A Theoretical Model for the Regulation of Sex-lethal, a Gene That Controls Sex Determination and Dosage Compensation in Drosophila melanogaster

Corresponding author: Matthieu Louis, EMBL Outstation, Cambridge CB10 1SD, United Kingdom., mlouis{at}ebi.ac.uk (E-mail)

Communicating editor: A. J. LOPEZ

	ABSTRACT

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Cell fate commitment relies upon making a choice between different developmental pathways and subsequently remembering that choice. Experimental studies have thoroughly investigated this central theme in biology for sex determination. In the somatic cells of Drosophila melanogaster, Sex-lethal (Sxl) is the master regulatory gene that specifies sexual identity. We have developed a theoretical model for the initial sex-specific regulation of Sxl expression. The model is based on the well-documented molecular details of the system and uses a stochastic formulation of transcription. Numerical simulations allow quantitative assessment of the role of different regulatory mechanisms in achieving a robust switch. We establish on a formal basis that the autoregulatory loop involved in the alternative splicing of Sxl primary transcripts generates an all-or-none bistable behavior and constitutes an efficient stabilization and memorization device. The model indicates that production of a small amount of early Sxl proteins leaves the autoregulatory loop in its off state. Numerical simulations of mutant genotypes enable us to reproduce and explain the phenotypic effects of perturbations induced in the dosage of genes whose products participate in the early Sxl promoter activation.

SOMATIC sex determination is the commitment of an embryo to either the female or the male developmental pathway. In Drosophila melanogaster, flies with the chromosome constitution 2X;2A (X, X chromosome; A, haploid autosomal set) are females and flies with the chromosome constitution XY;2A (Y, Y chromosome) are males. Therefore, the X-linked genes are in two doses in females and in one dose in males. This imbalance is essential to signal sexual identity and lasts for a short period of time after fertilization, after which the amount of products encoded by the genes located in the X chromosome is equalized in both sexes (dosage compensation). In Drosophila, the dosage compensation process is achieved through hypertranscription of the single X chromosome in males (reviewed in CLINE and MEYER 1996 ; LUCCHESI 1996 ).

In somatic cells of D. melanogaster, the X-linked Sex-lethal (Sxl) gene directs both sex determination and dosage compensation. The instruction for establishing sexual identity and dosage compensation is implemented by the absence or the presence of the Sxl gene product. Over the last three decades, experimental investigations have unraveled the regulatory mechanisms that determine the production state of Sxl protein (diagrammed in Fig 1). The Sxl gene has two promoters (SALZ et al. 1989 ), the establishment and the maintenance promoter (also called the early and the late promoter and denoted as Sxl_Pe and Sxl_Pm). For convenience, we adopt the following convention: the products of gene Sxl originating at the establishment promoter are referred to as early transcripts and proteins. The products of Sxl originating at the maintenance promoter are referred to as late transcripts and proteins.

View larger version (98K): In this window In a new window Download PPT slide   Figure 1. The determination of the production state of Sxl protein is a multistep process that converts the X/A ratio in a binary way. The dashed arrow indicates that positive autoregulation of the RNA splicing is launched in females only.

The primary genetic X/A signal (where X/A represents the ratio of X chromosome to autosomal sets) acts on the establishment promoter and controls Sxl expression at the transcription level (TORRES and SANCHEZ 1991 ; KEYES et al. 1992 ). Due to a twofold difference in the number of X chromosomes, Sxl transcription is efficiently initiated in females only. As a result, early Sxl protein is abundantly produced in females whereas it remains undetectable in males. The activation process of Sxl by the X/A signal is cell autonomous. The Y chromosome plays no role in Sxl activation.

Once Sxl transcription has been sex-specifically regulated, an event that occurs around blastoderm stage, the X/A signal is no longer needed and the production state of Sxl protein remains fixed (SANCHEZ and NOTHIGER 1983 ; BACHILLER and SANCHEZ 1991 ). The ability of the system to function as a stable genetic switch after the blastoderm stage and during the rest of development and adult life is due to the existence of a positive autoregulatory loop on Sxl expression (CLINE 1984 ).

After the blastoderm stage, the maintenance promoter Sxl_Pm starts functioning in both sexes, and production of the late transcripts persists throughout the remainder of the fly's life. Male transcripts differ from female ones by the inclusion of a male-specific exon that places stop codons in the open reading frame of mature mRNAs. The inclusion of this exon gives rise to truncated, nonfunctional Sxl proteins. In females, the male-specific exon is spliced out and functional Sxl protein is produced (BELL et al. 1988 ; BOPP et al. 1991 ). Therefore, the control of Sxl expression after the blastoderm stage occurs by sex-specific splicing of its transcripts. The autoregulatory function of Sxl takes place at the level of splicing since Sxl protein itself is the driving force for female-specific splicing of its own primary transcript (BELL et al. 1991 ). The role of the primary genetic X/A signal for sex determination and dosage compensation is thus to provide females with a transient early burst of Sxl protein sufficient to establish female-specific control of Sxl expression, once the maintenance promoter has become functional.

Like any other regulatory process, determination of the production state of Sxl protein can be viewed as a program. This program aims to sense small quantitative differences in the X/A signal and to amplify them into the final all-or-none production of Sxl protein. Experimental works have studied how this program is genetically encoded and a very good picture of the system has emerged. So far, the dynamical aspects of the program—the way the program code is executed—have been tackled through verbal models. The complex nature of the regulatory processes analyzed makes it desirable to unify the present knowledge within a theoretical framework. Quantitative models are often useful to clarify qualitative hypotheses based on intuition.

We present a theoretical model for determination of sex-specific production of Sxl protein. The regulatory process modeled is composed of three steps: the formation of the X/A signal, the activation of the establishment promoter Sxl_Pe by this X/A signal, and the effect of the early production of Sxl protein on the control of Sxl autoregulation (cf. Fig 1). The model focuses on the known molecular mechanisms operating at each step. As we shall see, this model clarifies the role of the system parts and allows testing working hypotheses. It emphasizes the importance of the molecular organization of the establishment promoter and shows that the decision-making process does not require all-or-none transcriptional regulation of Sxl_Pe by the X/A signal. Indeed, our simulations are not compatible with the total absence of early Sxl protein in males, and the model suggests that production of small amounts of early Sxl protein in males is not sufficient to switch on the autoregulatory loop on Sxl protein production. Numerical simulations of the model equations allow a thorough analysis of mutant genotypes and display the in silico effects of loss-of-function mutations and/or abnormal dosages of the X/A signal genes. Our results are in good agreement with experimental observations and shed insights into the mechanistic features that enable the system to buffer important variations in gene dosage.

	MODELS

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

The regulatory processes controlling the production of Sxl protein are both time and space dependent. Eukaryotic cells are highly organized milieus and major cellular functions have been shown to occur in specific compartments (for an analysis of the functional architecture of the cell nucleus, see DUNDR and MISTELI 2001 ). In a first attempt to partition the cell milieu, one can roughly distinguish the cytoplasm from the nucleus. While transcriptional regulation and RNA splicing occur in the nucleus, proteins are translated in the cytoplasm. For the sake of simplicity, the model developed herein does not integrate the distinction between nucleus and cytoplasm; diffusion and active transports between these two cell compartments are not considered. This simplification is motivated by the observation that all major regulatory events relevant to the transcriptional and the post-transcriptional regulation of Sxl take place inside the nucleus. In the forthcoming sections, the volume of the system implicitly refers to the volume of typical cell nuclei (for details see Appendix A).

Formation of the X/A signal
The X/A signal is polygenic. Genetic and molecular analyses have identified a set of zygotic and maternal genes that are necessary for activation of the establishment promoter: the zygotic numerators (X-linked), the zygotic denominators (autosomal), and the maternal genes. The numerator genes are scute [sc, also called sisterless-b (sis-b); TORRES and SANCHEZ 1989 , TORRES and SANCHEZ 1991 ; PARKHURST et al. 1990 ; ERICKSON and CLINE 1991 )], sisterless-a (sis-a; CLINE 1986 ), outstretched [os, also called sisterless-c (sis-c); JINKS et al. 2000 ; SEFTON et al. 2000 )], and runt (run; DUFFY and GERGEN 1991 ; TORRES and SANCHEZ 1992 ). The gene deadpan (dpn) is the only autosomal denominator gene of the X/A signal that has been found so far (YOUNGER-SHEPHERD et al. 1992 ). The maternal effect genes are daughterless (da; CLINE 1978 ), hermaphrodite (her; PULTZ and BAKER 1995 ), and extramacrochaetae (emc; YOUNGER-SHEPHERD et al. 1992 ).

