RESEARCH ARTICLE

Mathematical Model of MorphogenElectrophoresis Through Gap JunctionsAxel T. Esser,1 Kyle C. Smith,1 James C. Weaver,1 and Michael Levin2*

Gap junctional communication is important for embryonic morphogenesis. However, the factors regulatingthe spatial properties of small molecule signal flows through gap junctions remain poorly understood.Recent data on gap junctions, ion transporters, and serotonin during left–right patterning suggest a specificmodel: the net unidirectional transfer of small molecules through long-range gap junctional paths driven byan electrophoretic mechanism. However, this concept has only been discussed qualitatively, and it is notknown whether such a mechanism can actually establish a gradient within physiological constraints. Wereview the existing functional data and develop a mathematical model of the flow of serotonin through theearly Xenopus embryo under an electrophoretic force generated by ion pumps. Through computersimulation of this process using realistic parameters, we explored quantitatively the dynamics ofmorphogen movement through gap junctions, confirming the plausibility of the proposed electrophoreticmechanism, which generates a considerable gradient in the available time frame. The model made severaltestable predictions and revealed properties of robustness, cellular gradients of serotonin, and thedependence of the gradient on several developmental constants. This work quantitatively supports theplausibility of electrophoretic control of morphogen movement through gap junctions during earlyleft–right patterning. This conceptual framework for modeling gap junctional signaling—an epigeneticpatterning mechanism of wide relevance in biological regulation—suggests numerous experimentalapproaches in other patterning systems. Developmental Dynamics 235:2144–2159, 2006.© 2006 Wiley-Liss, Inc.

Key words: gap junctional communication; embryonic patterning; asymmetry; model; electrophoresis; serotonin

Accepted 16 May 2006

INTRODUCTION

The mechanisms that allow an em-bryo to reliably self-assemble the com-plex morphology, physiology, and be-havior appropriate to its speciesrepresent one of the most fundamen-tal and fascinating areas of researchin modern science. The events under-lying the generation and maintenanceof form require a complex web of in-formation flow between cell and tissuesubsystems during development. Re-ceptor-mediated signal exchange by

means of secreted messenger moleculeshas been studied extensively in the ageof molecular biology. However, anotherimportant system of signaling exists:the direct cell–cell exchange of smallmolecules through gap junctions (Good-enough and Musil, 1993; Lo, 1999).

GAP JUNCTIONALSIGNALING

Gap junctions can consist of proteinsfrom several gene families, and the

cell biology of gap junctions has beendescribed in detail in excellent recentreviews (Goodenough et al., 1996;Falk, 2000). Vertebrate gap junctionsare commonly thought to consist ofconnexins (Sohl and Willecke, 2003,2004). In contrast, invertebrate gapjunctions are made of innexins, a fam-ily formerly called OPUS (Barnes,1994). A variety of small moleculesand metabolites are thought to perme-ate gap junctional paths, includingcAMP (Burnside and Collas, 2002;

1Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, Massachusetts2Center for Regenerative and Developmental Biology, Forsyth Institute, and Harvard School of Dental Medicine, Boston, MassachusettsGrant sponsor: NIH; Grant numbers: GM-06227; CO6RR11244; GM-63857; Grant sponsor: NSF; Grant number: IBN-0234388; Grantsponsor: AFOSR/DOD: MURI*Correspondence to: Michael Levin, Center for Regenerative and Developmental Biology, Forsyth Institute, and HarvardSchool of Dental Medicine, 140 The Fenway, Boston, MA 02115. E-mail: mlevin@forsyth.org

DOI 10.1002/dvdy.20870Published online 19 June 2006 in Wiley InterScience (www.interscience.wiley.com).

DEVELOPMENTAL DYNAMICS 235:2144–2159, 2006

© 2006 Wiley-Liss, Inc.

Webb et al., 2002; Bedner et al., 2003),ATP (Bao et al., 2004; Pearson et al.,2005), and Ca�� (Toyofuku et al.,1998; Blomstrand et al., 1999; Paeme-leire et al., 2000).

Several human syndromes havebeen identified as mutations in con-nexin genes (Maestrini et al., 1999;Richard et al., 2002), and transgenicmice are beginning to allow the molec-ular dissection of gap junctional com-munication (GJC) function in thesecontexts (Krutovskikh and Yamasaki,2000), including the spread of electricwaves in cardiac tissue (Kimura et al.,1995; Severs, 1999) and the brain(Budd and Lipton, 1998; Momose-Satoet al., 2003, 2005), and the propaga-tion of signals through gland cells tosynchronize hormonal action and se-cretion (Meda, 1996). Regulation of tu-mor (Loewenstein and Rose, 1992; Ya-masaki et al., 1995; Krutovskikh andYamasaki, 1997; Li and Herlyn, 2000;Omori et al., 2001) and stem cell(Trosko et al., 2000; Burnside and Col-las, 2002; Tazuke et al., 2002; Gilboaet al., 2003; Cai et al., 2004; Wong etal., 2004) behavior is dependent upongap junction (GJ) -mediated signals.

When open, intercellular channelsare generally permeable to moleculesof less than 1 kDa (Loewenstein,1981), but the exact permeability is afunction of (1) precisely which con-nexin family members form the junc-tion (White and Bruzzone, 1996), (2)the charge and size of the permeantmolecule (Landesman et al., 2000; Ni-cholson et al., 2000), (3) the pH of cy-toplasmic and intercellular space(Morley et al., 1997), (4) transjunc-tional and membrane voltage of thecells (Brink, 2000), (5) the phosphory-lation state of the connexin proteinsubunits (Lampe and Lau, 2000), and(6) the activities of several chemicalgating molecules (Granot and Dekel,1998). Thus, functional GJC is depen-dant on the existence of compatiblehemichannels on the cells’ surfaces,the permeability of the hemichannelsto the substance, and the open statusof the gap junction. Regulated GJCpaths serve to establish patterns ofiso-potential and/or iso-pH cell fields(Sherman and Rinzel, 1991; Fitzhar-ris and Baltz, 2006; Rocheleau et al.,2006); this epigenetic “prepattern”overlaid upon the embryonic morphol-ogy, working in concert with the mo-

lecular fields defined by gene expres-sion domains, is an important controlelement in embryonic patterning(Levin, 2003a; Nuccitelli, 2003; Mc-Caig et al., 2005). This extremely ver-satile system for communication al-lows for rapid synchronization amongcells in a tissue and the passage ofsignals, both of which can be regu-lated at many levels. Thus, it is a per-fect conduit for information flow dur-ing development, which depends onthe ability of cells and tissues to com-municate on several time scales (Lo,1996; Levin, 2001). The converse,however, is also paramount—embryoscontain independent compartmentsthat must remain isolated with re-spect to key signaling molecules forproper morphology to result.

GJC is increasingly recognized asbeing involved in regulation of pat-terning signals in vertebrate and in-vertebrate pattern formation (Baueret al., 2001, 2002; Starich et al., 2003;Nogi and Levin, 2005; Lehmann et al.,2006). Such morphogenetic controlcould be exerted by GJC-mediatedchanges in differentiation (Zhang etal., 2002; Araya et al., 2003, 2005; Guet al., 2003; Hirschi et al., 2003), pro-liferation (Paraguassu-Braga et al.,2003; Pearson et al., 2005), or cell mi-gration (Minkoff et al., 1997; Huang etal., 1998; Lecanda et al., 2000; Oviedo-Orta et al., 2002; Kjaer et al., 2004).To integrate GJC-dependent signalsinto the systems that control complexmorphology, it is necessary to under-stand the factors regulating spatialflows of signals through gap junctions.

