Christiane Hilgardt
email: christiane.hilgardt@physik.uni-magdeburg.de

Jitka Cejkova

Marcus J B Hauser

Hana Sevcíkova

Jitka Čejková

Hana Ševcíková

Streamless aggregation of Dictyostelium in the presence of isopropylidenadenosin

Keywords: signal transduction, adenosine derivative, chemical inhibition, ACA activity, excitability, abnormal chemotaxis

Starving cells of Dictyostelium discoideum undergo a developmental cycle were cAMP is autocatalytically produced and relayed from cell to cell, resulting in the propagation of excitation waves over a spatially extended population. Later on the homogeneous cell layer transforms into a pattern of cell streams directed perpendicular to the cAMP waves. Here we chemically influence aggregation competent cells by isopropylidenadenosin (IPA), an adenosine derivative. It can be assumed, that IPA acts via specific adenosine binding sites localized in the cellular membrane. We find, however, that pattern formation and cellular aggregation under the influence of IPA differ considerably compared to experiments with adenosine. In particular, our observations point towards an inhibitory effect on adenylate cyclase (ACA), the key enzyme in the autocatalytic production process of cAMP inside the cell. Our results suggest the existence of a direct coupling (via intracellular affection) or indirect coupling (via inhibition of cAMP binding) of the specific adenosine receptors to the regulatory circuit that controls cyclic intra-and extracellular cAMP concentration.

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Streamless aggregation of Dictyostelium in the presence of isopropylidenadenosin

Christiane Hilgardt, Jitka Čejková, Marcus J.B. Hauser, Hana Ševcíková corresponding author: Christiane Hilgardt postal address: Otto-von-Guericke-University Magdeburg, Institute of Experimental Physics, Biophysics Group, Universitätsplatz 2, 39106 Magdeburg, Germany fax: ++49 (0)391 67 11181 telephone: ++49 (0)391 67 18334 e-mail address: christiane.hilgardt@physik.uni-magdeburg.de number of pages: 20 number of tables: 0 illustrations: 8

Introduction

Amoeboid cells of Dictyostelium discoideum live in soil as autonomous single cells. Upon nutrient deprivation they undergo a complex developmental cycle which includes the transition from uni-to multicellular organization through a succession of morphological states. Aggregation is the first process to occur during the morphogenetic development of a starving population into fruiting bodies. The aggregation is mediated by the relay of adenosine-3',5'-cyclic monophosphate (cAMP) that acts as a positive chemoattractant ensuring the coordinated aggregation of the cells [1; 2]. At the onset of the developmental cycle, individual cells synthesize cAMP autonomously and secret it into their environment. At neighbouring cells, the signal is detected via highly specific surface

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receptors [3; 4; 5], inducing various signal transduction cascades within the cells. Through the activation of membrane-bound adenylate cyclase (ACA) one of them leads to autocatalytic synthesis of cAMP and its secretion into the cellular environment [6; 7]. Binding of extracellular cAMP to the membrane receptors also triggers the production of cGMP that is involved in the chemotactic cell movement [8; 9; 10; 11; 12]. The excitation process is followed by a refractory period, where receptor phosphorylation and inactivation of ACA is involved, before the cells become deadapted and excitable again. During the refractory period the extracellular cAMP concentration is decreased by the external and membrane-bound phosphodiesterase (PDE), which is regulated by the phosphodiesterase inhibitor (PDI) [13; 14; 15]. These processes are manifested in the generation of aggregation centres, which periodically emit outwardly moving concentration waves of cAMP [16; 17; 6]. The waves propagate as concentric waves (target waves) or spiral waves due to wave interaction. The cells move chemotactically against the emitted waves, organize in characteristic streaming patterns and, at later stages of aggregation, accumulate in the aggregation centers. After passing through a mound and a motile slug stages, the aggregates transform into fully differentiated fruiting bodies, containing germinable spores.

In the present paper we report on the effects exerted by isopropylidenadenosin (IPA) in the course of pattern formation and cell aggregation in D. discoideum. IPA belongs to the group of adenosine derivatives that have been shown to affect binding of cAMP to the cAMP-specific surface receptors [START_REF] Van Lookeren Campagne | Specificity of adenosine inhibition of cAMP-induced responses in Dictyostelium resembles that of the P site of higher organisms[END_REF]. However, we have found that the course of aggregation in the presence of IPA differs considerably from that observed in the presence of adenosine [START_REF] Newell | Inhibition by adenosine of aggregation centre initiation and cyclic AMP binding in Dictyostelium[END_REF]. Although the cells on IPA agar relay the cAMP signal, and optical density waves can be observed in dark-field as in the case of adenosine [START_REF] Newell | Inhibition by adenosine of aggregation centre initiation and cyclic AMP binding in Dictyostelium[END_REF], the cells do never form any characteristic streaming patterns due to a concerted cell movement. Instead, the waves vanish after a certain time leaving the cell layer in a homogeneous state. Later, many aggregation centres emerge simultaneously all over the petri dish. These centres attract cells from their closest neighbourhood, however, again without visible stream formation. The aggregates then undergo morphogenesis leading to the formation of normally proportioned fruiting bodies.

Experimental

Growth conditions and initiation of starvation

Cells of the D. discoideum strain AX2 were cultured from frozen spores in axenic HL5-medium at 21 • C [START_REF] Sussmann | Cultivation and synchronous morphogenesis of Dictvostelium under controlled experimental conditions[END_REF] and harvested in the second half of the exponential growth phase. To remove all nutrients from the environment and thus to initiate starvation the cells were washed twice in phosphate buffer (16 mM KH 2 PO 4 , Na 2 HPO 4 , pH 6.14) and pelletized by low speed centrifugation. After re-suspension in phosphate buffer the cells were spread homogeneously onto agar plates (0.5 % Difco Bacto-Agar in phosphate buffer, 2 mM caffeine) at a density of 6.2 × 10 5 cells/cm 2 . IPA was dissolved in the buffer used to prepare the agar plates. After removal of the supernatant, the plates were incubated at 21 • C in darkness for 4 to 5 h until the first excitation waves appeared.

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Chemotactic assays

Single drops (0.5 µl) of nutrient-free cell suspension (2 × 10 7 cells/ml) were placed at a distance of a few millimetres around droplets (0.5 µl) containing cAMP (2.5 × 10 -8 M) on agar plates with either 0.25 mM IPA or IPA-free control plates. After a few hours the displacement of cells was evaluated.

Similarly, the chemotactic response to IPA gradients was examined.

Yield of spores

Washed cells were allowed to complete their developmental cycle on agar plates (2% Bacto-agar in phosphate buffer) with 0, 0.25, 0.5 and 1 mM IPA. The total number of developed spores out of a known number of vital cells was determined by microscopic observation.

Visualization techniques

The propagation of cAMP waves can be observed indirectly in dark-field [16; 2]), where optical density waves are caused by changes in cell shape and coherent cell movement depending on the excitation state of the cells. The optics of our dark-field set-up follows that of [START_REF] Gross | Signal emission and signal propagation during early aggregation in Dictyostelium discoideum[END_REF]. Images were taken at intervals of 3 s with a CCD camera (Hamamatsu C3077), digitized (DT-Open Layers DT 3155 Mach Series Frame Grabber), and stored as 8-bit images on hard disc. The behaviour of single cells within a population was observed with an inverse microscope (Zeiss Axiovert 200) equipped with a 10× PlanNeoFluar objective and a CCD camera (Hamamatsu C2400-07). Images were sampled at 1 min intervals.

Data analysis

To evaluate the temporal development of the wave parameters, space-time plots were constructed from the sequences of grabbed images according to the method described by [START_REF] Siegert | Digital image processing of optical density wave propagation in Dictyostelium discoideum and analysis of the effects of caffeine and ammonia[END_REF]. Changes in wave shape were visualized by extracting the grey values of one point in space along the time axis of the space-time plots. These profiles were normalized, and smoothed by moving average. Global trends were removed by polynomial regression. The standard deviation of all normalized data in a time series was taken as a measure of the wave amplitude. Power spectra were constructed by Fast Fourier Transformation of time series at single points in space.