Model components: For simplicity, only the most representative products of each class are taken into account to model the formation of the X/A signal: the numerator gene products Sc and Sis-a (denoted as SisA for notation clarity), the denominator gene product Dpn, and the maternal gene products Da and Emc (present in the same amount in male and female embryos; see above and a detailed discussion in SANCHEZ et al. 1994 ).

Among the X-linked genes required for Sxl activation, we retained the two predominant genes, sc and sis-a. This choice is justified by the observation that not all of the genes involved in the activation of Sxl_Pe play the same role. Indeed, despite the fact that increasing dosage of run alone is sufficient for promoting Sxl transcription in males (KRAMER et al. 1999 ), several lines of evidence point out weaker effects of run with respect to sc and sis-a. First, low levels of Sxl expression are observed in the absence of run activity, indicating that run is not absolutely required for the transcriptional initiation of Sxl (DUFFY and GERGEN 1991 ). By contrast, the absence of sc and sis-a activity completely eliminates Sxl expression (CLINE 1986 , CLINE 1988 ; TORRES and SANCHEZ 1989 ). Second, simultaneous duplications of run and sc show very low Sxl-dependent male-specific lethality (TORRES and SANCHEZ 1992 ) in comparison with the lethal effects produced by sis-a and sc duplications (CLINE 1988 ; TORRES and SANCHEZ 1989 ).

As far as sis-c is concerned, the following observations suggest that it plays a secondary role with respect to sc and sis-a: (i) while mutations in sis-a and sc strongly downregulate Sxl transcription, removal of sis-c activity has a significantly weaker effect and allows residual expression of the gene (JINKS et al. 2000 ; SEFTON et al. 2000 ), and (ii) the female-specific lethal synergistic effect between run and a deficiency of sis-c is weaker than that between run and either sc or sis-a and that between sis-c and either sc or sis-a (SANCHEZ et al. 1994 , SANCHEZ et al. 1998 ).

A molecular analysis of the interactions between the products required for early Sxl activation has been performed only for the gene products selected to model the formation of the X/A signal. Sc (VILLARES and CABRERA 1987 ; MURRE et al. 1989 ), Da (CRONMILLER et al. 1988 ; MURRE et al. 1989 ), and Dpn (BIER et al. 1992 ) proteins contain a basic-helix-loop-helix (bHLH) domain. The HLH domain allows these proteins to form homodimers or heterodimers, while the basic domain is required for their binding to specific DNA sequences (MURRE et al. 1989 ). Emc belongs to the HLH family and is devoid of a basic domain (ELLIS et al. 1990 ; GARRELL and MODOLELL 1990 ). Finally, SisA is a basic-leucine-zipper (bZIP) protein (ERICKSON and CLINE 1993 ). Members of the bHLH and bZIP protein families are usually involved in transcriptional regulation (for review see MASSARI and MURRE 2000 ).

Protein complexes: Interactions between the gene products selected in the model lead to the formation of complexes, which represent the molecular actors of the X/A ratio signal. Protein-protein interactions have been investigated by in vitro methods and yeast two-hybrid assays. Experimental evidence supports the formation of the following homo- and heterodimers: Sc-Da (CABRERA and ALONSO 1991 ; VAN DOREN et al. 1991 ; DESHPANDE et al. 1995 ; LIU and BELOTE 1995 ), Dpn-Dpn (WINSTON et al. 1999 ), Emc-Sc, and Emc-Da (VAN DOREN et al. 1991 ). Substantial levels of interaction were also reported between SisA and Da (LIU and BELOTE 1995 ) and between SisA and Dpn (LIU and BELOTE 1995 ). On this basis, we assume in this study that SisA can form stable heterodimers with Da and Dpn. Since very little is known about the formation of higher-order multimers, we restrict the present model of the X/A signal production to the formation of dimers. No interaction has been observed between Sc and Dpn (DESHPANDE et al. 1995 ; LIU and BELOTE 1995 ) and between Sc and SisA (LIU and BELOTE 1995 ).

The molecular characterization of the X/A signal components has clarified how they act on the establishment promoter Sxl_Pe. Sc-Da induces transcription at Sxl_Pe by binding to a set of regulatory sites within the promoter (ESTES et al. 1995 ; HOSHIJIMA et al. 1995 ; YANG et al. 2001 ). Similarly, Dpn-Dpn inhibits Sxl transcription by binding to a pair of adjacent regulatory sites (ESTES et al. 1995 ; HOSHIJIMA et al. 1995 ). It has been shown that SisA protein is required to efficiently induce transcription at Sxl_Pe (ESTES et al. 1995 ). Hence, we assume that SisA-Da is a transcription factor capable of activating transcription at the establishment promoter. In contrast, Emc-Sc and Emc-Da complexes do not interact with DNA (DAVIS et al. 1990 ; VORONOVA and BALTIMORE 1990 ). No interaction has been reported between SisA-Dpn and DNA. Therefore, Emc-Sc, Emc-Da, and SisA-Dpn are treated as inactive complexes that titrate Sc, SisA, Da, and Dpn and prevent the formation of the transcription factors mentioned above. Formation of the Emc-Sc complex impedes the activation of Sxl_Pe as it sequesters Sc. In contrast, formation of the SisA-Dpn complex has a dual effect: it simultaneously favors activation and repression of Sxl_Pe since it reduces the amount of both free SisA and free Dpn available for the formation of activator (SisA-Da) and repressor (Dpn-Dpn). The potential formation of Da-Da and Da-Emc complexes is not taken into account as it merely reduces in a non-sex-specific manner the amount of free Da and Emc. Table 1 contains the symbols used to denote single proteins and protein complexes, together with the supposed function of the individual complexes on the activation of Sxl_Pe.

Through the formation of inactive complexes, the primary signal realizes a balance between the gene products encoded by the X chromosome(s) and the autosomes. Because autosomal zygotic gene products and maternal gene products are present in equal amount in the two sexes, they have no discriminative power in sex determination. The sole difference between males (1X;2A) and females (2X;2A) is the number of X chromosomes and thus the dosage of the X-linked gene products. A decade ago, it was hypothesized that the competitive formation of positive and negative regulatory complexes could lead to a higher sequestration of activator molecules in males than in females and could thereby amplify the male-female differences (PARKHURST et al. 1990 ; PARKHURST and ISH-HOROWICZ 1992 ). By modeling the X/A signal formation, we were able to quantitatively estimate the amplification resulting from sequestration events.

Model assumptions: To model the formation of the X/A signal with ordinary differential equations, the following assumptions are made:

1. The concentration of the X/A signal proteins is assumed to be high enough so that a description in terms of concentration can be applied.

2. Since Sc, SisA, and Dpn are zygotically expressed shortly after fertilization, their production is modeled by constant influxes that are rapidly switched on and off, in agreement with the times of in vivo mRNA and protein appearance (cf. Table A22).

3. Da and Emc are present in nonlimiting concentrations in the two sexes. Whether or not maternal da and emc mRNAs are still translated at the time early Sxl activation occurs remains unclear. The existence of translation, though, should not affect the system dynamics as long as the proteins are sufficiently abundant. This is supported by numerical tests (data not shown). For simplicity, translation of the maternal mRNAs is ignored and Da and Emc proteins are given as initial conditions.

4. Individual proteins are degraded following first-order reactions.

5. The production rate of Sc and SisA is roughly proportional to the number of gene doses. This has been shown experimentally (ERICKSON and CLINE 1993 ; DESHPANDE et al. 1995 ).

6. Interactions between the protein pairs SisA::Dpn and SisA::Da lead to the formation of SisA-Da and SisA-Dpn heterodimers.

7. The formation of protein complexes is assumed to be reversible. No experimental fact contradicts this hypothesis.

8. Molecular dimers are assumed to be not directly degraded. This last hypothesis can be justified by the putative hiding of domains favoring degradation within oligomers. For instance, stabilization effects of oligomerization have been observed for spectrin assembly in erythroid development (LAZARIDES and WOODS 1989 ) and hyperthermophilic protein assemblies (VIEILLE and ZEIKUS 2001 ).

