty in response, let us take the standard deviation in the direction of pseudopod extension. For a T6–T7 amoeba this is 38.64° (Table 2). On comparing the two numbers, the directional information content of the signal exceeds that of the response by log2(38.6/12) or about 1.69 bits. A more accurate estimation of the direction of pseudopod extension would

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improve these figures slightly but not much. For example, suppose we had divided the field into 16 equiangular sectors instead of 8 and T6–T7 cells continued to extend a pseudopod within the correct sector 86.7% of the time (as they do now) and the rest of the distribution also remained as it is. Then the maximum information content in the pseudopodal response would go up by one bit from 2.2 to 3.2. Haldane and Spurway [23] estimated the directional information content in the waggle dance of a honeybee worker to be about 5 bits; a worker who used the dance in her search for food made use of about 50% of the information contained in the dance. Preston [37] found that the information contained in mutualistic signals between gobies and shrimps was about 2.3 bits per act, of which about 0.3 bits were transmitted usefully to the recipient. Gherardi and Pieraccini [21] studied 20 behaviours during agonistic interactions in crayfish and, after averaging over all pairs in which they were associated, found that most conveyed about 2 bits of information. In a seminal paper Shannon showed that the information content of English text, as judged by the ability of readers to guess, ranged from 0.6 to 1.3 bits per character [44]. Admittedly these are unrelated instances of information transfer in vastly different situations involving biological communication. It is interesting that the information content of the orientation of pseudopod extension in D. discoideum – an example of ‘behaviour without nerves or muscles’ [7] – compares favourably with them.

Acknowledgements The experiments were carried out about 20 years ago when the authors were in the Molecular Biology Unit, Tata Institute of Fundamental Research, Bombay, and the results formed part of an internal report. Thanks are due to Prof. O. Siddiqi for encouragement and suggestions made at the time. We are grateful to the anonymous referees who drew our attention to shortcomings in an earlier version of the manuscript.

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