Gap Junctions in Left–RightAsymmetry

A considerable amount of data on thefunction of GJC in vertebrate pattern-ing, including evolutionary conserva-tion in two species, is now available inthe field of left–right (LR) patterning(Burdine and Schier, 2000; Levin,2005). The vertebrate body is based ona bilaterally symmetrical plan; how-ever, the visceral organs and braindisplay marked and consistent asym-metries in their location or geometrywith respect to the embryonic midline.The LR asymmetry is a fascinatingexample of large-scale embryonic pat-terning and raises many deep theoret-ical issues thought to link molecular

stereochemistry with multicellularpattern control (Brown and Wolpert,1990; Wood, 1997). Because no macro-scopic force distinguishes right fromleft, a powerful paradigm has beenproposed to leverage large-scale asym-metry from the chirality of subcellularcomponents (Brown and Wolpert,1990; Brown et al., 1991). In this classof models, some molecule or organellewith a unique chirality is orientedwith respect to the anteroposteriorand dorsoventral axes, and its chiralnature, thus, is able to nucleate asym-metric processes. Thus, the first devel-opmental event that distinguishes leftfrom right is proposed to take place ona subcellular scale. However, it is nowknown that a pathway of multicellularfields of asymmetric gene expressiondirects the laterality of asymmetricorgans (Levin, 1998; Yost, 2001). Amechanism must then exist to trans-duce subcellular signals to cell fieldsand to transduce information about Lor R direction into a cell’s positionwith respect to the embryonic midline.

In light of evidence for preferen-tially directional and pH-dependentGJC-mediated transfer in the earlyXenopus embryo (Turin and Warner,1980; Guthrie, 1984; Guthrie et al.,1988; Nagajski et al., 1989), Levin andMercola tested the hypothesis thatGJC could play a role in the mecha-nism by which asymmetry at the levelof a cell can be transduced into em-bryo-wide asymmetry of gene expres-sion (Levin and Mercola, 1998, 1999).Spatially oriented multicellular GJCpaths could allow directional informa-tion derived within a cell to be im-posed differentially upon its neigh-bors, resulting in global positionalinformation. Using injections of a sys-tem of a small junctionally permeablefluorescent dye together with a largemolecule, junctionally impermeabledye (to mark the injected cell and ruleout false positives due to cytoplasmicbridges and incomplete cell cleavage),we showed that there is indeed a dor-soventral difference, with a zone ofisolation across the ventral midlineand good junctional coupling on thedorsal side (Fig. 1A). This asymmetryin ability to transfer dye could be al-tered in predictable ways by chemicalagents that are known to regulatejunction permeability. Using domi-nant-negative and constitutively ac-

MODEL OF GAP JUNCTIONAL SIGNALING 2145

tive constructs, it was shown that in-terruption of the circumferential GJCpath, as well as introduction of ectopicopen gap junctions across the zone ofisolation, both randomized earlyasymmetric gene expression and posi-tion of the asymmetric viscera in frogembryos (in the absence of other de-fects). Similar data were obtained inthe chick—an embryo with very differ-ent gastrulation architecture—usingantisense oligonucleotides and a vari-ety of surgical manipulations to inter-rupt the GJC zone (Fig. 1B). Takentogether, the results indicated that azone of circumferential connexin43 ex-pression around the primitive streak(which does not express Cx43) is re-quired for normal laterality.

While much remains to be learnedabout the individual connexins func-tioning in the frog embryo and thepermeabilities of the native gap junc-tions for different molecules (Dicaprioet al., 1975; Landesman et al., 2000,2003), these experiments revealed avery similar picture of the role of GJCin LR patterning in two embryonicsystems. In both chicks and frogs, acircumferential large-scale pattern ofGJC exists around a zone of isolation.The contiguity of this path is crucialfor normal expression of LR-asymmet-ric genes; interruption of this path bysurgical, pharmacological, or molecu-lar-biological methods specificallycauses LR randomization. The zone ofisolation is likewise crucial for LR pat-terning. Taken together, these obser-vations suggested a simple model:that some small molecule signal is ini-tially homogenously distributed, butthen traverses gap junctions and pref-erentially accumulates on one side ofthe zone of isolation when it arrives

there (Levin, 2003b; Adams et al.,2006b). This accumulation can theninduce asymmetric gene expressionusing canonical mechanisms. In thismodel, interfering with the open GJCpath results in LR randomization bypreventing the movement of the mor-phogens, whereas introducing GJCthrough the zone of isolation random-izes asymmetry because there is thenno midline barrier that can ensure anasymmetric accumulation of the mor-phogens.

The evolutionary conservation ofLR patterning mechanisms is highlycontroversial (McGrath and Brueck-ner, 2003; Tabin and Vogan, 2003;Levin, 2004b); early steps are espe-cially poorly understood, and it is nowimperative to integrate available ge-netic and cell-biological data into com-prehensive models that can be testedand applied to different model species.The class of models based on GJCraised two main lines of inquiry.

(1) What controls the unidirectional(chiral) flow of LR informationthrough the gap junctions? Why wouldmorphogens traverse the GJC pathcounterclockwise rather than equallyin both directions (resulting in no netgradient)? Proposing that an electro-phoretic mechanism might provide aforce for movement of charged deter-minants through the GJC field, wescreened for the involvement of iontransporters in asymmetry. Identify-ing several ion channels and pumpsrequired for normal laterality, weshowed that early frog and chick em-bryos contain the predicted asymme-tries in membrane voltage at the zoneof isolation (Levin et al., 2002; Adamset al., 2006b) and that asymmetry israndomized when these physiological

LR differences are abolished by equal-ization of the activity of the implicatedH� and K� transporters. Thus, onepossibility is that the cells on the L vs.R side of the zone of isolation developdifferent levels of polarization (steady-state membrane voltage level) by dif-ferential ion exchange with the out-side world. We have demonstratedrecently some of the cytoplasmic trans-port mechanisms that result in such anasymmetric localization of the rele-vant ion channel and pump proteins(Qiu et al., 2005). As long as this dif-ference was actively maintained, anelectrophoretic force would be exertedthrough the other part of the circuitformed by the GJC field (Fig. 1C).

(2) What is the molecular nature ofthe small-molecule LR signals thatmight be exchanged between cells onthe L and R sides? The ideal candidatewould be smaller than the size cut-offof gap junctions (� �1 kDa), be water-soluble (lipophilic molecules such asretinoic acid do not need gap junctionsto move between cells), and becharged (to enable regulation of move-ment by means of ion pump-depen-dent voltage gradients as proposed inLevin and Nascone, 1997; Levin et al.,2002; Levin, 2003b). Serotonin fitsthese criteria, has been demonstratedto go through gap junctions betweennervous system cells (Wolszon et al.,1994), and offers the considerable ad-vantage of a well-developed pharma-cological tool set (Gaster and King,1997).