The reaction scheme for the formation of the X/A ratio signal is given in (1) and discussed in more detail in Appendix A:

(1)

In (1), the three dots symbolize the degradation pathway. In agreement with assumption 5, the influx rate of the X-linked gene products is equal to a production rate per gene dose (denoted as F) multiplied by the number of gene doses (denoted as ). For wild-type flies, _p is equal to the number of autosomal sets while _a and _s are equal to the number of X chromosomes. As only the number of X-linked genes differs between males (1X;2A) and females (2X;2A), the initial X/A ratio difference corresponds to a twofold difference in the production flux of Sc and SisA.

Due to the lack of experimentally measured kinetic data, parameter values are either inferred from known kinetic constants of homologous proteins or deduced theoretically as explained in Appendix A. Although the exact amounts of proteins forming the X/A signal have never been precisely quantified over time, it has been experimentally observed that: (i) sc and sis-a mRNAs start being substantially transcribed during nuclear cycle 11; (ii) substantial amounts of dpn mRNAs are not detectable before cycle 11; and (iii) the production of Sc and SisA proteins correlates with the activation of early Sxl transcription in females (see Table A22 for details). For simplicity, it is assumed that Sc, SisA, and Dpn proteins appear in the nucleus simultaneously. The initial concentration of Da and Emc was chosen so that both proteins remain in nonlimiting concentrations throughout the X/A signal assessment.

Steady-state analysis: The kinetic equation system corresponding to the reaction scheme (1) consists of 10 ordinary differential equations (ODEs) given by (A1) in Appendix A together with their steady-state solutions. Under the aforementioned biochemical assumptions of the model, the following results are derived:

1. The putative lack of production of Da and Emc implies that both proteins are absent at steady state and therefore the amount of SisA-Da (ad) and Sc-Da (sd) activator complexes is nil at steady state. Consequently, the steady-state composition of the X/A signal hinders the expression of Sxl at the establishment promoter in both males and females.

2. The steady-state concentration of repressor Dpn-Dpn (p₂) complex is not a function of SisA flux (F_a) or SisA degradation (d_a). As a consequence, the amount of repressor complexes (p₂) is not influenced by the formation of the sequestration complex SisA-Dpn (ap) at steady state.

3. Numerical simulations show, for the realistic parameter set chosen, that the time needed for the system to relax toward steady state largely exceeds the developmental time window during which the X/A signal is formed and activates Sxl. It can be concluded that the X/A signal is not at steady state when it governs the transcriptional control of early Sxl.

Numerical simulations: To analyze the system outside steady state, the time evolution of the protein concentration is computed numerically by integrating the ODE system (A1) with the parameter set and initial conditions presented in Appendix A. Results are shown in Fig 2. Activator SisA-Da (Fig 2A), Sc-Da (Fig 2B), and the repressor Dpn-Dpn (Fig 2C) complexes are present in significant amounts in both males and females. The amounts of activators are higher in females than in males, whereas the amount of repressor is higher in males than in females. The primary signal is sensed through the relative amount of activators (Sc-Da and SisA-Da) vs. repressor (Dpn-Dpn). Interestingly, scanning parameter space suggests that an absence of activator in males and/or an absence of repressor in females are not achievable for parameter sets that are realistic biologically (data not shown).

View larger version (21K): In this window In a new window Download PPT slide   Figure 2. Time evolution of the number of proteins in females (solid curve) and males (dashed curve) for (a) SisA-Da, (b) Sc-Da, and (c) Dpn-Dpn. Numerical simulations of the ODE system (A1) were achieved with the parameter set given in the Appendix A and by using the free software XPPAUT (http://www.math.pitt.edu/~bard/xpp/xpp.html).

To quantify to what extent the twofold difference in X chromosome dosage is amplified by the reduction of the effective concentration of numerator factors through the formation of inactive complexes, the ratio of the number of activator complexes in males and females is computed over the time. The same is done for the ratio of repressor complexes. Fig 2 displays that maximum amplification is reached 40 min after the X/A ratio genes are expressed for both the activators and the repressor. Around that stage, the activator and repressor ratios are roughly equal to 3 and 1/3, respectively:

In conclusion, our model of X/A signal formation shows that the existence of the sequestering complexes SisA-Dpn and Sc-Emc leads to a significant but moderate four- to fivefold amplification of the initial difference in X-linked (numerator) products between males and females. Moreover, the complete absence of activators in males and/or of repressors in females is unlikely to be at the origin of the female-specific early activation of Sxl.

Activation of Sxl by the X/A signal
The molecular structure of the establishment promoter Sxl_Pe has been the object of different experimental studies (ESTES et al. 1995 ; HOSHIJIMA et al. 1995 ; YANG et al. 2001 ). It has been shown in D. melanogaster that a fragment of 1400 bp upstream of the Sxl_Pe transcription initiation site contains all the cis-acting sequences required for the control of early Sxl expression by the X/A signal (cf. Fig 3A). Several canonical and noncanonical nucleotide sequences, called E-boxes (MASSARI and MURRE 2000 ), lie within this region and show a wide range of DNA-binding affinities for the Sc-Da complexes (YANG et al. 2001 ). The E-boxes can be roughly clustered in two regions with distinct average affinities. Deletion experiments suggest that the upstream distal region (-1.4 to -0.8 kbp) is not essential for sex specificity, although it groups high or moderate affinity sites. In contrast, the proximal region (-390 bp to the transcription initiation site) is necessary and sufficient to induce transcription (in the presence of a sufficient amount of activators). The ostensible function of the distal region is to enhance transcription once it has been initiated by the proximal region; control of transcription at Sxl_Pe is exerted mainly through the proximal region (YANG et al. 2001 ).

View larger version (69K): In this window In a new window Download PPT slide   Figure 3. (a) Establishment promoter (SxlPe) structure following YANG et al. 2001 . The model identifies the establishment promoter with its proximal region. (b) Illustration of the regulatory mechanisms included in the transcription function rx(a, r). The left bracket (dark gray background) describes the binding of activator complexes to the independent E-boxes located upstream of the D-boxes. The right bracket (light gray background) describes the competitive binding of activator and repressor complexes to E-box 1 and the D-boxes. (c) Following the model assumptions, an example of an active configuration of SxlPe. (d) Example of an inactive configuration of SxlPe.

Binding sites controlled by the repressor are called D-boxes. Two D-boxes lie upstream and close to the transcription initiation site; they bind Dpn dimers (HOSHIJIMA et al. 1995 ; YANG et al. 2001 ). Due to their location close to the transcription start site, binding of the D-boxes is very likely to inhibit the formation of the transcription machinery.

Gene regulation is a process of an intrinsically probabilistic nature (FIERING et al. 2000 ). Stochastic aspects of gene expression have been recently pointed out (ZLOKARNIK et al. 1998 ; HUME 2000 ; ELOWITZ et al. 2002 ). Several theoretical studies have highlighted important implications of noise and randomness in gene regulatory systems that are not predictable by deterministic models (KO 1991 ; ARKIN et al. 1998 ; COOK et al. 1998 ; MCADAMS and ARKIN 1999 ; KIERZEK et al. 2001 ). Transcriptional regulation at the establishment promoter is clearly subject to molecular noise, inasmuch as it takes place at only one promoter composed of a few regulatory sites. Moreover, recent observations suggest that the activation of Sxl_Pe occurs independently on each X chromosome (ERICKSON and CLINE 1998 ).

Increasing evidence supports the view that enhancers/repressors stochastically regulate the probability that transcription occurs (WALTERS et al. 1995 ; FIERING et al. 2000 ; HUME 2000 ). Accordingly in our modeling, we consider that the activity of a promoter is described in terms of "on" states that allow transcription and "off" states that do not allow transcription. The dynamics of the transitions between the on and off states of the promoter are determined by the binding and unbinding of the transcriptional enhancers/repressors, and the rate of transcription is a function of the fraction of time that the promoter spends in its on states. We have therefore developed a probabilistic model for the control of Sxl_Pe activity by the X/A signal that takes into account the various molecular characteristics of the promoter.

Model assumptions:

1. Since only the proximal region (-390 bp to the transcription initiation site) of the establishment promoter is necessary and sufficient to induce transcription, the model identifies the establishment promoter with this proximal region, which also contains the D-boxes (see Fig 3B).