Results

The effect of IPA on pattern formation was first studied in dark-field. Figure 1 shows snapshots obtained in the presence of IPA and under control conditions, respectively. Under both conditions excitation waves can be observed about 5 h after the initiation of starvation. However, the early waves in the presence of IPA can hardly be seen in dark-field, and their visualization requires further image processing in order to recover the appearance of the first low-amplitude waves originating at different random positions (Fig. 1(a)). During the following 2 h these patterns develop into stable,

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well visible spiral waves (Fig. 1(b)-(c)). One of the most remarkable differences to the experiment under control conditions is the formation of huge spirals, whose diameters may attain a few centimetres (Fig. 1(d)). Indeed, in the IPA concentration range between 0.125 to 2 mM spirals with diameters greater than 4 cm can be observed. Their extension is typically limited by the borders of the petri dish. By contrast, the diameter of a spiral domain in control experiments is typically a few millimetres (Fig. 1(a)).

While the control cells have already reached the state of late aggregation with tight cell-cell contacts and stream formation, the cells on IPA-containing agar still form a homogeneous layer (Fig. 1(e)). A continuous change in homogeneity caused by directed cell movement towards the spiral centres leading to stream formation is not observed even after 10.5 h of starvation (Fig. 1(f)). This is also reflected in the faint territory borders where colliding cAMP-waves annihilate. Although the waves propagate as undisturbed, smooth fronts for at least 2 h the spirals suddenly and simultaneously disappear throughout the petri dish. Within each territory, this happens from inside to outside, i.e. spiral rotation around a core suddenly stops and the inner open end of the spiral propagates to the border of the petri dish where it is annihilated (Fig. 1(e)). During the following transition phase, which can last for up to 2 h, the cell layer seems to be inactive. However, image processing reveals the occurrence of concentric waves with large wave lengths at seemingly random positions in the yet homogeneous cell layer (Fig. 1(f)). Suddenly, many small oscillatory regions emerge, emitting short-range target waves that attract the cells in their closest neighbourhood. These waves cause a break-up of the cell layer into many miniature territories (Fig. 1(g)-(h)). Again, no visible stream-like structures evolve. Later, these clusters transform into mounds (Fig. 1(h)). After approx. 30 h the developmental cycle is complete. Many small, but normally proportioned fruiting bodies develop.

Occasionally it can be observed that after mound formation individual mounds connect by tube-like, more or less unbranched structures (Fig. 2). Within these "streams" adjacent mounds are absorbed by the attracting mounds. Further analysis shows that these mounds elicit high-frequency target waves inducing cell migration over distances up to 3 mm. In contrast to the observations in the wild type strain DH1 [22; 5] these mounds do not have ring-like forms, although the inverted representation of these structures in Figure 2 lead to that optical impression. Furthermore, these mounds do not break or change their position.

To extract the basic parameters of wave propagation, space-time plots were constructed. In general, wave profiles can give information about the coherence of a cell population in their chemotactic response. Figure 4(a) was constructed from Figure 3(a), which was obtained under control conditions. These waves show regular oscillations with a tendency to biphasic behaviour, as seen from the presence of a small secondary peak within one period. The corresponding power spectra consist of two harmonic frequencies with a periodicity of 5 and 2.5 min, respectively. The shape of individual waves during advanced spiral propagation in the presence of 0. In control experiments periods of rotating spiral waves stay roughly constant at values ∼ 5 min for the first 20 rotations, before the period begins to shorten. In this case, up to 39 rotations could be analyzed before the waves become invisible in dark-field due to tight cell-cell contacts. In contrast, the propagation velocity of waves on IPA-containing agar -represented by the slopes of each wave band in the space-time plots (Fig. 3)do not differ significantly from those obtained under control conditions, neither in magnitudes nor in their temporal evolution (Fig. 5(b)). For any condition the velocity of successive waves decreases continuously from ∼ 0.4 mm/min to 0.25 mm/min. Figure 6 shows the propagation velocity of subsequent wave fronts as a function of the rotation period, i.e. the time required for a full rotation of the spiral wave. Under control conditions the dispersion relation follows the general course known from other excitable media, i.e. the velocity decreases with decreasing period [START_REF] Siegert | Digital image processing of optical density wave propagation in Dictyostelium discoideum and analysis of the effects of caffeine and ammonia[END_REF]. In the presence of IPA the form of the dispersion relation is

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ACCEPTED MANUSCRIPT more complex. For the first few waves the curve follows the common trend (their propagation velocities decrease with the period) even though this occurs in a range of larger periods and with less steep course. In a transitory domain, the period remains more or less constant while the velocity still decreases. Before the waves vanish the period increases once more while the propagation velocity decreases, resulting in a "backward" turn of the dispersion curve.

To investigate both the capability of starving cells to sense a cAMP gradient and the chemotactic response in the presence of IPA, chemotactic assays were prepared. It was found, that cells incubated on IPA-containing agar are basically able to sense cAMP and move up a gradient. In contrast, IPA exerts no chemotactic attraction on the cells.

To specify the effect of IPA, the cells were allowed to starve under standard conditions until spiral waves were fully developed. Then a fine mist of IPA in phosphate buffer (2 mM) was sprayed homogeneously onto the cell layer. Experiments were conducted with varying IPA concentrations.

No immediate effect of IPA on spiral wave patterns was observed. The appearance of spiral waves was not disturbed and the sizes of the areas occupied by individual spirals did not change. Although growth of territory borders was observed, in most cases no streams were formed. As in the case of cells incubated on agar containing IPA, the spiral waves disappeared after some time, the primordial borders became washed out, and corresponding territories disintegrated into subsystems of many small cell clusters. Phase contrast microscopic investigations of the behaviour of cells within a population confirms the absence of streaming patterns and the lack of obvious morphological correlation between the first spiral waves and the resulting distribution of fruiting bodies. Figure 7 shows a time series of experimental results in the presence of 1 mM IPA. Image recording was started immediately after cell preparation. After 5 to 6 h a continuous increase in local cell density can be observed (Fig. 7(b)). Parallel investigations in dark-field reveal that this developmental state approximates the phase of early spiral propagation. Even when the cells start to assemble, neither local clustering nor stream formation can be observed. Furthermore, cell prolongation, i.e. indication of cellular polarization processes in response to cAMP waves, sets in after 8.9 h. 11 h after initiation of starvation, the cells are attracted by areas of particularly high cell density (Fig. 7(c)), upper left corner and darker areas in Figure 7(d), indicated by arrows). Periodic cell movement towards these centres as well as cell prolongation can now clearly be observed. This period can be associated to the state of aggregation corresponding to Figure 7(g). However, individual cells seem to respond to more than one high density centre and change their direction frequently. In Figure 7(e) a distinct aggregation centre has been established in the upper part of the image, which at a later stage forms a mound (Fig. 7(f)). Elsewhere smaller centres have dissolved. The mound reshapes many times and never forms a compact cell aggregate (Fig. 7(g)-(j)). Several thin, long fingers protrude as multiple tips. Fusion and reforming of the fingers repeats for a long time. Thicker fingers appear and separate from the aggregate (Fig. 7(i)-(k)). These motile slugs disrupt frequently into smaller slugs, which sometimes regress into mounds before converting into slugs again (Fig. 7(l)-(n)). Approximately 30 h after the initiation of starvation normally proportioned fruiting bodies were formed from slugs typically without any previous culmination phase (Fig. 7(o)). Surprisingly, the yield of spores from

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fruiting bodies developed on agar containing IPA is significantly about 2.5 times higher than under standard conditions (Fig. 9).

Discussion

Excitability of D. discoideum cells during the developmental cycle is mainly determined by their sensitivity to cAMP, depending on the amount and affinity of receptors, and by the level of cAMP output, based on ACA activity [START_REF] Sawai | An autoregulatory circuit for long-range self-organization in Dictyostelium cell populations[END_REF]. Thus, it was shown, that an overall decrease in excitability by changing the affinity to extracellular cAMP [START_REF] Dormann | cAMP receptor affinity controls wave dynamics, geometry and morphogenesis in Dictyostelium[END_REF] (e.g. via modification of the receptor composition) or by external chemical influence [24; 21; 25] (e.g. via the addition of caffeine, resulting in an inhibition of the cAMP-dependent activation of the ACA and in altered chemotactic adaptation) can lead to a wave length increase and, at the same time to a dramatic enlargement of the aggregation territories and an inhibition of centre initiation.