Thus, taking advantage of the largenumber of well-characterized re-agents available to test and character-ize its role in LR asymmetry in chickand frog embryos, serotonin’s role inasymmetry, before the formation of

Fig. 3. Dependence of final serotonin concentration on voltage difference. The stationary serotonin concentration cS(x) (given by Eq. (4), final state that canbe produced by the mechanism) across the Xenopus embryo is an exponential function and is given here for voltage differences between �10 mV and �40mV. Medial circumferential length indicates position along the path from L to R border in an embryo roughly 1 mm in diameter (see Fig. 2).

Fig. 4. Dependence of overall gradient gain on the electrical charge of morphogen. The serotonin right–left gain, which follows from the stationaryconcentration given in Eq. (4), exhibits an exponential dependence (shown here on a logarithmic plot) on both the electric charge of the morphogenmolecule and the voltage difference. Thus, similar right–left gains (in different biological systems) may be obtained by different combinations of themorphogen charge and potential differences.

Fig. 5. The dependence of overall gradient gain (left–right [L–R] steepness) on voltage difference. The stationary electrophoretic serotonin gain RS(x)within the embryo also follows an exponential function. It is shown here for the same voltage differences as in Figure 1. The morphogen gain measureshow much more serotonin we expect to find within the embryo compared with the left side. Thus, RS(x � 0) � 1 at the left side, and increases towardthe right side. The morphogen’s right–left gain RS(x � L), that is, the maximum expected ratio of the serotonin concentration at the right side of theembryo with respect to the left side is approximately 2-fold, 5-fold, 10-fold, and so on, for a voltage difference of �10 mV, �20 mV, and �30 mV,respectively. Medial circumferential length indicates position along the path from L to R border in an embryo roughly 1 mm in diameter (see Fig. 2).

2146 ESSER ET AL.

Fig. 1. Proposed circumferential movement of morphogens through gap junctional communication (GJC) paths. A: In the animal pole tier ofblastomeres of the 16–32 cell frog embryo, a ventral zone of junctional isolation (red line) exists around which the dorsal cells form a long-range cellpath coupled by gap junctions. Both the barrier at the ventral midline and the open GJC path are required for normal left–right asymmetry (Levin andMercola, 1998). Green line, path of junctional communication; blue conduits, gap junctions. B: A similar situation exists with respect to the primitivestreak in the early chick blastoderm (Levin and Mercola, 1999). p.s., primitive streak. C: These observations suggested a model wherebyasymmetry results from the net unidirectional movement of small molecule morphogens through the circumferential path; the gradient is schematizedin blue:red in B and C, and the movement of the morphogen (schematized in yellow) is indicated by the red line in A. One possibility is that thismovement is driven by left–right (LR) voltage differences generated across the zone of isolation by differential ion exchange of those cells with theoutside medium. D–F: A maternal pool of serotonin is initially symmetrically (homogenously) distributed throughout the early embryo (D; Fukumoto etal., 2005b), but becomes present in a gradient (E), and eventually is restricted to one cell adjacent to the ventral embryonic midline (F). The spatialdetection of serotonin was performed using immunohistochemistry (Levin, 2004a).Fig. 2. Schematic of model of electrophoresis of small molecule morphogens. A,B: The animal tier of blastomeres in the frog can be modeled as alinear array of eight cells coupled by gap junctions (A), if the circular arrangement is visualized as being pulled apart at the ventral zone of isolation(B). We simulate the eight cells, connected by gap junctions, having at each end either the left (L) or right (R) blastomere (cell1, cell8) that terminatesat the zone of isolation (red line). These end cells are different with respect to the ion pumps functioning within them; as a result of differential K� andH� transport with the outside medium (excess export of positive charges out of the right-most blastomeres), they acquire a consistently biasedmembrane voltage that is estimated to be 10–20 mV in the frog and chick systems. C: The path can be generalized as an open circuit driven by abattery, along which charged molecules can be electrophoresed.

Fig. 2.

Fig. 1.

Fig. 3. Fig. 4. Fig. 5.

MODEL OF GAP JUNCTIONAL SIGNALING 2147

neurons, was probed recently in chickand frog (Fukumoto et al., 2005a,b). Us-ing analysis of endogenous localizationof serotonin and its receptors, as well asgain- and loss-of-function experimentsusing pharmacological serotonergicblockers and dominant-negative andwild-type expression constructs, it wasshown that serotonin, and serotonergicsignaling through receptor subtypes R3and R4, were crucial for normal asym-metry. In early Xenopus embryos, sero-tonin is distributed in a striking radialpattern, beginning homogenously dis-tributed (Fig. 1D), but eventually form-ing a gradient (Fig. 1E) and coalescinginto a single blastomere on the rightside of the midline by the 32/64-cellstage (Fig. 1F), precisely as predicted inthe original model (Fig. 1A; Levin andNascone, 1997). While direct movementof serotonin through gap junctions inthis system has not been demonstratedin vivo (serotonin is too small to be mod-ified by fluorescent tags without signif-icantly changing its properties), it hasbeen shown that the ability of serotoninto localize asymmetrically through theGJ-connected blastomeres was depen-dent upon open gap junctions and uponthe function of the H,K-ATPase, and V-ATPase ion pumps. Moreover, evidencewas presented for a novel intracellulartransducing mechanism for serotoninactivity—this is a requirement of themodel because serotonin arrivingwithin the target cells’ cytosol has to beable to activate receptor mechanismsthere.

Taken together, these data sug-gested a model that provides a possi-ble answer to the chirality of the mor-phogen movement: that serotoninmoves asymmetrically through thefield of GJC-connected cells under anelectrophoretic force provided by dif-ferential membrane voltages in cellsat opposite ends of the circumferentialcell field (Levin, 2003b; Levin et al.,2006).

RESULTS

Could It Really Work? AQuantitative Model forElectrophoretic Movement ofSerotonin

Roles for electrophoretic movementof morphogens (Cooper, 1984; Coo-per et al., 1989; Fear and Stuchly,

1998a–c) have been proposed previ-ously, in the context of follicle– eggsystems (Woodruff and Telfer, 1980;Telfer et al., 1981; Bohrmann andGutzeit, 1987; Woodruff et al., 1988;Woodruff and Cole, 1997; Adler andWoodruff, 2000), self-electrophoresisin Fucus symmetry-breaking (Jaffeet al., 1974; Jaffe and Nuccitelli,1977), and regeneration in both ver-tebrate and invertebrate systems(Rose, 1966, 1970; Smith, 1967;Lange and Steele, 1978). The data onLR patterning suggest a mechanismconsistent with this class of models,as applied to a GJC-coupled cellfield. Specifically, this mechanism ismotivated by the dependence of cor-rect laterality on the GJC path(Levin and Mercola, 1998, 1999), theasymmetric localization of the pro-posed morphogen (Fukumoto et al.,2005a,b), and the differential bioel-ectrical properties of the cells on ei-ther side of the zone of isolation(Levin et al., 2002; Adams et al.,2006b). Of interest, theoretical anal-ysis indicates that gap junctionalcoupling increases the sensitivity ofcells to electric fields produced intheir milieu (Cooper, 1984; Cooper etal., 1989). However, having proposedthis mechanism, we sought to an-swer the question: Are physiological-strength endogenous electric fieldsactually sufficient to produce ameaningful gradient in serotonin inthe time provided, given the knownproperties of embryonic cells? Theavailable data on early asymmetrymechanisms in Xenopus provide therelatively rare opportunity to con-struct a quantitative model using re-alistic values for parameters. Be-cause of the rich data set available inthis field and because of the impor-tance of integrating mechanisticmodels of morphogenetic events intoquantitative, predictive, syntheticmodels that include both physiologi-cal and genetic components (Raya etal., 2004; Fischbarg and Diecke,2005), we have begun to develop amathematical model of these eventsto ask whether this mechanism isreally plausible to generate a long-range gradient in the available timeand to make predictions that can betested in future experimental work.