2. As no evidence supports the existence of cooperativity among E-boxes, E-boxes are also considered to be independent. Furthermore, cooperativity does not exist among D-boxes (WINSTON et al. 1999 ).

3. Because the first E-box is located adjacent to the D-boxes and the transcription initiation site (YANG et al. 2001 ), it is treated differently than the other E-boxes. Steric hindrance is assumed to exist between E-box 1 and the two D-boxes, since the activator and repressor molecules are relatively large (X/A gene products are made up of 250 amino acids). This would lead to a competition for the binding of activators and repressors to their respective regulatory sites. We thus assume that the binding of any D-box prevents the subsequent binding of E-box 1 and vice versa (see Fig 3C and Fig D).

4. It has been demonstrated that mutations in only one of the E-boxes do not prevent expression at the establishment promoter Sxl_Pe. Combinations of several mutated E-boxes, however, can substantially reduce the promoter activity (YANG et al. 2001 ). These results suggest that a minimum number of E-boxes must be occupied to efficiently activate Sxl_Pe. We assume that transcription is induced if and only if no D-box is occupied and at least six E-boxes (including E-box 1) are occupied (cf. Fig 3C and Fig D).

5. Corepression mechanisms are ignored, as they are mediated by a maternal factor present in the same concentration in males and females (e.g., role of Groucho; DAWSON et al. 1995 ).

6. The measured dissociation constant K_r = l_r/k_r of the reaction between Dpn-Dpn and the D-boxes,

7. 
   

   (2)

   has been experimentally estimated to be 2.6 nM (WINSTON et al. 1999 ).

7. For simplicity, the affinities of the E-boxes are chosen to be equal. The measured dissociation constant K_r for the reaction between D-boxes and repressor complexes is one to two orders of magnitude lower than the values reported for most of the bHLH proteins that bind DNA. The dissociation constant K_a = l_a/k_a for the binding of the E-boxes is thus expected to be higher than K_r:
   

   (3)

   We arbitrarily set K_a = 50 nM so that it is larger than the dissociation constant of Dpn-Dpn and the bHLH protein E47 and smaller than the dissociation constant of MyoD (experimental values of dissociation constant for bHLH proteins are given in SUN and BALTIMORE 1991 ).

8. It is assumed that Sc-Da and the putative SisA-Da complexes bind to the same regulatory sites and have similar effects on transcription. From now on, the numbers of activator complexes Sc-Da and SisA-Da are combined and denoted as A (activator acting upon the establishment promoter). The number of repressor complexes is denoted as R (repressor acting upon the establishment promoter).

9. The number of activator and repressor molecules is assumed to be sufficiently large to systematically neglect the number of molecules bound to the promoter in comparison to the number of molecules in free solution. Furthermore, numerical simulations support the idea that the formation of the X/A signal evolves on a timescale slower than the dynamics of the interactions between the promoter and the regulatory factors (data not shown).

10. For simplicity, the times separating association and dissociation events of the transcription factors are modeled as a random variable that follows a Poisson distribution.

Probabilistic model for the transcriptional regulation of a single gene: Let us define the configuration of a promoter as the state of occupancy of all its individual binding sites. The effects of site deletions suggest that promoter configurations can be clustered into the active ones able to induce transcription and the inactive ones that show very little transcriptional induction. As a first approximation, it is sound to assume that the transcription rate is proportional to the average fraction of time the promoter spends in its active configurations. We aim to justify this statement and calculate the average fraction of time the promoter is active. Below, we briefly outline the basic concepts underlying the methodology used in the model.

Given the structure of a promoter, configurations can be symbolized by Boolean vectors where each bit encodes the state of occupancy of a particular binding site (by convention, the state occupied is denoted as one and the state unoccupied as zero). If we suppose that the probability that two binding sites undergo a change at the same time is negligible, configuration changes can be viewed as transitions between Boolean vectors where only one bit is allowed to flip at each change. Given that the promoter can be in a finite number of states (maximum 2ⁿ, where n is the number of binding sites), and assuming that association and dissociation of activators and repressors occur following a stochastic process, transitions between configurations can be formalized as a time-continuous Markov chain. As we shall see, the Markov chain theory provides a useful framework to calculate the probability of being in any promoter configuration at any time. In this formalism, the vector containing all the configuration probabilities obeys a master equation that can be solved analytically for small and rather simple systems (MCQUARRIE 1967 ) or numerically in every case (GILLESPIE 1977 , GILLESPIE 1992 ). Hence, probabilities instead of concentrations are handled, thus avoiding the pitfall of having to define "configuration concentrations" for a single gene system.

On the basis of the above assumptions 2 and 3, we divide the establishment promoter into two functional independent domains: the upstream domain contains the independent E-boxes 2–7 and the downstream domain contains E-box 1 and the two D-boxes (cf. Fig 3B). Analysis of the regulatory characteristics of the two domains is done separately in the following two paragraphs.

Competitive binding for D-boxes and E-box 1 (downstream domain): Let us represent the state of the domain that contains E-box 1 and the D-boxes by a Boolean vector [₁, ₁, ₂], where ₁ denotes the state of occupancy of E-box 1 and _i the state of occupancy of the ith D-box. As mentioned above, ₁ and _i = {0, 1}. In theory the system admits 2³ different states. Nevertheless, competitive bindings (assumption 3) allow only 5 of them (denoted as G, GE, GD¹, GD², and GD¹D²). In the current model, inhibition of promoter activation is restricted to short-range effects occurring through steric hindrance between E-box 1 and the D-boxes (assumption 3). Even though the requirement of E-box 1 for the operation of Sxl_Pe remains unclear (YANG et al. 2001 ), transcription activation is assumed to occur, in our model, if and only if the E-box 1 is occupied by the activators, implying that the two D-boxes are empty of repressors.

The subsystem states are listed in Table 2 together with their notation and effect on transcription. Let us define the probability distribution vector as

where P_G, P_GE, ... denotes the probability that the promoter is in configuration "G," "GE," ...

Let A_t be the number of activator molecules and R_t be the number of repressor molecules at time t. If A_t and R_t are changing slowly enough, they can be considered as transiently constant. Therefore, the explicit time dependence of A_t and R_t is suppressed in the rest of this paragraph. The binding of activator and repressor molecules to the promoter is modeled as a Poisson process (cf. assumption 10) with kinetic constants given in reactions (2) and (3). The probability that one activator molecule binds to E-box 1 during the infinitesimal time interval dt is thus equal to (k_aAdt). Similarly, the probability that the complex activator::E-box 1 dissociates during dt is equal to (l_adt). The same holds for the repressor with the on- and off-rate k_r and l_r. The dynamics of vector (t) are described by a system of ODEs called the master equation,

(4)

represents a (5 x 5) matrix of transition probabilities (for the expanded form of , see Appendix B). Equation 4 can be solved analytically and numerically. The steady-state solution of (4) is given in Appendix B. It can be shown that the relaxation time of the chain is relatively fast for the set of parameters chosen (in the order of a few seconds). For simplicity, it is then sufficient to focus on the calculation of the steady-state distribution ^st rather than calculating the time-dependent distribution (t). As derived in Appendix B, the probability that the promoter domain is in the active state GE is

(5)

Relation (5) gives the average fraction of time that the promoter spends in state GE at steady state.

Multiple E-box (upstream) domain: The upstream domain is made up of six independent E-boxes. Their location is assumed to not influence their role in promoting transcription. Since we are dealing with six E-boxes, the number of possible different configurations (2⁶) is obviously too large to enable us to proceed as we did for the downstream domain. This problem can be bypassed by the clustering of states in classes of equivalence that have the same number of occupied binding sites, whichever they are. Equivalence classes are defined in Table 3.