In the present paper we examined the influence of IPA, an adenosine derivative, on cAMP pattern formation and on the developmental cycle. Adenosine is a physiological degradation product of cAMP and has been shown to inhibit several cAMP-depending processes by inhibition of cAMP binding [START_REF] Newell | Inhibition by adenosine of aggregation centre initiation and cyclic AMP binding in Dictyostelium[END_REF] and postaggregative differentiation [START_REF] Van Lookeren Campagne | Specificity of adenosine inhibition of cAMP-induced responses in Dictyostelium resembles that of the P site of higher organisms[END_REF]. Adenosine reduces oscillatory cAMP activity [26; 19; 27] and plays a role in cell type regulation at later states of the developmental cycle [28; 29]. In spite of the structural similarity, adenosine does not compete with cAMP for the cAMP surface receptors [26; 19]. Instead, adenosine acts non-competitively and indirectly though β-receptors as one of the cellular adenosine binding sides [26; 30; 18]. Alterations in the ribose moiety of adenosine in most cases increases the inhibitory effect on cAMP binding [START_REF] Van Lookeren Campagne | Specificity of adenosine inhibition of cAMP-induced responses in Dictyostelium resembles that of the P site of higher organisms[END_REF]. It has been shown that IPA as an adenosine derivative decreases the cAMP binding to the cAMP surface receptors dramatically [START_REF] Van Lookeren Campagne | Specificity of adenosine inhibition of cAMP-induced responses in Dictyostelium resembles that of the P site of higher organisms[END_REF]. Here wo show, that similarly to the effect of adenosine, IPA has an inhibitory action on the initiation of aggregation centre formation, leading to oversized spiral wave domains (Fig. 1). In addition, the appearance of coherent spiral patterns is delayed, though IPA does not prevent signal release. Although one can assume that the effect of IPA is mediated by binding to the adenosine specific surface receptors, our results on the effect of IPA on developmental processes differ clearly from the results obtained from experiments with adenosine [START_REF] Newell | Inhibition by adenosine of aggregation centre initiation and cyclic AMP binding in Dictyostelium[END_REF]. Moreover, our observations point towards a strong inhibitory effect of IPA on ACA, the key enzyme of cAMP production, which couples to the regulatory circuit that controls cyclic increases of intra-and extracellular cAMP [31; 23; 32]. In general, the inhibition of ACA results in a decrease in excitability of the system influencing in turn the extension and distribution of aggregation territories.

In the presence of IPA no visible stream-like structures evolve during the whole developmental cycle (Fig. 1, 3, 7). Streamless aggregation in D. discoideum has also been observed in null mutants of acaA which encodes ACA [START_REF] Kriebel | Adenylyl cyclase localization regulates streaming during chemotaxis[END_REF]. It has been shown that acaA -cells have polarity defects and fail to aggregate in stream-like structures. Stream formation results from a head to tail arrangement of the cells, which is regulated by the asymmetric ACA localization at the uropod of the cells [START_REF] Kriebel | Adenylyl cyclase localization regulates streaming during chemotaxis[END_REF]. ACA-deficient cells are able to polarize and move up a cAMP gradient when stimulated, but are unable to form streams. In the presence of IPA we have observed a long-lasting homogeneity of the

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cell layer, which later transforms into streamless accumulation of cells at the aggregation centres. Microscopical observations confirm that this impression results from directed cell movement proceeding without head to tail arrangement, which results in a local cell density increase around the attracting aggregation centres (Fig. 7). Moreover, the ACA independent polarization process of the cells, induced by external cAMP, and required for the chemotactic cell movement, is delayed (Fig. 7). In [START_REF] Stepanovic | Intracellular role of adenylyl cyclase in regulation of lateral pseudopod formation during Dictyostelium chemotaxis[END_REF] it has been shown that acaA -cells are able to orient in spatial and temporal cAMP-gradients, but exhibit chemotactic defects. In particular, these cells show a reduced supression of lateral pseudopod formation and cell turning, resulting in highly inefficient chemotaxis and randomly appearing effective direction.

A close correlation exists between the frequency of spiral rotation and the formation of stable signaling centres in D. discoideum [START_REF] Dormann | cAMP receptor affinity controls wave dynamics, geometry and morphogenesis in Dictyostelium[END_REF]. Mutants in cAMP receptors that show low frequency of wave initiation and wave length, respectively, have difficulties in forming stable aggregation territories, because stable centre formation seems to require an equilibrium between the period of cAMP waves, chemotactic response and cell dispersion due to random cell movement [START_REF] Dormann | cAMP receptor affinity controls wave dynamics, geometry and morphogenesis in Dictyostelium[END_REF]. In the case of IPA the potential role of random cell movement is suggested by the delayed and faintly developing territory borders, which reveal undirected cell movement. It appears that IPA forces the cells into a parameter regime, where the effect of random cell movement between successive waves does no longer allow the formation of stable aggregation centres.

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  Figure 3(a) and (b) correspond to wave patterns observed within one experiment, where patterns in (a) evolved on control agar and (b) on agar containing 0.25 mM IPA.The evolution of wave propagation within the two cell layers can be followed directly along the time axis, as well as the respective temporal changes in cell density (i.e. in the homogeneity of the medium). While in Figure3(a) chemotactic cell aggregation progresses continuously, yielding distinct, separated aggregation territories, in 3(b) wave initiation suddenly ceases. The homogeneity of the cell layer does not show any pronounced changes with time caused by directed cell movement towards the centres. Merely the core of the spiral shown in Figure 3(b) increases during the last three propagating waves. The extension of the spiral domain exceeds the area of the spatial axis, A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT whereas under control conditions several individual spiral centres evolve at the same spatial scale. Figure 3(c) was constructed from sections through the right images of Figure 1. It covers a time interval of more than 9 h summarizing the different developmental states in the presence of IPA (early patterns with huge spirals (I), disappearance of spiral waves and transition phase (II), secondary patterns with small clusters emitting target waves (III)). Typical wave profiles obtained from grey values at given fixed positions in space plotted as function of time are shown in Figure 4. High values correspond to the bright bands of the original wave pattern and are caused by prolonged cell shapes due to the excitation process and chemotactic movement.

  [START_REF] Siegert | Analysis of optical density wave propagation and cell movement in the cellular slime mould Dictyostelium discoideum[END_REF] mM IPA (corresponding to the time-frame shown in Figure1 (c)-(d), right-hand side) is shown in Figure4(b). These waves possess a plateau-like form at high grey values indicating the pronounced cellular response to passing waves. Power spectra reveal several frequencies with a periodicity between 1.4 to 6.0 min. The amplitudes of the waves were not affected by the presence of IPA (0.125 -2 mM) in the agar. Secondary target waves emitted after mound formation (Figure1(g)-(h), right-hand side) lead to very regular oscillations for approx. 1.5 h with a periodicity of about 7.5 min (Fig.4(c)). The change in the rotation period of successive waves under control conditions and in the presence of IPA is shown in Figure5(a). The values for the experiments with IPA have been obtained from fully developed spiral waves occurring in early wave patterns (developmental state in Figure1(c)-(e), right-hand side). The change in period of successive waves does not show any significant concentration dependency. Even 0.125 mM IPA causes initially long periods. With each propagating wave the period shortens and eventually reaches values around 6.5 min. After a phase of stagnation the period increases again to values close to the initial values before the waves die out (according to Figure1(e), right-hand side). Approximately 25 rotations can be observed on IPA-containing agar before wave cessation in a still homogeneous cell layer.

Figure 1

 1 Figure 1Effect of IPA on the course of aggregation as observed in dark-field. The left half of each image corresponds to control conditions, the right half to 0.25 mM IPA in agar. Both patterns appeared inside the same petri dish divided into two chambers. If indicated by an asterisk the visibility of wave patterns was enhanced by image substraction of two successive images. Times corresponds to the time after the initiation of starvation.