The Nernst–Planck equation con-siders the simultaneous presence of

diffusion and electric gradients andmay be used to describe the spatio-temporal morphogen distributionwithin an embryonic cell array. TheXenopus topology and cell-to-cell com-munication suggested by Figure 1Aallows unrolling into a linear descrip-tion of serotonin along the medial cir-cumferential length L of the cell array(schematized in Fig. 2), and we con-sider the serotonin concentration CS,in mol/cm3, as function of time t andposition x from the continuity equa-tion

�cS

�t �1

zSF�IS

� x . (1)

Here, zS is serotonin’s �2 valenceand F is the Faraday constant. Thus,the time-dependent change of the se-rotonin concentration is broughtabout by spatial gradients in the sero-tonin current density IS. The total se-rotonin current density, in A/cm2, con-tains both a diffusive component andan active component (Ohm’s law), thatis

IS � � zSF�DS

�cS

� x � uSzSFcS

��

� x� .

(2)

In the Nernst–Planck equation, DS

and US are serotonin’s diffusion coef-ficient and electrical mobility respec-tively, which are related by the Ein-stein relation Ds � RTUs, where R isthe gas constant and T is the temper-ature of the medium in which the em-bryo is immersed. Furthermore, �(x)is the electric potential as function ofposition. To gain initial insight, wefirst consider a hypothetical com-pletely open path between the cells,representing the existence of so manyGJ at the cellular interface that thediffusion coefficient DS has a constantvalue throughout the embryo along L.The voltage difference across the em-bryonic cell array �� is assumed to beconstant in time and to vary linearlyalong the embryo’s circumferencesuch that the cytoplasmic electric fieldE is given by

E � ���

� x � ���

L (3)

everywhere in the cell array of lengthL � 1.5 mm. This corresponds to aninternal electric field of E � 13.3 V/mfor the voltage difference of ��

2148 ESSER ET AL.

��right � �left � �20 mV between theright and left side (Ito and Hori, 1966;Woodruff and Telfer, 1973; Telfer etal., 1981; Borgens et al., 1994; Levinand Mercola, 1999). Electrodiffusiveequilibrium, that is, a time-indepen-dent stationary serotonin profile fol-lows from Eq. (1) by setting the leftside equal to zero. It follows that theserotonin current density IS is con-stant everywhere, and exploiting a no-flux boundary condition, this yieldsIS � 0. Eq. (2) then is an ordinarydifferential equation that can bereadily integrated and has the ana-lytic solution

cS x � cS0zSF��

RT

exp� �zSF��x

RTL �1 � exp� �

zSF��

RT �.

(4)

This is an exponential serotoninprofile, as found in other morphogenmodels (Driever and Nusslein-Vol-hard, 1988; Fosslien, 2002; Eldar etal., 2003; Veitia, 2003), caused by aconstant electric force and is shown inFigure 3 for �� values between �10mV and �40 mV (Levin et al., 2002;Adams et al., 2006b). The normaliza-tion constant cs

0 � ns/LAcc in Eq. (4) isgiven by the measured total amount ofserotonin nS � 2 pmol (Fukumoto etal., 2005b). The total amount of sero-tonin at the 8 animal-tier cells consid-ered here has a constant value in theembryo. Serotonin is homogeneouslydegraded by monoamine oxidase (Sjo-erdsma et al., 1955; Weissbach et al.,1957; Baker, 1971) starting aroundthe 16-cell stage, and the completedegradation takes approximately 40minutes (Fukumoto et al., 2005b). Thestationary morphogen profile given byEq. (4) exhibits an exponential depen-dence on the electric charge of the rel-evant morphogen molecule and thevalue of the voltage difference �� (seeFig. 4). Thus, the model predicts thatsimilar gradients in other systemsmay be obtained by different combina-tions of �� and morphogens withother electric valence. The serotoninmolecular mass has no influence onthis stationary profile, but is impor-tant for the time constant of its estab-lishment, as will be discussed later. Assuch, higher-valenced morphogenswith presumably higher molecular

masses theoretically may be too slowto establish a significant gradient onthe relevant time scales of embryonicdevelopment, and there may be anoverall optimum in the choice of va-lence, voltage difference, and morpho-gen mass.

The Nernst equation used through-out the biophysical literature definesan electric field from a stationary con-centration ratio of two cellular com-partments. The situation is reversedhere as the morphogen profile in theembryo exhibits a series of differentserotonin concentrations in each indi-vidual cell, brought about by the volt-age difference between the L and Rblastomere. Rather than consideringconcentrations, it is more instructiveto look at the electrophoretic serotoningain, defined as the concentration ra-tio RS(x) � cS(x)/cS(0) inside the cellarray. At the left side of the embryo(containing the lowest serotonin con-centration), we have RS(0) � 1, and,everywhere else, larger values(RS(x)�1) will be found as shown inFigure 5. It is a metric of how muchmore serotonin we expect to find atposition x with respect to the left side(at x � 0). The electrophoretic right–left gain RS(L) is a particular choice ofthat function,

RSL �cS x � L

cS x � 0� exp� �

zSF��

RT �(5)

and measures how much more seroto-nin can be found on the right side ofthe embryo with respect to the leftside. As can be observed from Eq. (5),the final steady-state right–left gaindepends only on the voltage difference�� and not on the system size L. TheXenopus and chick embryos both sup-port left–right voltage differences ofapproximately �� � �20 mV, sug-gesting similar serotonin gradients forthe stationary states. We consider avalue of RS(L) � 10 for the right–leftgain as a significant spatial profile forembryonic development (Shvartsmanet al., 2002; Pribyl et al., 2003), forwhich the patterned signal for embry-onic development needs to decay byone order of magnitude over the devel-oping field. For zS � �2, a value ofRS(0) � 10 indeed occurs for a voltagedifference of �� � �29 mV. This value

is remarkably close to the measured�� � �20 mV.