According to assumption 4, the rate of transcription from Sxl_Pe will depend on the active classes C₅ and C₆ solely. Following the methodology depicted in Appendix B, the steady-state probability that the promoter domain is active is given by

(6)

Transcription rate as a function of activator and repressor molecules (full promoter analysis): The results of the two previous paragraphs can be combined to compute early Sxl transcription rate as a function of the number of activator and repressor molecules present in the system. A simple model for the transcriptional regulation of the full promoter can be constructed on the basis of relations (5) and (6). Let us denote the class of active configurations of the whole promoter as G* and the class containing the other nonactive configurations as G. To calculate the number of transcripts produced per unit of time, we need to estimate the number of transcription rounds induced per unit of time. Each transcription round starts with the successful engagement of the polymerase machinery. The binding of the polymerase machinery requires the promoter to be in state G*. Once the binding of the polymerase complex has occurred, a transcription round starts and ends up with the synthesis of an early mRNA molecule (denoted as rx). The reaction scheme (7) illustrates the process schematically:

(7)

In the reaction scheme, the production of an mRNA molecule and the activation/deactivation kinetics of the promoter are represented by different arrows to emphasize that the promoter is not transformed into rx but conditions the production of rx. The binding of the polymerase is modeled as a Poisson process of parameter k_t. Under this scheme, transcription initiation is supposed to arise from the competition between the binding of the polymerase complex to the active promoter (i.e., when the promoter is in state G*) and the deactivation of the promoter from state G* to state G.

In a first approximation, the average number of early Sxl mRNAs (_rx) can be estimated as being the fraction of time spent by the promoter in its active state G* divided by the average time required to induce a transcription round. Given the independence of the upstream and downstream domains, the fraction of time searched can be calculated as the probability of having E-box 1 and at least five other E-boxes occupied simultaneously, i.e.,

When the promoter is active, the average time separating two bindings of the polymerase is 1/k_t (general property of Poisson process). It then follows that

(8)

The numerical value of k_t is fixed as discussed in Appendix B.

Sex-specific promoter activation: At this stage, it is useful to introduce the scaled variable a = A/K_a for the activator and r = R/K_r for the repressor [where K_a and K_r represent the dissociation constant of reactions (3) and (2), respectively]. Fig 4 displays two different graphs of _rx as a function of variables a and r. For the range of values considered in Fig 4B, we observe that the response of the transcription rate (a, r) to increases in the amount of activators at a fixed amount of repressor is either very weak, when the amount of activators is low, or almost linear when the amount of activators is sufficiently high (cf. Fig 4B). This behavior resembles a threshold phenomenon where the activator leaves transcription off until it reaches a certain value and then induces transcription, though it is less pronounced. The contour plot of (A, R) in the plane (A, R) is presented in Fig 5A. Fig 5A suggests that two conditions need to be simultaneously fulfilled to induce transcription efficiently: (i) the amount of activators must be sufficiently high and (ii) the amount of repressor must be sufficiently low. On this basis, the plane (A, R) can be separated in four qualitatively distinct quadrants (cf. Fig 5A). The borderline between quadrants is arbitrarily placed, as no obvious threshold values can be defined from (8):

• Quadrant I is characterized by a (relatively) low number of activator molecules and a (relatively) high number of repressor molecules. In this quadrant, none of the conditions that allow transcription are fulfilled and repression is safely ensured.

  
  

  View larger version (31K): In this window In a new window Download PPT slide   Figure 4. Graphs of the transcription rate rx(a, r) as a function of the scaled variables a = A/Ka and r = R/Kr. rx is given by relation (8) and the parameter set used is discussed in the Appendix A. (a) Three-dimensional plot of rx(a, r) as a function of a and r; (b) two-dimensional plot of rx(a, r) as a function of a at fixed r = 2.

  
  

  View larger version (38K): In this window In a new window Download PPT slide   Figure 5. (a) Contour plot of the transcription rate rx(A, R) in the state space (A, R), where A and R represent the number of activator and repressor molecules. rx(A, R) is given in molecules per second. The roman numerals I, II, III, and IV symbolize the four distinct quadrants discussed in the text. Each of them clusters states that are qualitatively similar. Abundant transcription is induced only in quadrant IV. Contour intervals correspond to 1/4, 1/8, 1/16, 1/32 of the maximum transcription rate. (b) Trajectories of the males (red curve) and the females (white curve) plotted from t = 0 sec to t = 20,000 sec and superposed to the contour plot of rx(A, R). (c) Masculinizing effects of lowering the dosage of X-linked gene products in females. Color codes: white curve, wild-type female; green curve, mutant (g1) with one dose of sis-a; blue curve, mutant (g2) with one dose of sc; and red curve, mutant (g4) with one dose of sc and sis-a. Note that our simulations took into account that the viability of mutant (g1) has been experimentally quantified for a hypomorphic mutation (i.e., a mutation that does not lead to a complete loss of function). (d) Feminizing effects of lowering the dosage of dpn in males. Color codes: red curve, wild-type male; green curve, mutant (g13) with one dose of dpn; and blue curve, mutant (g14) with almost zero dose of dpn. (e) Rescue experiments in females. Color code: white curve, wild-type female; gray curve, mutant (g7) with zero dose of sis-a; blue curve, mutant (g8) with zero dose of sis-a and three doses of sc; and green curve, mutant (g9) with zero dose of sis-a and four doses of sc.

• Quadrant II is characterized by a low number of activator molecules and a low number of repressor molecules. One condition allowing transcription is true as the amount of repressor is low enough. However, the low number of activator molecules impedes transcription.

• Quadrant III is characterized by a high number of activator and repressor molecules. The balance between the number of activators and repressors favors binding of D-boxes over binding of E-box 1. As a result, transcription is inhibited by the repressor, even though the absolute amount of activator is high.

• Quadrant IV is characterized by a high number of activator molecules and a low number of repressor molecules. In this quadrant the two conditions for activating transcription are both true.

These results suggest that female fate is determined in quadrant IV whereas male fate is totally secured in quadrant I. Numerical simulations confirm this idea.

In Fig 5B, the time trajectories of male and female are plotted in the (A, R) plan after numerical integration of the model equations. The female trajectory (white curve) visits quadrant IV for a relatively long time while the male one (red curve) remains in quadrants I and II, mainly visiting quadrant I. These trajectory differences reflect the fact that the activator/repressor ratio is constantly higher in females than in males. Accordingly, the number of early Sxl proteins produced in males and females differs dramatically. Let us denote as the number of early Sxl proteins in the system before the maintenance promoter is constitutively turned on (i.e., at time t = 12,000 sec in our model; see next section). From the simulated trajectories depicted in Fig 5B, the ratio ^female/^male of Sxl protein in females vs. males is estimated to be 80. We conclude that the transcriptional control of the establishment promoter by the X/A signal leads to almost a 100-fold difference in the number of early Sxl proteins present in the two sexes. We observe that the amount of early Sxl proteins is not nil in males, even though it is low compared to the amount produced in females. From relation (8) and Fig 5A, we learn that the establishment promoter is fully off only if the amount of activators is nil.

In summary, the net activation of the establishment promoter by the X/A signal depends on both the relative amount of activator A and repressor R molecules. An amplification effect of Sxl transcription exists in females vs. males that results in the production of substantially more early Sxl proteins in females than in males.

Establishment of Sxl autoregulation
Differences between the transcripts derived from the establishment promoter Sxl_Pe and the maintenance promoter Sxl_Pm are due mainly to usage of different promoters and alternative splicing. The early Sxl primary transcripts originating at Sxl_Pe follow a fixed splicing pattern where the late exon L2 and the male-specific exon L3 are not included and the early specific exon E1 is directly spliced to exon L4. Exon L4 and the exons downstream from it are present in both early and late Sxl mRNAs. Splicing of the early Sxl primary transcripts is constitutive and does not require Sxl protein (HORABIN and SCHEDL 1996 ), whereas the presence of Sxl protein is necessary to drive the female-specific splicing of the late Sxl primary transcripts originating at Sxl_Pm (i.e., skipping of the male-specific exon L3).

The action of early Sxl proteins on the splicing of the late transcripts is essential for the establishment of Sxl autoregulatory function. The mechanism by which Sxl protein controls the skipping of exon L3 is not totally understood. Notwithstanding, it has been observed that Sxl cooperatively binds to the late transcripts at several poly(U) sequences (WANG and BELL 1994 ; KANAAR et al. 1995 ). The present model aims to quantitatively assess the conditions that switch on the state of the positive feedback loop controlling the splicing pattern of the late Sxl primary transcripts.

Model assumptions:

1. The different subunits of the splicing machinery are not explicitly modeled. Only Sxl protein is taken into account since it is sex-specifically produced before Sxl_Pm becomes active.