Figure 2

 2 Figure 2 Fusion of mounds observed on agar with 0.25 mM IPA (reversed representation). Mounds connect by thick cell "streams". The development from (a) to (d) required 8.5 hours.

Figure 3

 3 Figure 3 Space-time plots corresponding to control conditions (a) and 0.25 mM IPA in agar (b) and (c). The patterns of (a) and (b) were observed within one petri dish, divided into two chambers. Arrows indicate the spiral core (1) and the open wave ends when wave initiation ceases (2). (c) originates from the right-hand side of the images in Figure 1. The positions of the sub-images (c)-(h) of Figure 3 on the time axis are indicated. Particular developmental steps in the presence of IPA are indicated in (c): early spiral waves (I), disappearance of spiral waves and transition phase (II), secondary wave patterning period (III).

Figure 4

 4 Figure 4 Profiles of dark-field waves under control conditions (a), and with 0.25 mM IPA in agar (b) and (c). Early wave propagation is represented by (b), whereas the waves corresponding to (c) appeared 7.8 h later during mound formation.

Figure 5

 5 Figure 5 Rotation period (a) and propagation velocity (b) in the presence of IPA. Between 3 and 8 spiral waves were analyzed for each IPA concentration. Arithmetic mean values were formed and slightly smoothed by moving averaging over time.

Figure 6

 6 Figure 6 Dispersion relation based on the data of Figure 5. The time course is indicated by arrows.

Figure 7

 7 Figure 7 Developmental cycle in the presence of 1 mM IPA observed by phase contrast microscopy over a time interval of 30 h. The exact times correspond to the duration after starvation has been initiated. (a) homogeneous cell layer before cell migration, (b)-(e) cell movement without stream formation, (f)-(j) mound state, development of multiple tips, (k)-(m) slug state, disruption into smaller

Figure 8

 8 Figure 8Yield of spores in depencene of the IPA concentration in agar with error bars as standard deviation. The initial cell density was 6.2 × 10 5 cells/cm 2 . The values correspond to the percentage of the initial number of cells which transformed into spores.

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Acknowledgements C.H. is indebted to C.J. Weijer for the information about IPA. C.H. was supported by the Deutsche Forschungsgemeinschaft (grant MU 884/18-2). J.C. and H.S. acknowledge the financial support of Ministry of Education of the Czech Republic (Research project MSM 6046137306) and of the Grant Agency of the Czech Republic (grants 204/04/1421 and 104/03/H141).


M. S A Caduff

Y Talary

A Feldman

Tura
email: tura@isib.cnr.it

PhD Andrea Tura

Reply to Comment on experimental methods and conclusions of Impedance spectroscopy of solutions at physiological glucose concentrations by

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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The comment to the manuscript [START_REF] Tura | Impedance spectroscopy of solutions at physiological glucose concentrations[END_REF] Second criticism to our study is that the measurement probe was not adequate for electro-impedance spectroscopy (EIS), because it was designed for conductivity measurements. It is right that the probe is usually sold coupled with a conductivity meter, but this is not enough to conclude that the probe cannot be used for EIS. Caduff et al. just claim that the probe and its electrodes are not adequate, but they fail to explain why. Moreover, it must be noted again that our major findings were related to differences in the impedance modulus at frequency values where the phase was small, or even almost zero. This is particularly verified in blood and sodium chloride samples (Fig. 3 and4), but also for water (Fig. 2) in a still relatively wide frequency interval. Clearly, when the phase was zero the impedance reduced to conductance, and hence the probe was certainly used properly at those frequency values.

The third criticism is that our impedance measurements may be unreliable because of some electrode polarization phenomena. First, we performed four-electrode measurements: Caduff et al.

should be aware that this technique strongly reduces the problem of electrode polarization [START_REF] Mirtaheri | Electrode polarization impedance in weak NaCl aqueous solutions[END_REF][START_REF] Schwan | Four-Electrode Null Techniques for Impedance Measurement with High Resolution[END_REF].

Furthermore, it is well known that the electrode polarization is more relevant at low frequency values. In the study [START_REF] Gabriel | The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz[END_REF], the electrode polarization phenomena in the analysis of some biological

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tissues were considered relevant below 100 Hz and still present up to 1 kHz, but negligible at higher frequencies. In other studies electrode polarization was observed also for higher frequency values, but it is unlikely at frequencies above a hundred of kHz. Moreover, electrode polarization has effects more pronounced on the capacitance rather than the conductance of the investigated medium [START_REF] Gabriel | The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz[END_REF], and in our study the major findings were observed at frequencies where the capacitance was small or also negligible. Finally, we used platinum electrodes, which are less prone to electrode polarization phenomena compared to other materials independently from black platinum covering [START_REF] Mirtaheri | Electrode polarization impedance in weak NaCl aqueous solutions[END_REF][START_REF] Ragheb | The polarization impedance of common electrode metals operated at low current density[END_REF][START_REF] Rigaud | Bioelectrical impedance techniques in medicine. Part I: Bioimpedance measurement. Second section: impedance spectrometry[END_REF], though such covering may further reduce the amplitude of the electrode polarization phenomena by a factor of 4 or more [START_REF] Rigaud | Bioelectrical impedance techniques in medicine. Part I: Bioimpedance measurement. Second section: impedance spectrometry[END_REF].

Despite all the considerations above, let's suppose that we underwent some electrode polarization phenomena that affected our results. There are two possible cases:

1) Electrode polarization is independent by the glucose concentration of the samples. This is likely, since electrode polarization should occur only in the presence of ions within a solution, and glucose does not dissociate into ions in solutions (therefore, it does not add new ions to those already present in the solution). In this case, for each series of experiments in our study, i.e. water, blood and sodium chloride, the amplitude of the electrode polarization had to be the same for all the samples. Thus, we might have found inaccurate absolute values in the impedance of the samples at different glucose concentrations (due to the possible electrode polarization systematic error), but still the impedance differences between samples would be right. In fact, possible impedance differences between samples were of special interest in our study, and not the absolute impedance values.

2) Electrode polarization is dependent by the glucose concentration of the samples. In this case not only the absolute impedance values, but also the impedance differences between samples might be inaccurate. However, though the impedance differences might be overestimated they still need to be present to some extent. Otherwise, there is no reason for the glucose dependent electrode polarization. In any case, this possibility is unlikely for the reason explained at case 1).

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Thus, even in the hypothesis that some of the criticisms by Caduff et al. were right, the main conclusions of our study would remain correct, i.e. glucose directly affects the impedance (at least the conductance) of the investigated solutions at relatively low frequencies, independently from other possible mechanisms triggered by glucose. It is true that the glucose effect may be small; for this reason, we are certainly extremely cautious to suggest that our findings can be used for clinical applications, as we already clearly claimed in the Discussion of our study.

  by Caduff et al. discusses some possible limitations of the study. The first criticism is related to the reported phase values. In science, experiments are carried on to validate theories, and not the opposite. Caduff et al. claim that the phase values are not convincing as they are not in agreement to what one would expect for a RC sensor. The experiments

	were performed with commercially available equipment of proven quality, especially the impedance
	analyzer. Thus, obvious reply is that the RC model may be oversimplified for complex systems such
	those investigated in our study, especially for blood. Furthermore, even in the case that phase values
	outside ±p/2 were incorrect, it must be noted that such values were found only at the low and high
	ends of the studied frequency band. In our study the findings of major interest were observed in
	frequency intervals where the phase values were almost zero, thus certainly within what expected

from the RC sensor theory (see Fig.

2, 3 and 4

).


Christopher J R Illingworth

Kevin E Parkes

Christopher R Snell

Philip M Mullineaux

Christopher A Reynolds

Criteria for confirming sequence periodicity identified by Fourier transform analysis: application to GCR2, a candidate plant GPCR?