The time-dependent solution of Eq.(1) has a lengthy analytical solution,but with regard to the later more im-portant involvement of GJ also may befound from a numerical integration.Starting from a homogenous serotoninprofile (Fig. 1D) throughout the sys-tem at t � 0, Figure 6A shows thetime-dependent change of the seroto-nin profile for the measured voltagedifference of �� � �20 mV. With themodel parameters given in Table 1, ittakes approximately 1 hr to reach thestationary state given by Eq. (4). Thisis consistent with the developmentaltime available to reach the 32-cellstage, at which time the serotonin lo-calization is complete (Levin and Mer-cola, 1998; Fukumoto et al., 2005b).Figure 6B shows the time-dependentevolution of the right–left gain RS(L)for both the serotonin diffusion coeffi-cient DS as well as for a comparativedidactic study with a smaller diffusioncoefficient of DS/√5. The latter uses aD � 1/√M relationship between thediffusion coefficient and the molecularmass M (Weiss, 1996), which is validfor GJ-permeable molecules (�1 kDa)and is meant to represent a hypothet-ical molecule with five times the mo-lecular mass of serotonin. The station-ary serotonin profile in that casewould be established only after ap-proximately 3 hr. The time constant stat to reach the stationary state isdiffusion-limited and, hence, scaleswith L2, which for the chick embryo isa factor of 25 longer time scale com-pared with Xenopus. The stationarymorphogen profile, although expectedto be equal in both systems as dis-cussed above, is reached in the modelonly after 25 hr in the chick embryo,which is longer than the time avail-able for the LR patterning step in thechick embryo (Levin, 1998; Raya andIzpisua Belmonte, 2004; Raya et al.,2004). The stat �L2/DS scaling law isa lower temporal bound, as GJC canonly further delay the formation of thegradient. In particular, stat dependson the number of GJC sites in theembryo (or equivalently on the num-ber of cells, of which there are approx-imately 60,000 in the chick embryo).Thus, we expect from this analysisonly a weak serotonin gradient in thechick gradient after the available time

MODEL OF GAP JUNCTIONAL SIGNALING 2149

of 10 hr. In striking agreement, a se-rotonin gradient has not been ob-served experimentally so far in thechick embryo (Fukumoto et al.,2005b). Thus, if the system size is toolarge, then the stationary state givenby Eq. (5) may not be reached in theavailable time. However, the most im-portant metric probably is whether RS

reaches a critical (yet unknown) valueRs

critical, sufficient for triggering theonset of asymmetric LR gene expres-sion.

What is the relevance of GJC to thespatiotemporal generation of the mor-phogen profile? Without GJs, each cellis isolated and the serotonin profileevolves only over each individual cellbut not over the whole embryo. GJsopen the path for morphogens to beelectrophoretically transported be-tween the cells (Safranyos andCaveney, 1985; Safranyos et al., 1987).With GJC, the time course of establish-ing the stationary serotonin profile isdelayed in comparison with the com-pletely open situation discussed in Fig-ure 6 because of finite GJ density andhindered serotonin diffusion through aGJ pore. But how much longer, andwhat properties of the GJ are the mostrelevant for this delay? For simplicity,we use here a single generic connexin-type GJ, which represents a homotypicnonselective GJ. The Xenopus embryocontains different types of gap junc-tions, including Cx31, Cx38, Cx43, andCx43.3 (de Boer and van der Heyden,2005), but the stoichiometric contribu-tion of each type is not known. The sin-gle GJ conductance �GJ is estimatedfrom the GJs geometrical shape, which

is given by the pore radius aGJ and thegap junction length IGJ (Hille, 2001)

�GJ ��aGJ

2

�cyt

1

lGJ ��aGJ

2

, (6)

and includes both the internal poreresistance and the spreading resis-tance. For the gap junction parametergiven in Table 1 and a cytosolic resis-tivity of �cyt � 0.83 �m, we have �GJ �83 pS. This is a typical value for con-nexin-specific gap junctions that ex-

hibit �GJ values in the range from 30pS to 300 pS (Hille, 2001). The seroto-nin current density through a singleGJ in our model is given by the Gold-man–Hodgkin–Katz equation (Veen-stra, 2000)

iS,GJ �zS

2F2DS,GJ��GJ

lGJRT

� �cl � crexpzSF��GJ/RT

1 � expzSF��GJ/RT �, (7)

which follows from Eq. (2) under thesimplifying assumption of a symmet-rical membrane without any internalstructure along the GJ pore. As such,the ionic mobility and the electro-chemical potential can be assumed tobe linear over the diffusional distance.Thus, the voltage difference seen bythe serotonin concentrations on theleft (cl) and right (cr) side of the GJ is��GJ � E dGJ � 21 nV. This is anextremely small value with respect tovoltage ranges at which nonlinearitiesand rectifying properties of GJ aretypically observed (Veenstra, 2000).The serotonin partition coefficientfrom cytosol into the pore of the GJ isassumed to be unity. However, wetake into account the frictional sterichindrance of the serotonin molecule

TABLE 1. Serotonin, Gap Junction, and Model Parameters

Parameter Value

Diffusion constant DS 3•10�10 m2/sec (Mastro et al., 1984)Serotonin valence zS �2 (Kema et al., 2000)Serotonin radius rS 0.3 nmSerotonin molecular mass MS 175 DaVoltage difference �� 20 mVGap junction length lGJ 1.6 nmGap junction pore radius aGJ 0.6 nmGap junction density nGJ 6•1011 m�2 (Hanna et al., 1980)

2•1010 m�2 (Spray et al., 1981a)System length L 1.5 mm (medial circumference, based on

�1 mm embryo diameter, calculatedas L�2�(d/4)�1.5 mm)

Temperature T 293 KInterfacial area Acc 1.5•10�5 m2

Fig. 6. The temporal development of the gradient. The time-dependent development of theserotonin concentration in the Xenopus embryo, calculated from Eqs. (1,2), in a completely openpath. A: Starting from a constant profile at t � 0 and a constant voltage difference of �� � �20 mV,the exponential stationary profile is reached after approximately 1 hr. B: The serotonin right–leftgain RS(L) as a function of time shows that the stationary right–left gain is reached after approxi-mately 1 hr. A hypothetical smaller diffusion coefficient (smaller by a factor 1/√5 and intended torepresent a five times greater morphogen mass), for comparison, leads to a slower generation ofthe morphogen gradient (almost 3 hr here). Medial circumferential length indicates position alongthe path from left (L) to right (R) border in an embryo roughly 1 mm in diameter (see Fig. 2).

Fig. 7. The influence of gap junctions (GJs) on gradient development. The time-dependent devel-opment of the serotonin concentration in the Xenopus embryo. Gap junctional transport includedby Eqs. (7–9) shows distinct effects. A: Starting from a homogenous profile at t � 0 (not shown here)and a constant voltage difference of �� � �20 mV, the initial profiles appear ragged at each cellularinterface such that the serotonin gradient across each individual cell is larger compared with thecompletely open embryo discussed in Figure 4, but evolves into the smooth stationary profile forlater times. The number of GJs at each cellular interface here is NGJ � 105. B: The serotoninright–left gain RS(L) as function of time for different GJ densities. In comparison with a completelyopen syncytium, the GJ-mediated stationary profile is established on a slower time scale. Thisretardation depends strongly on the number of GJs at each cellular interface. C: The right–left gainis shown at t � 2 hr for different GJ densities.

Fig. 8. Serotonin gradient across individual cells in the cell field. A,B: Space–time plot (A) and timecourse (B) of the individual cell serotonin gradients defined as concentration difference within eachcell divided by the cell size vs. the embryonic serotonin gradient, defined here as difference ofconcentrations on the left and right side of the embryo divided by the embryo size. The individualcell gradients show heterogeneous behavior (compare, e.g., cell 8 [right side, gradient increases forall times] vs. cell 1 [left side; initial gradient increase but later decreasing]). The overall embryogradient is initially smaller than all individual cell gradients but evolves into an intermediate value ofall individual gradients.