2. Early and late Sxl proteins differ exclusively in their N-terminal exons encoding two dozen amino acids (KEYES et al. 1992 ). It is assumed that these differences do not alter the RNA-binding properties of the proteins.

3. Sxl proteins cooperatively bind to the transcript splicing sites and form homodimers (SAKAMOTO et al. 1992 ; WANG and BELL 1994 ; WANG et al. 1997 ; SAMUELS et al. 1998 ). Since the formation of dimers is important for the stabilization of the splicing factors on the primary transcripts (WANG et al. 1997 ; SAMUELS et al. 1998 ), it is assumed that alternative splicing is mediated by dimers. The putative assembly of higher-order Sxl complexes is not considered. In the absence of clear evidence that strong Sxl::Sxl interactions occur in vivo, we furthermore consider that homodimerization of the proteins does not occur in free solution. However, all our results would hold qualitatively when Sxl dimerization in free solution is included in the model.

4. Splicing is regulated by the synergistic action of Sxl dimers on multiple sites lying upstream and downstream of exon L3 (SAKAMOTO et al. 1992 ; WANG and BELL 1994 ; PENALVA et al. 1996 ). For simplicity, only one splicing site is taken into account and thus synergistic effects are not considered.

5. In Drosophila, dosage compensation is triggered by the presence or the absence of Sxl protein (reviewed in LUCCHESI 1996 ). We assume that the initial transcription from the maintenance promoter Sxl_Pm is not dosage compensated. Since the Sxl gene is located on the X chromosome, the production of late Sxl primary transcripts is supposed to be twice as high in females (XX) as in males (XY).

Reaction mechanism: Let us denote Sxl proteins as x, Sxl primary transcripts as h (h for late heterogeneous nuclear RNA), and Sxl mRNA spliced in its female mode as rx. Let us define the state space of the system as = {(x, h, hx, hx₂, rx), where x, h, hx, hx₂, and rx ⁺}. A particular state of the system represents a point in . The reaction scheme below displays the reaction mechanisms of the model:

(9a)

(9b)

(9c)

(9d)

The three dots in (9a) and (9d) symbolize the degradation pathway. On the basis of assumption 5, reaction (9a) represents the production of the primary transcript h following a constant influx _x.F_h, where _x denotes the number of Sxl gene copies and F_h the production rate of transcripts per gene. Reaction (9b) describes the binding of a Sxl monomer to the "naked" primary transcripts. The second binding of Sxl to the primary transcripts is represented by the left reversible reaction of (9c). The irreversible reaction on the left of (9c) represents the splicing step where exon L3 is removed from h.Sxl₂ so that it becomes a messenger RNA (rx). The splicing step is accompanied by the release of two Sxl monomers. Reaction (9d) describes the production of Sxl protein. The instantaneous translation rate of Sxl mRNA is set equal to constant rate _tsl per primary transcripts. Further information about the kinetic scheme is given in Appendix C with the corresponding ODE system.

Generic properties of the alternative splicing mechanism: As shown in Appendix C, the kinetic equation (C1) describing the reaction scheme (9a–9d) admits two stable (z₀ and z₊) and one unstable steady state (z_-). These steady states are points within the five-dimensional space with their respective number of Sxl protein x such that

(10)

Except for the unique unstable steady-state z_-, all the points of the state space dynamically evolve toward either z₀ or z₊. The set of points that tends to one particular steady state constitutes its so-called basin of attraction. Reaction scheme (9a–9d) is characterized by the existence of two antagonistic trends that compete for the control of Sxl production. On the one hand, Sxl protein concentration tends to decrease as the protein and its (functional) mRNA undergo a natural turnover. Once the Sxl protein concentration has fallen to zero, it remains nil as the female-specific splicing cannot be achieved anymore. On the other hand, the concentration of functional protein increases each time a primary transcript is spliced and subsequently translated. The initial abundance of Sxl protein determines which trend is the strongest. The two trends pull the system toward two different steady states, one with Sxl protein concentration equal to zero and another where the Sxl concentration is high and balances production and degradation.

Among the points belonging to , we are interested in a subset that corresponds to the initial conditions of the system just before alternative splicing starts, i.e., all the states such that

wherein denotes the concentration of early Sxl protein present in the system when the constitutive production of late Sxl primary transcripts is about to be launched. Let us denote the projection of the basin of attraction of z₀ and z₊ on the subspace (⁺, 0, 0, 0, 0) as B(0) and B(x₊), respectively. As illustrated in Fig 6A, B(0) and B(x₊) are separated by a threshold value (not to be confounded with x_-). The value of can be accurately computed by numerical simulations. As seen in relation (10), x₊ is a function of the late Sxl production fluxes _x.F_h and differs between sexes since _x is twice as large in females as in males. Similarly, the value of depends on _x.F_h as well; hence it is denoted as in males and in females. Numerical simulations enable us to estimate the value of and as 1100 and 2700 molecules, respectively.

View larger version (60K): In this window In a new window Download PPT slide   Figure 6. (a) Illustration of the projection of the basin of attraction of states z0 and z+ on the subspace (R+, 0, 0, 0, 0). White arrows over the axis indicate the dynamical trends of the states following in the corresponding regions. (b) Typical dynamics of an initial condition (, 0, 0, 0, 0) such that < . (c) Typical dynamics of an initial condition (, 0, 0, 0, 0) such that > . Note that the value of is different in males and females (see text).

In males, the system unavoidably evolves toward the complete absence of Sxl protein production when the initial number of early Sxl proteins is smaller than the threshold , i.e., if belongs to B(0) (cf. Fig 6B). In contrast, stable production of Sxl protein is ensured if the number of early Sxl proteins is larger than , i.e., if belongs to B(x₊) (cf. Fig 6C). It can be hypothesized that fates will be robustly specified when is significantly smaller than in males and significantly larger than in females. In this respect, it is interesting to note that the size of B(0) is larger for males than for females. Numerical simulations of the whole system show that / = 82/2700 = 0.03 in males and / = 6700/1100 = 6.1 in females.

The difference in the threshold value in females and in males ensues from the assumption that the initial transcription at Sxl_Pm is not dosage compensated. While so far no experimental observation suggests that the single gene Sxl is hypertranscribed in males when the maintenance promoter becomes active, gene run, however, is already dosage compensated when the X/A ratio is measured (GERGEN 1987 ). The early dosage compensation mechanism of run being independent from the main dosage compensation pathway mediated by the male-specific lethal genes (BERNSTEIN and CLINE 1994 ), we have assumed that Sxl_Pm transcription remains unaffected by dosage compensation during the establishment of Sxl autoregulation.

If it turns out that Sxl_Pm transcription is dosage compensated from the start, the threshold value , in the model, would become equal for males and females, due to X chromosome hypertranscription in males (_x = 2 for both males and females). The main conclusions of the model would, however, hold since the number of early Sxl proteins in males ( = 82) would still be far below this threshold value ( = = 1100).

In outline, we have formally shown that the positive feedback loop that is involved in the Sxl RNA splicing process can generate a bistable switch for both males and females. The state of this switch is triggered by the residual concentration of early Sxl protein present when Sxl_Pm becomes active. This result is in agreement with the observation (CLINE 1980 ; KEYES et al. 1992 ; BERNSTEIN et al. 1995 ) that activation of the autoregulatory function of Sxl requires that the concentration of early Sxl protein be higher than a threshold value. Our model clarifies the nature of this threshold phenomenon. Indeed, for both males and females, a threshold separates the basin of attraction of a steady state where the concentration of Sxl protein is nil (male phenotype, lethal state for females) from the basin of attraction of a steady state where the concentration of Sxl protein remains nonzero in a self-sustained manner (female phenotype, lethal state for males; Fig 6A). This threshold is of a different nature than the thresholds usually involved in the transcriptional activation of a cooperative promoter (see, for instance, HERSCHLAG and JOHNSON 1993 ; CAREY 1998 ). In the absence of a positive feedback circuit, such thresholds do not imply the existence of bistability. Rather, they express that the activity of a given gene is low or high when the concentration of an activator is below or above a threshold value, respectively.

The model also sheds insight into the relationship between the kinetic properties of the system (degradation rate of Sxl transcripts and Sxl protein, splicing efficiency, etc.) and the existence of a bistable behavior. Equation 10 or C3 shows, for instance, that when the degradation rate of the primary transcripts (d_h) significantly increases, bistability is lost and the only remaining steady state is the zero state.