Keywords: Fourier transform, periodicity, codon pairs, codons, G-protein coupled receptors, hydrophobicity

come

A C C E P T E D M A N U S C R I P T Introduction

Some degree of symmetry and periodicity is occasionally observed in protein sequence and structure and this is often related to function. Here we present criteria to ensure that periodicity inferred from Fourier transform approaches is not over reported, and apply these methods to Gprotein coupled receptors (GPCRs). Discrete Fourier transformation is one of many methods that can be used to infer structure and function from the physical properties associated with a protein sequence. The importance of such methods arises from the need to analyze the ever growing wealth of protein sequence data arising through genome projects. Fourier analysis is particularly well-suited to looking for patterns within the amino-acid sequences. In one of the earlier predictions of symmetry, Zimmerman used Fourier transforms and an autocorrelation function to search for periodicities in residue properties such as volume and interchangeableness, and inferred a 5-residue repeating pattern in the polarity of the residues in the bakers yeast cytochrome c sequence [START_REF] Zimmerman | The characterization of amino acid sequences in proteins by statistical methods[END_REF] , which may be related to stretches of small amino acids in the alpha helices of the protein structure. MacLachlan and Stewart used Fourier transforms to find a 14fold periodicity in α-tropomysin, and demonstrated the statistical significance of this result through a mathematical analysis of the Fourier transform method [START_REF] Mclachlan | The 14-fold periodicity in alpha-tropomyosin and the interaction with actin[END_REF] ; the 14-fold periodicity was later confirmed by X-ray crystallography [3 ; 4]. Statistically significant periodicities have also been found through the application of Fourier transform methods to DNA sequences [START_REF] Chechetkin | Search of hidden periodicities in DNA sequences[END_REF][START_REF] Chechetkin | Nucleosome units and hidden periodicities in DNA sequences[END_REF][START_REF] Lobzin | Order and correlations in genomic DNA sequences. The spectral approach[END_REF][START_REF] Silverman | A measure of DNA periodicity[END_REF]. At a more local level of protein structure, Fourier transforms have been successfully applied in conjunction with hydrophobicity scales to reveal amphiphilic secondary structures in protein sequences [START_REF] Cornette | Hydrophobicity scales and computational techniques for detecting amphipathic structures in proteins[END_REF][START_REF] Donnelly | An analysis of the periodicity of conserved residues in sequence alignments of G-protein coupled receptors. Implications for the three-dimensional structure[END_REF][START_REF] Donnelly | Modeling alphahelical transmembrane domains: the calculation and use of substitution tables for lipid-facing residues[END_REF][START_REF] Donnelly | The prediction and orientation of alphahelices from sequence alignments: the combined use of environment-dependent substitution tables, Fourier transform methods and helix capping rules[END_REF][START_REF] Eisenberg | The hydrophobic moment detects periodicity in protein hydrophobicity[END_REF][START_REF] Leonov | A periodicity analysis of transmembrane helices[END_REF].

More recently, weaknesses in the Fourier transform method have been identified, for example in the potential loss of periodicity when a protein is converted into a numerical sequence [15 ; 16], and this has led to the development of other methods of determining periodicity in protein sequences [17 ; 18]. However, the Fourier transform remains a valid method for searching for periodicity in a particular property of a sequence, especially if the property is not necessarily related to the individual amino acids in a simple 1-to-1 fashion (see below).

Another approach to searching for periodicity in amino acid sequences involved the application of Fourier transform methods to sets of proteins, and processing the transformed sequences of individual proteins to generate a combined signal measuring periodicities which are common to the set. Periodicities predicted by this method have been related to both structural factors [START_REF] Rackovsky | Hidden" sequence periodicities and protein architecture[END_REF] A C C E P T E D M A N U S C R I P T

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4 and protein function [START_REF] De Trad | Protein sequence comparison based on the wavelet transform approach[END_REF]. In these applications of Fourier transforms across sets of proteins, particularly large peaks in the combined signal were taken as indicative of common periodicities, however, there was no attempt to quantify the certainty with which this periodicity can be inferred. Here, Monte Carlo envelopes are used to give, for the first time, indications of the statistical significance of these methods. In an initial test, Fourier transform methods are applied to random protein sequences to study previously reported significance levels. The same methods are then used to study hydrophobicity, and the factors believed to govern the speed of codon translation, in sets of G-protein coupled receptor (GPCR) sequences.

The results highlight potential pitfalls of the method, and suggest that previous predictions of periodicity may have been over-interpreted, though they also illustrate cases in which the method can be very useful, for example in uncovering genuine low frequency periodicities. To illustrate the power of the method, we have applied the method to the GCR2 family.

GCR2 [START_REF] Liu | A G protein-coupled receptor is a plasma membrane receptor for the plant hormone abscisic acid[END_REF] and the homologous Lanthionine synthetase C-like proteins [START_REF] Li | Structure and mechanism of the lantibiotic cyclase involved in nisin biosynthesis[END_REF][START_REF] Mayer | Organization and chromosomal localization of the human and mouse genes coding for LanC-like protein 1 (LANCL1)[END_REF][START_REF] Bauer | Characterization of p40/GPR69A as a peripheral membrane protein related to the lantibiotic synthetase component C[END_REF][START_REF] Hirayama | Perception and transduction of abscisic acid signals: keys to the function of the versatile plant hormone ABA[END_REF], have been reported to be GPCRs [21 ; 26 ; 27], primarily because they have been identified by transmembrane helix prediction algorithms as having 7 transmembrane helices, but in the case of the LANCL1 protein, motifs such as putative glycosylation sites were also identified [26 ; 27]. There is currently much interest in GCR2 because it has been proposed as the receptor for abscisic acid [START_REF] Liu | A G protein-coupled receptor is a plasma membrane receptor for the plant hormone abscisic acid[END_REF], an important plant hormone. However, the LANCL1 protein was later re-classified by the original authors as a peripheral membrane protein with enzyme activity [START_REF] Bauer | Characterization of p40/GPR69A as a peripheral membrane protein related to the lantibiotic synthetase component C[END_REF], an observation now justified by the recent X-ray crystal structure of a lantibiotic cyclase (PDB codes 2g02, 2g0d). Recently the status of GCR2 as a GPCR has also been queried [START_REF] Gao | Genetic characterization reveals no role for the reported ABA receptor, GCR2, in ABA control of seed germination and early seedling development in Arabidopsis[END_REF] and in an attempt to understand the origin of this confusion, we have analysed the GCR2 -Lanthionine synthetase C-like protein family using Fourier transform analysis, using hydrophobicity as the transformed property.

Methods

Discrete Fourier transform

From any one-dimensional sequence of amino acids of length l, a numerical sequence f(k) can be derived, by assigning numerical values, for example hydrophobicity scores, to the amino acids in the sequence. Given such a numerical sequence, the discrete Fourier transform is the sequence

F(n),
where k is the position along the numerical sequence and n is the frequency, given by

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 5 m l F (n) = ∑ f (k )e 2πikn / l k =1 (1)
F(n) is usually complex, and can be separated into a real cosine series, and an imaginary sine series, as follows:

l l F (n) = C (n) + iS (n) = ∑ f (k )cos(2πkn / l ) + i ∑ f (k )sin(2πkn / l ) (2)
Signal-to-noise ratio, S / N

k =1 k =1
Given a numerical sequence f(w) with m elements, the signal-to-noise ratio is the maximum absolute value of an element of the sequence divided by the mean absolute value of the sequence

= { ( )} ∑ f (w) S / N max w f w / w =1 m (3)
The signal-to-noise ratio has a minimal value of 1, in the case of all elements of f(w) having the same absolute value, and a maximal value of m, in the case of all but one elements of f(w) having an absolute value of zero.

Random sequence generation

A number of different sets of sequences were used in the study. In order to study the significance values quoted by de Trad et al. [START_REF] De Trad | Protein sequence comparison based on the wavelet transform approach[END_REF], and Cosic , a large number of random sequences were generated.

A set of 100 000 proteins was randomly selected from the UniRef50 database [START_REF] Wu | The Universal Protein Resource (UniProt): an expanding universe of protein information[END_REF], which consists of protein sequences clustered such that no two sequences in the database have more than 50% sequence identity. A random residue was chosen in a random sequence, and the protein sequence was read off from that point. If the end of the sequence was reached, a jump was made to another random sequence from the 100 000, starting at the residue int(p 2 L), where L was the length of the new sequence, and p was a uniform random variable on the interval [0,1].