2150 ESSER ET AL.

with radius rS within a finite-size GJpore. For � � rS/rGJ, we adopt fromLevitt (Levitt, 1991)

DS,GJ

DS�

1 � 2.1054� � 2.0805�3

� 1.7068�5 � 0.72603�6

1 � 0.75857�5

(8)

and find for the serotonin radius ofrS � 0.3 nm a reduction of the cytoso-lic diffusion coefficient in the GJ ofDS,GJ � 0.17 DS. The total gap junc-tion current density IS,GJ through thecell-to-cell interfacial area Acc is givenby

IS,GJ � NGJiS,GJP0. (9)

Here, the total number of gap junc-tions NGJ is found from the observedGJ density nGJ (see Table 1) and theinterfacial area, and P0 is the proba-bility to find the gap junctions to beopen. For simplicity, we assume P0 �1in our model. Many specific and im-portant details of gap junction trans-port such as ionic selectivity, ionic im-pedance, and the internal structure ofthe GJ pore are neglected in this sim-plified description, in which we seekan educated initial understanding ofthe spatio-temporal properties of themorphogen distribution within thewhole embryo, and ask if these resultsare plausible with respect to the ex-perimental observations.

Figure 7A shows the spatiotemporaldependence of the serotonin concen-tration as it is influenced by GJC. Theprofile appears ragged initially, as thegap junctions represent a barrierthrough a lower effective cross-sec-tional area NGJ AGJ compared with

Fig. 6. Fig. 7.

Fig. 8.

Acc. As a result, the serotonin gradientin each individual cell is stronger atearlier times compared with a hypo-thetical completely open cytoplasmicsyncytium (Fig. 8), but the gradientover the whole embryo is weaker. Forlater times, we find a smoothening ofthe embryonic serotonin profile to-ward the stationary profile describedby Eq. (4). In total, the creation of thestationary profile is delayed comparedwith the completely open situation.This retardation depends strongly onthe total number of GJ at each cellularinterface, as detailed in Figure 7B,C,and thus represents a significant bot-tleneck for the embryonic system. Dif-ferent values for the gap junctionaldensity nGJ can be found or inferredfrom the literature (see Table 1). Twoexamples are shown in Figure 7B. ForNGJ �105 at the cellular interface, avalue estimated by counting GJplaques on electron microscopy pic-tures (Hanna et al., 1980), the station-ary profile is obtained in 3 hr. This isa plausible time scale in accordancewith experimental observations. ForNGJ � 3*103, a number estimated bytotal conductance measurements be-tween amphibian blastomeres (Sprayet al., 1981a), the profile is notreached in 10 hr and appears implau-sible. Figure 7C shows the right–leftgain at 2 hr as a function of the GJnumber over 4 orders of magnitude.The right–left gain exhibits nonlinearbehavior. It starts from RS(L) � 1.22for a negligibly small number of GJs.The marginal gain in that case is theconsequence of the morphogen gradi-ent in the individual cell and can becalculated from Eq. (5) for the intra-cellular voltage difference ��cell ��2.5 mV. For larger GJ numbers be-tween NGJ � 103 to NGJ � 105 at thecellular interface, RS(L) increases rap-idly and then for NGJ�106 saturatesat the completely open right–left gain.

These results indicate that an elec-trophoretic mechanism acting upon aGJ-coupled cell field can produce arich but temporally limited set of dis-tinct local gradients (Figs. 7, 8). TheGJs initially hinder the electro-phoretic serotonin redistribution suchthat the gradient across each GJ in-terface has the opposite sign com-pared with the intracellular gradi-ents. However, while evolving towardthe time-independent smooth steady-

state distribution, a sign-reversal oc-curs and the transcellular gradientsacross the GJs of two neighboring cellsexhibit a value in-between the con-nected two intracellular gradients.The existence of serotonin gradientsacross individual cells that differ fromthe overall transembryonic gradient isa fascinating consequence of thismodel and supports a previous conjec-ture that gradients in neurotransmit-ter concentrations across cells canprovide directional signals to cellsthrough differential activation of re-ceptors and transporters on oppositesides of the cell (Levin et al., 2006).

DISCUSSION

The present model is a first step to-ward a quantitative understanding ofbiophysical processes involved in theembryonic pattern formation. In par-ticular, we quantified the spatiotem-poral serotonin profile, which is ex-pected to result from the measuredvoltage difference between the leftand right sides. The influence of GJCin the development of the morphogengradient is explored. GJs are the keyfor mediating morphogen signals be-tween adjacent cells and creating arobust and significant morphogen pro-file within the embryo. In particular,the rate at which the morphogen pro-file can be established dependsstrongly on the number of GJs at thecellular interface.

This model reveals that, given rea-sonable estimates for all of the bio-physical constants, a left–right gradi-ent plausibly can be created by theproposed mechanism. Some of the con-sequences of this model have alreadybeen confirmed. For example, it pre-dicts that the L and R sides of the zoneof isolation should have differentmembrane voltage levels. This predic-tion further suggests that introducinggap junctions in the isolation zoneshould short-circuit the battery and,thus, randomize the LR axis. Both ofthese have been observed experimen-tally (Levin and Mercola, 1998; Levinet al., 2002). This model makes sev-eral other predictions that will betested in the context of left–right pat-terning in the future, because theamount and distribution of serotonin,gap junctions, and ion transporterscan be experimentally varied in this

system. However, the analysis pre-sented here can be used to modelmany other signaling systems in addi-tion to serotonin; for example, othersmall molecules such as cAMP, ATP,and so on are known to pass throughgap junctions (Cotrina et al., 1998,2000; Braet et al., 2003; Bao et al.,2004) and have been suggested as can-didate morphogens participating inthe establishment of positional infor-mation (Schiffmann, 1989, 1991).

This analysis also provides an ex-planation for the extraordinary ro-bustness (Eldar et al., 2002; Englandand Cardy, 2005; Houchmandzadeh etal., 2005; Mizutani et al., 2005) of theleft–right system. Under normal con-ditions, less than 1% of frog embryoshas any laterality defect, and this pro-portion is stable against a large num-ber of pharmacological, physiological,and surgical manipulations (Nasconeand Mercola, 1997; Levin et al., 2002).Indeed, short pulses of reagents thatfunctionally interfere with the knownearly steps do not result in randomiza-tion, revealing the ability of the sys-tem to regulate and restore LR signalswhen perturbed (Levin and Mercola,1998). Our results indicate that themagnitude of the final right–left gra-dient gain for a specific morphogendepends only on the voltage differ-ence, not on the number of GJs, two-dimensional (2D) geometry, or on anassumption of a constant electric fieldeverywhere in the path. This indepen-dence of a large number of parametersand molecular details reveals a novelmethod of generating a gradient thatdoes not rely on feedback loops ortranscriptional cascades for robust-ness and stability (Freeman and Gur-don, 2002; Eldar et al., 2003, 2004).