	INTEGRATED MODEL

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

The results presented in the three previous sections enable us to write the kinetic equations governing the number of Sxl mRNA (rx) and Sxl protein (x)

In (11a), the transcription rate _rx(A_t, R_t) is given by relation (8). It is a function of the number of activator molecules A_t and repressor molecules R_t present at the time considered. The bracket denoted as "early" contains the kinetic terms related to the expression of Sxl originating at the establishment promoter. Similarly, the bracket denoted as "late" contains the kinetic terms for the expression of Sxl at the maintenance promoter. The meaning and value of the parameters appearing in the late brackets are discussed in Appendix C. It should be stressed that transcription from the establishment promoter is shut off long before the maintenance promoter becomes active (see Table A22). Thus, function _rx and hx₂ are never simultaneously non-nil in Equation 11a as expressed by the existence of a pause in Fig 1.

The system of ODEs composed of (A1) in Appendix A, (C1) in Appendix C, and (11a–11b) fixes the time evolution of the number of Sxl proteins. By numerically integrating this ODE system from time t = 0 sec when the formation of the X/A signal is initialized until some time after the maintenance promoter has been switched on (say t = 20,000 sec), the expected behaviors of males and females can be simulated. The time evolution of Sxl mRNA and Sxl protein molecules is presented in Fig 7, a and b.

View larger version (26K): In this window In a new window Download PPT slide   Figure 7. (a) Dynamics of the number of mRNA molecules rx in males (dashed curve) vs. females (solid curve). (b) Dynamics of the number of Sxl proteins x in males (dashed curve) vs. females (solid curve). (c) Sketchy dynamics of the sex-specific control of Sxl expression in the somatic cells of Drosophila for females (top) and males (bottom). The abscissa represents the time and the ordinate the concentration of early Sxl transcript (dashed curve) and Sxl protein (solid curve). Action of the X/A signal on SxlPe is symbolized as a box whose color intensity is proportional to the relative abundance of activator molecules. The late Sxl primary transcript refers to the unspliced form of the pre-mRNA originating from the maintenance promoter.

Numerical simulations:
During the time interval 0–6000 sec, the X/A signal differentially activates the establishment promoter in both sexes. This results in a significant peak of Sxl mRNA (rx) in females (Fig 7A, solid line), whereas in males activation remains very low (Fig 7A, dashed line). According to experimental observations, the production of X/A gene products is shut off after 40 min (2400 sec) and the early Sxl mRNA subsequently disappears due to its natural turnover (cf. Table A22). When the maintenance promoter starts functioning (time interval 12,000–20,000 sec), the residual amount of early Sxl proteins present in females is sufficient (i.e., higher than ) to kick start the production of mature functional Sxl mRNAs. Abundant production of late Sxl protein ensues (Fig 7B, solid line). In males, the amount of early Sxl proteins is too low to efficiently set in motion the female-specific splicing of late Sxl primary transcripts. Consequently, the amount of Sxl mRNA (rx) and protein (x) falls to zero (Fig 7B, dashed line).

Multistep nature of the regulatory cascade:
Sex determination is essentially an amplification process that senses subtle differences in the composition of the X/A signal and converts it into the all-or-none production of Sxl proteins. Amplification is mediated by the combined actions of multiple mechanisms. Rather than isolating a single regulatory step that would be responsible for the whole amplification, our model suggests that each step of the cascade is necessary to achieve a switch-like response.

The primary difference between sexes is the zygotic chromosomal composition. As already mentioned, there is a ratio difference X of two between the number of X chromosomes in females and males. We saw that this difference and the existence of sequestering inactive complexes (Emc-Sc and SisA-Dpn) induce a moderate amplification of the X/A ratio so that A_t/A_t 3 and R_t/R_t 1/3 for several minutes. The ratio between (A_t/R_t) and (A_t/R_t) is then roughly equal to nine. In other words, the formation of the X/A signal achieves a four- to fivefold amplification of the initial ratio X. This amplification is transient since the presence of the X/A signal products lasts for a narrowly defined window of time. Numerical simulations show that the number of early Sxl proteins produced through the transcriptional control at Sxl_Pe greatly differs between males and females (see Fig 7B). As a result, the ratio ^female/^male of early Sxl proteins in females vs. males is roughly equal to 80 before Sxl_Pm is switched on. We conclude that the effect of the X/A signal upon Sxl_Pe induces a transient 40-fold amplification of X. Next, bistability is generated at the level of Sxl RNA splicing control. Depending on the basin of attraction in which lies, the feedback loop on late Sxl expression is driven into the on or off state. This final step converts and maintains the initial difference X into the all-or-none production of Sxl protein.

Simulation and analysis of mutations affecting the components that make the X/A signal
The state of the autoregulatory loop on Sxl expression is determined by the composition of the X/A ratio signal. To test the robustness and consistency of the present model, the effects of variations in the dosage of the X/A signal genes are analyzed.

Once sexual fates have been specified, it is vital that the production of X-linked genes is equalized in males and females (dosage compensation). Molecular analyses have shown that the process of dosage compensation is achieved through hypertranscription of the single X chromosome in males and can take place only in the absence of Sxl proteins. Therefore, changes in the X/A signal composition favoring the activation of Sxl_Pe cause male-specific lethality as they disrupt the X chromosome hypertranscription. Likewise, changes in the X/A signal composition favoring inactivation of Sxl_Pe cause female-specific lethality. From now, the terms females and males are used to designate 2X;2A and 1X;2A individuals.

The consequences of varying the dosage of genes have been studied by the assessment of mutant viability (or lethality). In the following paragraphs, the degree of specific lethality of either males or females is used as a semiquantitative criterion to measure the agreement of simulation results with experimental observations. The deterministic nature of our global model allows only a one-to-one correspondence between a given genotype (i.e., a particular X/A signal) and the corresponding simulated outcomes. To overcome this problem, our mutant analysis is based on the idea that the / ratio provides information on the robustness of the decision. By establishing a link between the viability of mutants and the simulated / ratio, experimental and simulated results can be compared.

Previously, we hypothesized that female lethality arises when < whereas male lethality arises when > . Thus, / = 1 represents a critical ratio. The / ratio is expected to significantly decrease below 1 when the percentage of female viability decreases. Similarly, / should increase above 1 when the percentage of male viability decreases. The results of our numerical simulations are shown in Table 4 and reveal a good correlation between the experimental and simulated results: female viability is 100% if / > 1 and <100% if / < 1, with only two exceptions out of 10 cases. Male viability is 100% if / < 1 and near 0% if / > 1 in all 6 cases.

Effects on female viability of lowering the doses of X-linked components of the X/A signal: Females are able to buffer important variations in gene dosages without upsetting the final outcome of the regulatory cascade. It has been demonstrated that females heterozygous for either sis-a (g1 in Table 4) or sc (g2) and females carrying one-half the amount of maternal Da products (g3) are fully viable. In contrast, female viability is drastically reduced when loss-of-function mutations are combined (genotypes g4–g6). These mutants illustrate the existence of synergism between the X/A signal components and between the individual X/A signal components and the maternal Da product (recall that Sc and SisA form the activator complexes with maternal Da). X-linked gene products clearly do not act in a simple additive way. The model accounts for these observations since the / ratios corresponding to single mutants (g1–g3) are all considerably above one while the ratios of double mutant females fall below one.

Explanation for the effects of loss-of-function mutations at sc and sis-a can be drawn from the numerical simulations. In Fig 5C, the trajectories of females heterozygous for sc (g2), or sis-a (g1), or both (g4) are plotted in the (A, R) plane with the contour plot of function _rx in background. It is predicted that decreasing the dosage of X-linked genes "masculinizes" the X/A signal. Reduction in the number of gene doses of sc lowers the amount of Sc-Da complexes. Reduction in the number of gene doses of sis-a has the following two consequences: (i) it lowers the amount of SisA-Da activators and (ii) it lowers the amount of inactive SisA-Dpn complexes thereby raising the formation of Dpn-Dpn repressor. As seen in Fig 5C, the trajectory of mutant (g1) (green curve) and (g2) (blue curve) is shifted toward lower amounts of activator with respect to the wild-type female. In addition, the trajectory of mutant (g1) is shifted toward higher values of repressor due to the reduced formation of inactive complexes. Both trajectories are characterized by an incursion in quadrant IV that brings their / ratio above one. In mutant (g4) (red curve), a reduction of both sc and sis-a gene doses considerably amplifies the effects of single-dose reductions. Consequently, the trajectory shift toward quadrant I is dramatically amplified. As quadrant IV is no longer visited, the amount of early Sxl proteins remains insignificant and the splicing autoregulation is left in its off state. The resulting imbalance between X-linked and autosomal products leads to female lethality.