Here the p 2 term biases the choice of residue towards the start of the protein, in order to minimise the number of jumps between sequences. Generating random sequences from real protein data, rather than on a residue by residue basis, incorporates into the random sequences more of the autocorrelations found in real protein sequences than would the generation of sequences on a residue-by-residue basis. Thus, results from specific protein sequences can be compared to a backdrop of what would be expected from a protein sequence that was chosen by chance.

Residues from these random sequences were converted into their EIIP (Electron Ion Interaction Potential) values [START_REF] Cosic | Macromolecular bioactivity: is it resonant interaction between macromolecules?--Theory and applications[END_REF], nominally ranging from 0 to 0.1263. The EIIP value, sometimes referred to as the PEII value [START_REF] Cosic | Prediction of 'hot spots' in SV40 enhancer and relation with experimental data[END_REF], is based on a pseudopotential method [START_REF] Veljkovic | Simple General Model Pseudopotential[END_REF], and was claimed to

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correlate with properties of organic molecules such as carcinogenicity, toxicity, and antibiotic activity [START_REF] Veljkovic | Correlation between the carcinogenicity of organic substances and their spectral characteristics[END_REF][START_REF] Veljkovic | A Theoretical Approach to the Preselection of Carcinogens and Chemical Carcinogenesis[END_REF][START_REF] Veljkovic | Is it possible to analyze DNA and protein sequences by the methods of digital signal processing?[END_REF][START_REF] Veljkovic | Theoretical prediction of mutagenicity and carcinogenicity of chemical substances[END_REF]. Though this claim was the subject of some controversy [38 ; 39], the method is used here for consistency with the previous key studies in this area. The experiment was also carried out using random sequences generated on a residue-by-residue basis, which by definition have no inherent autocorrelation. Results from these latter tests are contained in supporting information.

GPCR sequence generation

A second set of sequences was used to study potential periodicity in codon and codon pair data and in hydrophobicity. The alignment of olfactory proteins from the GPCR database [START_REF] Horn | GPCRDB: an information system for G proteincoupled receptors[END_REF] was edited to extract a set of human olfactory proteins with no gaps or insertions, each of 314 residues in length, and these were filtered using the program Jalview [START_REF] Clamp | The Jalview Java alignment editor[END_REF] to remove redundancy, so that no two sequences had more than 50% sequence similarity. DNA sequences for each of the resulting 12 proteins were taken from the EMBL nucleotide database [START_REF] Kulikova | EMBL Nucleotide Sequence Database in[END_REF] and converted into χ 2 values describing the frequency of the DNA codons, using information from the codon usage database (http://www.kazusa.or.jp/codon/), and the extent of over-representation of codon pairs, derived from the work of Gutman [START_REF] Gutman | Nonrandom utilization of codon pairs in Escherichia coli[END_REF], to give two numerical sequences describing each protein.

χ 2 values for codons that occur less frequently than average, and for codon pairs that were overrepresented, were arbitrarily given a positive sign. Other work, not reported here, has suggested that codons and codon pairs which are translated slowly may have a role in protein folding. Rare codons, and over-represented codon pairs [START_REF] Buchan | tRNA properties help shape codon pair preferences in open reading frames[END_REF] that are reportedly translated slowly [START_REF] Irwin | Codon pair utilization biases influence translational elongation step times[END_REF] and which have positive χ 2 values were the focus of this project, so χ 2 -values which were below zero were set to equal zero, representing the assumption that the speed of translation is unimportant when the translation happens quickly. Further sets of sequences were used to study periodicity in the hydrophobicity of residues. Twenty six proteins were randomly selected from the multiple sequence alignment of archaeal bacteriorhodopsins in the GPCR database [START_REF] Horn | GPCRDB: an information system for G proteincoupled receptors[END_REF], and another thirty proteins randomly selected from the multiple sequence alignment of all rhodopsin vertebrate sequences in the same database. These were converted into numerical hydrophobicity values according to a measure of the hydrophobicity of each residue [START_REF] White | Hydrophobic interactions of peptides with membrane interfaces[END_REF]. Forty six proteins from a GCR2 alignment were also studied with this hydrophobicity method (see below).

RRM method

In order to analyze each of the numerical sequences, the RRM method described by Cosic [START_REF] Cosic | Macromolecular bioactivity: is it resonant interaction between macromolecules?--Theory and applications[END_REF] was applied to the sequences f(k). In-house code was used to generate the real and imaginary

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parts of a Fourier transform (see Eq. 2), which in turn were used to generate the real sequence

R(n) R(n) = C(n) 2 + S (n) 2 (4)
Given a protein length of L, the sequence R(n) takes values for 0≤n≤L/2, as any pattern found in the protein could not have a wavelength shorter than 2 residues. Where R(n) i is the sequence corresponding to protein j, repeating this process for all of the proteins in the set and multiplying together generated the cross-spectral function

M P(n) = ∏ R(n) j j =1 (5)
where the product is taken over all of the M transformed sequences from the protein set. This multiplication identifies frequencies n which have high values of R(n) for most values of j -if for a few values of j, R(n) is small, the product will also be small. Hence the multiplication step identifies common frequencies in the data [START_REF] Cosic | Macromolecular bioactivity: is it resonant interaction between macromolecules?--Theory and applications[END_REF]. As a measure of the significance of the resultant signal, the signal-to-noise ratio, S/N, was calculated for P(n) using Equation 3.3. Where the RRM method was applied to codon and codon pair data, and to residue hydrophobicity scores, inhouse code was used to calculate the discrete Fourier transforms, rather than the Fast Fourier Transform (FFT) method used by Cosic [50]. This allows for easier interpretation of frequency results.

Significance tests

In order to provide an estimate of significance levels for the EIIP method, sets of between 1 and 30 random sequences were generated, with lengths varying from 100 to 400 residues. For each length and set size, 10 000 sets of proteins were randomly generated. Calculating the S/N ratio for each of these sets, and ordering them, gave statistically derived estimates for significance at the 50%, 95%, and 99% levels.

In order to test the significance of the result from the olfactory protein DNA data, the RRM method was applied to 10 000 sets of 12 random DNA sequences. Random sequences were generated from DNA taken from proteins in the human genome, from the EMBL nucleotide human coding sequence database, [START_REF] Kulikova | EMBL Nucleotide Sequence Database in[END_REF] edited to remove duplicate sequences, partial sequences, and sequences with bases other than G, C, A, or T. These sequences were filtered to remove any sequences less than 315 amino acids in length, to give a set of 57 825 proteins. The χ 2 information for the codons and for the codon pairs were derived as above. To generate a random

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sequence, a random protein was chosen from the set, and the χ 2 scores for the codons and for the codon pairs were read along the DNA sequence, starting from the first codon of the sequence.

In a similar manner, 10 000 sets of random sequences equal in number and length to the sequences in the hydrophobicity test sets were generated and converted into hydrophobicity scores, to give a statistical indication of the significance of the results obtained.

GCR2 transmembrane helix prediction and sequence alignment

The transmembrane regions of GCR2 and related GPCRs were predicted using TMHMM [51 ; 52] and the Kyte-Doolittle method [START_REF] Kyte | A simple method for displaying the hydropathic character of a protein[END_REF]. The results indicating that GCR2 is not a GPCR are shown in supporting information. The alignment of GCR2 with the lantibiotic cyclase (PDB codes 2g02) [START_REF] Bateman | Pfam 3.1: 1313 multiple alignments and profile HMMs match the majority of proteins[END_REF] and the PFAM [START_REF] Bateman | Pfam 3.1: 1313 multiple alignments and profile HMMs match the majority of proteins[END_REF] seeded alignment of the Lanthionine synthetase C-like proteins (pfam code LANC_like/PF05147) was generated using a profile alignment with clustalX [55 ; 56]. This homology of GCR2 to the Lanthionine synthetase C-like proteins has been reported elsewhere [START_REF] Hirayama | Perception and transduction of abscisic acid signals: keys to the function of the versatile plant hormone ABA[END_REF]. We note that all of the key Lanthionine synthetase C-like GXXG motifs [START_REF] Bauer | Characterization of p40/GPR69A as a peripheral membrane protein related to the lantibiotic synthetase component C[END_REF] are aligned in both GCR2 and 2g02, along with the catalytic residues of the lantibiotic cyclase (PDB code 2g02) [START_REF] Li | Structure and mechanism of the lantibiotic cyclase involved in nisin biosynthesis[END_REF]. Hydrophobicity values were assigned to each position, as above [START_REF] White | Hydrophobic interactions of peptides with membrane interfaces[END_REF]. The Fugue [START_REF] Shi | FUGUE: sequence-structure homology recognition using environment-specific substitution tables and structure-dependent gap penalties[END_REF],

genTHREADER [START_REF] Jones | GenTHREADER: an efficient and reliable protein fold recognition method for genomic sequences[END_REF] and Phyre [START_REF] Nett-Lovsey | Exploring the extremes of sequence/structure space with ensemble fold recognition in the program Phyre[END_REF] fold recognition servers all identified lantibiotic cyclase as a high scoring hit for GCR2 (see supporting information).