Comparison to OtherMorphogen Gradients

Our model takes a somewhat widerview of the concept of “morphogen,”because in this system, serotonin islikely to signal to downstream eventsin early blastomeres before the gener-ation of specific “cell types” by differ-entiation (Ashe, 2006 #8011). Never-theless, it is useful to compare thegradient produced by the electro-phoretic system to morphogen behav-ior in other systems to ask what mag-nitude of gradient is likely to be

2152 ESSER ET AL.

developmentally relevant. Whilequantitative studies of morphogendistributions are not many, one mole-cule presents itself as a good candi-date for comparison. Auxin, a planthormone that bears a striking similar-ity to serotonin, is a positional signalin several plant patterning events (Sa-batini et al., 1999; Vroemen et al.,1999; Baluska et al., 2003; Friml,2003; Barlow, 2005) and is involved inestablishment of bilateral symmetryin plants (Lee and Evans, 1985; Liu etal., 1993; Zgurski et al., 2005). Indeed,many of the same players that havebeen implicated in left–right asymme-try (K� channels, plasma membraneH� flux, cell membrane transporters,and gradients of serotonin/auxin, reg-ulatory roles of pH, fusicoccin, etc.)are now known to be crucial compo-nents of auxin signaling (Arend et al.,2002; Coenen et al., 2002; Pasternaket al., 2002; Hager, 2003; Rober-Kle-ber et al., 2003; Wind et al., 2004). Aswith serotonin, there is thought to bea soluble auxin receptor (Dharmasiriet al., 2003; Woodward and Bartel,2005a,b). Most interestingly, auxinhas been suggested to move underelectrophoretic control (Goldsworthyand Rathore, 1985; Rathore and Gold-sworthy, 1985; Rathore et al., 1988;Rathore and Robinson, 1989; Wang etal., 1989; Rathore et al., 1991; Fischeret al., 1997; Fischer-Iglesias et al.,2001; Rober-Kleber et al., 2003).

Paponov et al. (2005) proposed anauxin-flow canalization hypothesis:auxin flow, starting by diffusion, in-duces formation of polar transportsystem. This system in turn promotesauxin transport, leading to canaliza-tion of auxin flow along a narrow col-umn of cells, ultimately controllingdifferentiation. This kind of positivefeedback amplification loop (Jaffe etal., 1974) parallels the feedback loophighlighted in Fukumoto et al. (2005),due to the interaction between 5HTand K� flow through 5HT-R3. Likeserotonin (Hirdes et al., 2005), auxinalso activates K� channels (Philipparet al., 1999, 2004; Fuchs et al., 2003).The many molecular and biophysicalsimilarities between the auxin and se-rotonin pathways may represent aprofound evolutionary conservation ofpatterning mechanisms, consistentwith the conservation of fusicoccin sig-naling between plants, fungi, and ver-

tebrate asymmetry (Bunney et al.,2003), or may reveal a convergence ofmechanisms to achieve similar pur-poses from different molecular geneticstarting points. The comparison withauxin is particular interesting for themodel as it has a singly negativecharge. Apart from the different signin electric charge that would reversethe morphogen profile with respect tothe anatomical polarity of the embryo,Figure 4 shows that for comparabletotal serotonin and auxin profiles thevoltage difference in the auxin systemshould be approximately twice aslarge as in the serotonin system.

Is the serotonin gradient gain ourmodel provides physiologically rele-vant? The magnitude of the serotoningradient predicted from our modelmatches well with the reported auxingradients, which are thought to be ap-proximately 10-fold (Edlund et al.,1995; Uggla et al., 1996). We also com-pared the gradient with another likelymorphogen (although one which is notGJC-related): retinoic acid. Gradientsof retinoic acid have been measuredbetween 2.5- and 2.7-fold in the ante-rior vs. posterior side of the develop-ing limb and proximal vs. distal cellsin the regenerating limb blastema(Eichele and Thaller, 1987; Scaddingand Maden, 1994). Protein morpho-gens such as activin can induce differ-ent gene expression levels at receptoroccupancy levels that differ by approx-imately 3 to 1 (Dyson and Gurdon,1998; Gurdon et al., 1998, 1999;Shimizu and Gurdon, 1999). Sonichedgehog protein can switch cells be-tween alternative fates within a rangeof two- or threefold changes in concen-tration (Ashe, 2006 #8011) and a sim-ilar range is observed for DPP in Dro-sophila (Ashe, 2000 #8012). Thus, theproposed electrophoretic mechanismappears at least as efficient as thatproducing other kinds of instructivedevelopmental gradients and, thus, islikely to be sufficient to account fordifferential downstream effects incells along the GJC path (Wolpert,1969, 1971). The molecular character-ization of the intracellular serotoninreceptor will allow further testing ofthis model as its behavior at differentligand concentration ranges is inves-tigated.

Testing the Model: FutureProspects

The model developed above is a start-ing point to understanding early de-velopmental physiology. For the firsttime, it enables specific testable pre-dictions to be made regarding the dis-tribution of serotonin and ultimatelaterality following perturbationssuch as manipulation of GJC levels,voltage gradient, serotonin levels, etc.It can also provide useful analysis ofother morphogens that may be pro-posed by future discoveries. Themodel will be refined by subsequentwork; direct measurements stillneeded are the density of gap junc-tions at blastomere interfaces, and thequantification of serotonin levels inthe blastomeres as a function of time.Moreover, several layers of complexityand feedback loops need to be incorpo-rated. The ability of serotonin to local-ize to a specific blastomere depends onGJC and H,K-ATPase function, which(in concert with a potassium channel)may provide an electromotive force formoving charged small molecules (suchas serotonin) between cells (Levin,2004b). Conversely, several ion trans-porters are controlled by 5HT, mostnotably 5HT-R3 and SERT (Maricq etal., 1991; Quick, 2003).

Additionally, the increased under-standing of pH gradients in embryos(Turin and Warner, 1980; Gillespieand McHanwell, 1987; Gillespie andGreenwell, 1988; Guthrie et al., 1988;Dickens et al., 1989; Grandin andCharbonneau, 1991) and their contri-bution to GJC in LR patterning (Ad-ams et al., 2006b) must be included, asmust possible unidirectional gap junc-tions (Robinson et al., 1993; Xin andBloomfield, 1997). These increasedlayers of complexity can be accommo-dated in the model through the P0

parameter in Eq. (9).The next important step will be to

develop a detailed model that includesthe activity of the ion pumps respon-sible for the generation of the voltagegradient and the complex inter-rela-tionship between pH and membranevoltage, and then to self-consistentlysolve for the generation of the morpho-gen profile. This can be approachedusing recently developing modelingmethods (Gowrishankar and Weaver,2003; Gowrishankar et al., 2006),

MODEL OF GAP JUNCTIONAL SIGNALING 2153

which are straightforwardly extend-able to the creation of more realistic2D and 3D models of embryonic mor-phology. pH is a crucial component,because gap junctions are sensitive tointracellular pH (Spray et al., 1981b;Francis et al., 1999); however, asym-metry in Xenopus is stable againstchanges in pHex ranging from 11 to 5(Adams et al., 2006a), providing addi-tional opportunities for exploring ro-bustness aspects of the electrophoreticmechanism.

The model makes several specificpredictions. First, the influx of the LRmorphogens into cells (as distinctfrom the well-understood serotonin re-ceptor activation on the cell surface)must be able to control cell fate. Thisappears to be borne out in other sys-tems, as the function of the SERT im-porter is involved in tumor growth(Nordenberg et al., 1999; Serafeim etal., 2002), as is serotonin (Dizeyi et al.,2004). An important aspect is theidentification of gene transcriptionmodulated by the intracellular arrivalof GJC-dependent morphogens intocells. Although these studies are justbeginning in our laboratory in the con-text of the asymmetric gene cascade,others have begun to characterize, us-ing transcriptome analysis, the ge-netic targets of gap junction-mediatedsignaling (Iacobas et al., 2003, 2004,2005; Kim et al., 2005). It is now abun-dantly clear that a large number ofimportant gene networks are poten-tially controlled by GJC signaling andthese networks represent excellenttargets for future molecular dissec-tion. For example, recent analysislinks the expression of Syndecan-2with connexin43 in mouse embryos(Iacobas et al., 2005), which is the firstglimpse of conservation of regulatorycircuits among the known roles of GJCand Syndecan-2 in lower vertebratesand mammals in establishment ofleft–right asymmetry (Kramer et al.,2002; Kramer and Yost, 2002; Fuku-moto and Levin, 2005). The model pre-dicts that microarray analysis of Xe-nopus embryos receiving intracellularinjections of serotonin on the ventralside will reveal genes up- and down-regulated by this signal. Such tran-scripts will be an essential link be-tween the serotonergic morphogenand downstream asymmetric genecascades (Levin, 1998).

In light of the current controversyover early LR mechanisms, this classof models needs to be extended toasymmetry in the chick, which is moresimilar to the embryonic architectureof most mammals, and where the cellsare far smaller but more numerousthan the early frog blastomeres. Thearchitecture of chick embryos, and thedetails of serotonin localization, differsignificantly between frog and chick(Fukumoto et al., 2005b), providing afertile ground for assessing the gener-ality of this model within the evolu-tion of developmental mechanisms.Indeed, even among amphibian em-bryos, early embryo sizes can varysharply (Berger and Roguski, 1978;Ninomiya et al., 2001; Rasanen et al.,2005). Our model (Eq. 5) suggests thatthe magnitude of the final left–rightgain at a given voltage is independentof system size; however, starting ini-tially from a homogenous morphogendistribution, the time scale on whichthe final right–left gain is reacheddoes depend on the system size. Fur-thermore, the generation of a constantvoltage gradient in much larger sys-tems will be progressively more diffi-cult. Thus, the investigation of thetiming, voltage gradient, and final se-rotonin distribution in other amphibi-ans provides an important opportu-nity for testing this class of models.

As this field matures, the modelmust incorporate more details aboutthe ion flux at the ends of the zone ofisolation, to be able to simulate thebidirectional relationship between5HT movement in an electric field andthe regulation of ion transporters (andthe resulting electromotive forceacross the field) by serotonin. The re-sulting ion flux and field gradient dis-tribution’s effect on GJC permeabilityshould also be incorporated. Suchmodels, when augmented by empiricaltesting, will allow a deep understand-ing of whether the activation of asym-metric ion flux by serotonin and theunidirectional movement of serotoninand other morphogens due to an elec-trophoretic force, thus, may be a pos-itive-feedback loop that could magnifysmall asymmetries on the cellularlevel into asymmetry on the scale ofcell fields. Other possible developmen-tal systems where such models may berelevant include Drosophila imaginaldisks and the chick limb (Weir and Lo,

1984; Coelho and Kosher, 1991). Theproposed electrophoretic mechanismmay also be related to dipolar align-ment of long, thin molecules largerthan the usual 1-kDa limit to facili-tate their passage through gap junc-tion channels (Woodruff, 2005) andwill have to incorporate knowledge ofthe specific types of GJ present in agiven tissue as well as the nature ofputative morphogens, as radically dif-ferent permeabilities of gap junctionsto different types of molecules havebeen reported (Bevans et al., 1998;Goldberg et al., 1999, 2002; Nicholsonet al., 2000). Ultimately, it will be nec-essary to incorporate not only up-stream mechanisms responsible forgenerating the voltage gradients butalso downstream mechanisms bywhich gradients are sensed by cellfields and transduced into stable cas-cades of differential gene expression.Detection of such biological signals atthresholds in light of developmentalnoise represent future efforts of con-siderable general importance (Weaveret al., 2000; Eldar et al., 2003).

Significant molecular genetic workremains to be done to address the fasci-nating and likely widely-relevant ques-tion of how intracellular movement ofsmall signaling molecules couples totranscription cascades, such as theNodal–Lefty–Pitx cassette known to bea conserved element of asymmetry invertebrates (Yost, 2001; Palmer, 2004).Consistent with our model, which relieson the cytoplasmic activity of the LRmorphogen, there is some evidence for anovel intracellular receptor of serotonin(Tamir and Gershon, 1981; Fukumotoet al., 2005b). The molecular machineryresponsible for linking cytoplasmic se-rotonin to downstream activations ofgene expression is an important area offuture investigation and may reveal ad-ditional candidates for electrophoreticmovement of morphogenetic signals.

This electrophoretic signaling sys-tem is a powerful and versatile pat-terning mechanism. Its analysis is thefirst step in providing quantitative,testable, extensible models that cansynthesize the available molecularand physiological data in a wide rangeof systems. These data provide specifichypotheses that will drive exciting ex-perimental work as patterning roles ofgap junctions continue to be uncov-ered.

2154 ESSER ET AL.

EXPERIMENTALPROCEDURES

The time-dependent continuity Eq. (1)together with the Nernst–Planck Eq.(2) represent a partial differentialequation (PDE), which can be solvedby standard numerical methods(Press et al., 1994). A discretization oftime and space allows representationof the PDE in a Finite DifferenceScheme, in which the first (time andspace) and second (space) derivativesof the serotonin concentration can beexpressed. We use a nonhomogeneousdiscretization in space with a smallspacing �x � lGJ across the gap junc-tional region and a larger spacing �x

�� lGJ in the intracellular space.The time step dt is basically deter-

mined by a stability criteria and thatcriterion is related to the discretiza-tion size in physical space. We typi-cally use dt � 0.1 sec. When the sim-ulation is run for 3 hr (that is the realphysical time for the events describedin the model), 3 hr/dt iteration stepsare needed. The real time it takes thecomputer to simulate those 3 hr isonly a couple of minutes on a standardworkstation.

The simulation starts off at t � 0with a homogenous serotonin distri-bution, and iteratively evolves in timeuntil the final stationary serotoninstate is reached. The serotonin con-centration is saved at many timepoints in-between and thus allows dis-play of the time-dependent change ofthe distribution in Figures 6–8.

ACKNOWLEDGMENTSThis paper is dedicated to the memoryof Alexander Gurvich, whose mathe-matical analysis of the biophysics ofembryonic development was a firstcrucial steps on this journey. Wethank Jose F. Ek Vitorin, Edgar Spal-ding, Winslow Briggs, Malcolm Ma-den, Jiri Friml, Daniel Goodenough,Ken Robinson, and Richard Borgensfor useful discussions on these topicsand William Baga for assistance withmanuscript preparation. Part of thisinvestigation was conducted in a For-syth Institute facility renovated withsupport from Research Facilities Im-provement Grant no. CO6RR11244from the National Center for ResearchResources, National Institutes ofHealth.

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