Results

Results of the measurement of S/N values in the random EIIP sequences are shown in Table 1.

The level of significance is dependent on both the number of sequences in a set, and on the length of those sequences. The figures obtained contrast with the S/N value of 20 which, following the work of Veljkovic et al [START_REF] Veljkovic | Is it possible to analyze DNA and protein sequences by the methods of digital signal processing?[END_REF] has widely been assumed as being significant [START_REF] Cosic | Resonant recognition model and protein topography. Model studies with myoglobin, hemoglobin and lysozyme[END_REF][START_REF] Cosic | Macromolecular bioactivity: is it resonant interaction between macromolecules?--Theory and applications[END_REF][START_REF] Cosic | Virtual spectroscopy for fun and profit[END_REF][START_REF] Cosic | Bioactive peptide design using the Resonant Recognition Model[END_REF][START_REF] De Trad | Protein sequence comparison based on the wavelet transform approach[END_REF] -for sets of 30 proteins, this value would in fact be below average. In order to demonstrate statistical significance at the 95% level, much higher values would often be needed. Thus to infer periodicity in the EIIP values with a 95% certainty for a set of 20 proteins of length 300 amino acids, a single to noise ratio in excess of 98.2 is required.

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9

Applied to the codon data, the RRM method found a spike in the cross-spectral function with a signal-to-noise ratio of 22.7, at a (very high) frequency of 116 (i.e. there are 116 repeating units in the 314 residues, corresponding to a wavelength of about 2.7 residues). Applied to the codon pair data, the RRM method found a spike with signal-to-noise ratio of 27.3, again at a (very high) frequency of 136. (These two frequencies are 0.3694 and 0.4331 in Cosic's measure.) In both cases, the signal-to-noise ratio was above the value of 20, identified by Cosic as being the threshold for significance. However, application of the RRM method to random DNA sequences suggested another picture with regards to significance. For the codon and codon pair data, the median signal-to-noise ratios from 10 000 sets of random DNA sequences were 24.8 and 23.8 respectively. Within the set of results from random protein sequences, the 5% and 1% high values were, respectively, 59.7 and 85.9 for the codon data, and 57.3 and 82.0 for the codon pair data. Thus, the signal-to-noise values obtained for the olfactory protein DNA data do not appear to be significant, and it is likely that the spikes obtained are the product of chance, rather than any interpretable pattern in the sequence data. Applying the RRM method to the set of bacteriorhodopsin sequences gave a spike in the cross-spectral function with S/N ratio of 114 at a frequency of 7. Applying the same method to the 10 000 sets of random proteins of the same length gave a 99% significance level of 107, indicating that the observed peak is significant. The frequency of 7 corresponds to the 7 hydrophobic alpha helices in the bacteriorhodopsin structures, thus demonstrating a clear link between the Fourier transform results and structure. Application of the same method to a set of vertebrate rhodopsins also gave a significant S/N ratio, but at a peak frequency of 8. The presence of a significant S/N ratio suggests that in this case, the frequency might correspond to a genuine hydrophobicity-related property of the sequence, although it is known that the sequence has only seven distinct hydrophobic regions, corresponding to the transmembrane helices. A possible explanation of this result is the common existence of irregular length loops and of additional amino acids at the start and end of the sequence, illustrated schematically in Figure 1. These additional amino acids have the effect of increasing the frequency that is observed, as the Fourier transform method effectively fills in another peak to fit the periodicity to the hydrophobic regions that do exist. We suggest that in this case, non-periodic insertions to the periodic sequence have distorted the frequency at which periodicity is found.

When applied to the GCR2 sequences, the RRM method identified a periodicity at a frequency of seven, with a S/N value of exactly 270 (to 7 significant figures). Given a sequence length of 539, this is equal, within machine accuracy, to the maximum theoretically obtainable value for this

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ACCEPTED MANUSCRIPT 10 alignment (hence there is no need to compare to random sequences). The seven fold hydrophobicity could be interpreted as giving strong support to the idea that GCR2 is a GPCR.

However, a blast search [START_REF] Altschul | Gapped BLAST and PSI-BLAST: a new generation of protein database search programs[END_REF] of the G-protein coupled receptor sequence database (GPCRDB) did not yield any significant hits; a search of the NCBI non-redundant database yielded hits from the Lanthionine synthetase C-like protein family and a putative class B GPCR (XP_318705.3; EAA13819.3; E value 1E-45) that was probably also wrongly characterized as it also aligned well to the Lanthionine synthetase C-like protein family (results not shown). Likewise, the TMHMM and Kyte-Doolittle transmembrane helix prediction algorithms did not given any clear indication that GCR2 is a GPCR. TMPro did identify 5 of the 7 transmembrane hydrophobic regions, but TMPro only highlights transmembrane regions, it does not determine whether these are sufficiently long to span the membrane. The results of the transmembrane prediction algorithms are given as supporting information and are similar to those given elsewhere [START_REF] Gao | Genetic characterization reveals no role for the reported ABA receptor, GCR2, in ABA control of seed germination and early seedling development in Arabidopsis[END_REF]. Given the negative results from the BLAST search and the transmembrane prediction algorithms, it is difficult to see why GCR2 has been proposed as a GPCR, particularly given its alignment to the Lanthionine synthetase C-like protein family. However, the origin of the confusion is apparent from the application of the RRM method to the GCR2 multiple sequence alignment. The signal to noise ratio of 270 (maximum possible = 270) compares very favourably with the signal to noise ratio of 114 (maximum possible = 115) for the bacteriorhodopsin family. Visual inspection of the GCR2 multiple sequence alignment using the hydrophobic display in jalview [START_REF] Clamp | The Jalview Java alignment editor[END_REF] shows 7 hydrophobic stretches which generally commence near the C-terminal end of each of the seven inner helices, they encompass the 7 GXXG motifs and end near the N-terminal region of he helix.

The length of these stretches is somewhat subjective as they differs slightly for each Lanthionine synthetase C-like sequence and contains hydrophilic residues (such as His in the zinc binding site), but a conservative estimate is 15±2 and so they are generally too short to span the membrane. These regions are plotted onto the structure the lantibiotic cyclase (PDB codes 2g02) (Figure 3A) and a homology model of GCR2 created using Phyre [START_REF] Nett-Lovsey | Exploring the extremes of sequence/structure space with ensemble fold recognition in the program Phyre[END_REF] (Figure 3B). The corresponding space-filling model of 2G02 show that the exposed regions of these hydrophobic stretches map onto a single face of the protein near to the active site (Figure 3C).

Discussion

In the application of Fourier transform methods to the detection of underlying periodicities in protein sequences, some apparently useful results have been obtained [67 ; 68]. However, care needs to be taken to ensure that such results are a product of the sequence data, rather than simply

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11 being an artefact of the mathematics. Here we have shown that empirically-derived statistical tables can be drawn up to serve as a measure of the significance of any one given result.

The significance values for the signal-to-noise ratio derived from random sequences were much higher than might be expected. In some cases a signal-to-noise ratio of 100 (that is, a signal of 100 times greater than average magnitude) would not be significant. This can best be understood by considering what happens when large quantities of numbers are multiplied together, as occurs when the Fourier transforms are multiplied together in Equation 3.5. Where numbers between 0 and 1 are multiplied together many times, those numbers that are close to 1 remain of roughly the same magnitude, while numbers closer to zero become very small very quickly. In the case of Equation 3.5, this leads to a very high variance in the numbers produced, such that the maximal value of the sequence P(n) is much larger than the mean value, simply as an artifact of the method. Regardless of the sequence data that is fed in, very large S/N values are produced as a matter of course, and great care needs to be taken in assuming significance. This is illustrated by Figure 2, which shows mean signal-to-noise ratios for sets of 100 products of uniform random variables. This gives the equivalent of the expected signal-to-noise ratio found in the crossspectral function (Equation 3.5) if the sequences R(n) (Equation 3.4) were 100 units long, and were distributed as uniform random variables on the interval [0,1]. From entirely random numbers, the mean signal-to-noise ratio rises above 20 for products taken across just nine sequences.

This caution about presuming levels of significance does not invalidate the method itself. Where proper care is taken to establish significance, results can be found that relate to genuine periodicities in the properties of protein sequences, as in the case of the bacteriorhodopsin set, where the seven-fold pattern in hydrophobicity reflects the seven-transmembrane helical structure of the protein. In this case also, however, care must be taken in the interpretation of results. As was demonstrated in the case of the rhodopsin sequences, factors such as insertions or deletions in the protein sequence can distort the periodicity that is found. Alignments of sequences are resistant to mutation, as long as insertions or deletions are not made in an uneven way throughout the sequence.

In another set of experiments (results not shown), a 90% rate of random residue mutation was applied to an alignment of identical, artificially constructed perfectly periodic sequences of length 300 residues (see supporting information), but the periodicity was still detectable with the RRM method, even though only 10% of the original residues remained. Similarly, random insertions made at random points in the sequence lowered the frequency at which periodicity was

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found, but periodicity was still recoverable at high rates of mutation of up to 83% (here residues were removed from the end of the sequence to maintain the fixed length of 300 residues). When deletions were made from the sequence, and random residues added at the end of the sequence (again to maintain the length of 300 residues), the frequency at which periodicity was found increased, and no significant periodicity was found at a level greater than 60% mutation. Similar experiments have shown that when insertions and deletions are made in a non-regular manner then periodicity is readily destroyed. These experiments contribute towards the observation that Fourier transform methods can detect low frequency periodicity more readily than high frequency periodicity. The experiments also support the hypothesis that insertions between the helices of rhodopsin sequences can cause a distortion of the frequency at which periodicity is found, in that addition and deletion of non-periodic residues has been shown to change the frequency of periodicity.

Where insertions are made between hydrophobic regions in a sequence, and the length of the sequence is not kept constant, the effect would be to increase the number of wavelengths that could be fitted into the sequence as a whole.

We note that periodicity as discussed here is a different concept to auto-correlation. Periodicity as discussed here implies a regular repeating pattern of residues, or of properties of residues, extending throughout the length of an entire sequence. This is a sufficient, but not necessary condition for autocorrelation, which simply measures the propensity for residues separated by a fixed-length gap to have similar properties. An example of this from mathematics would be the binary sequence {a i } which equals one if i is a prime number or if (i -23) is a prime number, but which equals zero otherwise. Such a sequence would have a strong autocorrelation at a distance of 23, but absolutely no periodicity. Elsewhere we will report autocorrelation in the codon and codon-pair χ 2 data even though it clearly has no statistically verifiable periodicity.

In an alternative approach to determining significance, Rackovsky [START_REF] Rackovsky | Hidden" sequence periodicities and protein architecture[END_REF], carrying out a Fourierbased method, compares results for protein sequences against a set of results for permutations of sequences. This gives a more accurate measure of significance than choosing a fixed value for all lengths of sequences, however, as we show elsewhere, permutations of sequences can be statistically different in nature to real protein sequences. Because of the use of multiplication to compare results, small differences between real protein sequences and their permutations can be magnified, and this has the potential to generate misleading results. In this case too, care must be taken to establish accuracy.

A C C E P T E D M A N U S C R I P T

An illustration of a protein where statistically meaningful periodicity was identified is GCR2.

Here the Fourier transform results reveal the origin of the confusion as to whether GCR2 and its lanthionine synthetase C-like homologues belong to the GPCR family. GCR2 does indeed have 7 fold hydrophobic periodicity that resides in the inner helical regions of the α-barrel and this was identified more strongly by the RRM method than the corresponding property in other wellcharacterized 7TM proteins such as bacteriorhodopsin and rhodopsin. These genuine hydrophobic stretches are too short to give a significant signal in most TM prediction algorithms but their presence is sufficient to yield a weak signal in some algorithms. However, the homology of GCR2 to lantibiotic cyclase [START_REF] Li | Structure and mechanism of the lantibiotic cyclase involved in nisin biosynthesis[END_REF] for which there is a crystal structure should be sufficient evidence to close the debate on the molecular nature of GCR2. Indeed, it is worth noting that Moriyama et al. used hidden Markov and related methods to identify novel plant GPCRs but they did not detect GCR2 [START_REF] Moriyama | Mining the Arabidopsis thaliana genome for highly-divergent seven transmembrane receptors[END_REF]. Nevertheless, while some aspects of the original report that GCR2 is the GPCR receptor for abscisic acid [START_REF] Liu | A G protein-coupled receptor is a plasma membrane receptor for the plant hormone abscisic acid[END_REF] have been seriously questioned [START_REF] Gao | Genetic characterization reveals no role for the reported ABA receptor, GCR2, in ABA control of seed germination and early seedling development in Arabidopsis[END_REF], there remains the option that GCR2 may retain an indirect role in signaling in plants since not all of the experiments have been disproved in all plant tissues.

A C C E P T E D M A N U S C R I P T

Fig. 1 .

 1 Fig. 1. Location of helices (marked as black blocks along the sequence) in bacteriorhodopsins and vertebrate rhodopsins, (sequences from the GPCRDB). The bacteriorhodopsins (top) have 7 essentially equally-spaced hydrophobic helices, leading to a significant spike related to hydrophobicity score at a frequency of 7 in the cross-spectral function. In the rhodopsin sequences (below), extended non-hydrophobic regions, e.g. between helices 4 and 5 lead to a peak in the cross-spectral function at a frequency of 8.

Fig. 2 .

 2 Fig.2. Mean signal-to-noise ratios calculated from 100 000 sets of 100 products of t U[0,1] random variables.

Fig. 3 .

 3 Fig. 3. (A) The structure of 2G02, with the 7 hydrophobic regions mapped onto the 7 inner helices in shown black (or various shades of green online), that contain the 7 GXXG motifs (cyan online); the key residues of the active site are displayed in space-filling mode. (B) The structure of GCR2, with the 7 hydrophobic regions mapped onto the 7 inner helices shown in black (or various shades of green online) that contain the 7 GXXG motifs (cyan online); the key residues of the active site are displayed in space-filling mode. Residues 255-260 are omitted. (C) The structure of 2G02, shown in space-filling mode, indicating that the 7 hydrophobic regions, shown in black (or various shades of green online), map onto a single surface. The 7 GXXG motifs are shown in cyan in the online version and the key residues of the active site are displayed in spacefilling mode.

Table 1 .

 1 Significance levels for signal-to-noise ratios calculated from EIIP data, derived from trials of random protein sequences. The table gives the signal-to-noise ratio that would be required for a sequence to have statistically validated genuine periodicity according to the RRM method. The signal of peak amplitude may occur at any frequency. Note that these values only apply where the protein sequences are transformed into numerical sequences according to their EIIP value. These numbers may differ if values other than EIIP are used.

			Sequence length (residues)
		100	200	300	400
	Mean value
	10 sequences	11.6	16.6	20.1	23.5
	20 sequences	19.1	30.4	40.9	49.8
	30 sequences	24.3	41.3	56.2	72.5
	95% significance
	10 sequences	25.7	38.8	47.2	57.3
	20 sequences	40.0	71.0	99.4	128.5
	30 sequences	45.5	85.7	124.8	165.3
	99% significance
	10 sequences	33.6	55.6	71.0	88.7
	20 sequences	46.0	87.2	123.8	167.7
	30 sequences	48.8	95.9	140.7	190.1