In the case of mutant females carrying half the amount of maternal Da products and heterozygous for either sis-a (g5) or sc (g6), reduction in viability can be explained by similar rationales. Note that the observed moderate lethality of these mutants qualitatively matches the fact that the corresponding / ratios get close to 1.

Effect on male viability of increasing the doses of X-linked components of the X/A signal: Males, like females, are able to buffer single-dose variations in the dosage of X/A signal genes without disrupting their viability. Males with a duplication of either sis-a (g10) or sc (g11) are fully viable. Numerical simulations of these genotypes give a value for the ratio / < 1. In contrast, combining an increase in gene doses for sis-a and sc (g12) is fully lethal. The corresponding increase of / matches the experimental results.

These mutant behaviors highlight the synergistic nature of the action of the X/A signal proteins on the promoter Sxl_Pe; the model accounts for them. Simulations show that increasing production of Sc and/or SisA proteins results in the formation of more activator complexes and the simultaneous formation of less repressor complexes (mediated by existence of sequestering complex SisA-Dpn). The X/A signals are thus displaced toward female-like compositions and males are feminized. Increases in both sc and sis-a doses are necessary, however, to ensure efficient activation of promoter Sxl_Pe and establish the autoregulatory function of Sxl.

Effects on male viability of lowering the doses of autosomal components of the X/A signal: Dpn is the single known repressor protein acting on the establishment promoter. Reductions in the dosage of dpn are thus intuitively expected to cause male-specific lethality. Experimental observations corroborate this prediction but stress at the same time the complexity underlying the action of repressor complexes on Sxl_Pe. It has been assessed that (i) males with one dose of dpn are fully viable and (ii) males with no dose of dpn are fully lethal although they display patchy expression of Sxl (YOUNGER-SHEPHERD et al. 1992 ). These results can be explained by analyzing the mutant trajectories. As shown in Fig 5D, reduction of dpn dosage has two consequences: it lowers the amount of repressor Dpn-Dpn complexes and inactive SisA-Dpn complexes and, as a result, increases the amount of activator SisA-Da complexes. The X/A signal is thus feminized and more early Sxl proteins are produced. For the heterozygous dpn mutants (g13) (green curve), the amount of early Sxl proteins, however, remains far below the threshold value and viability is ensured. The trend is much more pronounced for homozygous dpn mutants (g14) (blue curve) as no Dpn at all is produced. The corresponding trajectory is then confounded with the R = 0 axis. For visual convenience, the trajectory plotted in Fig 5D is obtained for a number of dpn doses equal to 0.2. Mutant (g14) extensively visits quadrant IV and produces a substantial amount of Sxl proteins. As a consequence, the / ratio increases and lethality of these mutants follows.

As sex determination is a cell-autonomous process in the somatic cells of Drosophila, the existence of patchy expression of Sxl reveals that the autoregulation of Sxl is not switched on in all the cells of mutant (g14). Mosaicism suggests that for these mutants the ratio / is closer to the threshold value 1.0 than to the simulated value 2.0. This phenomenon might be due to the existence of other repressor factors that are not taken into account in the present model.

Compensatory effects among the X-linked components of the X/A signal: In the previous paragraph, we investigated the consequences of lowering the amount of repressor in males. The effect of increasing the amounts of activators at a fixed amount of repressor in females is depicted in Fig 5E. Female mutants homozygous for sis-a (g7) are fully lethal. However, the viability of these females recovers by increasing doses of sc. Females with three doses (g8) or four doses (g9) of sc are characterized by a viability of 22 and 70%, respectively. Accordingly, the simulated / ratios increase. In Fig 5E, mutants (g7), (g8), and (g9) are compared to wild-type females by displaying their trajectories in the (A, R) plane. While mutant (g7) (gray curve) mainly resides within quadrant I, mutant (g8) (blue curve) visits quadrant III for a significant time and gets close to the borderline region with quadrant IV. Accordingly, mutants (g8) are characterized by a moderate percentage of viability. With respect to mutants (g7) and (g8), the trajectory of mutants (g9) (green curve) is further displaced from quadrant I to the borderline region between quadrant III and quadrant IV. The relatively high viability of mutants (g9) ensues. Thereupon, we conclude that the gradual recovery of female viability is underlain by a shift of trajectories from quadrant I toward quadrant IV at fixed amount of repressor. This shift accounts for the mutual compensatory effects of increasing dosage of either sc or sis-a and decreasing dosage of the other.

The simulated ratios for mutants (g8) and (g9) are above one. These values are obviously too high to explain the 20 and 70% viability. The discrepancy between simulation results and experimental observations may be due to disruptions in the non-sex-specific functions of the SisA protein in females homozygous for sis-a.

Sxl mutants: Females heterozygous for Sxl are fully viable, whereas females doubly heterozygous for Sxl and either sis-a or sc show very low viability. Conjointly, males that carry a duplication of Sxl but have a wild-type single dose of sis-a and sc are fully viable while those that carry a duplication of Sxl and either sis-a or sc are lethal. Within our modeling framework, the reduction of Sxl dosage by one in females is technically equivalent to the addition of one dose of sc and sis-a gene in males (mutant g12). As the simulated / ratio is equal to 1.24 for the male mutant (g12) and indicates a complete feminization, the model predicts that females with one dose of Sxl are fully viable. Similarly, males with a duplicated dose of Sxl can be modeled as females heterozygous for sis-a and sc (g4). As the simulated ratio / of mutant (g4) is equal to 0.15, the full viability of males carrying an extra dose of Sxl is inferred.

Following the same rationale, females with a reduced dose of Sxl and either sis-a or sc are technically equivalent to male mutants (g11) and (g10), respectively. As both these mutants have a simulated ratio / significantly below one, the lethality of the corresponding female mutants ensues. Males with an extra dose of Sxl and either sis-a or sc are equivalent to females having lost a dose of either sc (g2) or sis-a (g1), respectively. The simulated ratios / are both above one for these female mutants, which implies the lethality of the corresponding male mutants.

Triploid intersexes: Triploid intersexes present an interesting genotype with two X chromosomes and three sets of autosomes (BRIDGES 1921 ; CLINE 1983 ). These individuals have an X/A ratio of 0.67 and develop as sexually mosaic flies, formed partly by pure female cells that synthesize Sxl proteins and pure male cells that do not synthesize Sxl. This situation is caused by the ambiguous nature of the X/A signal that randomly switches on the autoregulatory feedback loop on Sxl in some cells, thereby initiating the female developmental pathway, while in the others it leaves Sxl autoregulation off and initiates the male developmental pathway.

The development of mosaic flies clearly illustrates the cell autonomy of sex determination in the somatic cells of Drosophila. The triploid intersex phenotype can be explained by postulating that its / ratio is close to the threshold value 1.0 with taking its values below and above the threshold as a consequence of intracellular fluctuations. This working hypothesis corroborates Cline's idea that the ambiguous X/A signal sets the triploid intersex cells on a threshold for stable Sxl activation (CLINE and MEYER 1996 ).

A thorough analysis of this working hypothesis is beyond the scope of a deterministic model. However, to test the plausibility of this hypothesis, the (2X;3A) genotype is numerically simulated as wild-type females with increased doses of dpn (the autosomal component of the X/A signal). The putative existence of unidentified autosomal elements suggests that the number of gene doses for dpn should be taken slightly >3 so that it includes the contribution of other repressors. In Fig 8, the evolution of the number of Sxl proteins is displayed for triploid intersexes with 3.0, 3.5, or 4.0 gene "doses" for dpn. The curve corresponding to 3 dpn doses falls in the basin of attraction of the non-nil steady state while that corresponding to 4 doses falls in the basin of attraction of state zero. It can be concluded that intermediate conditions might correspond to (2X;3A) triploid intersexes.

View larger version (19K): In this window In a new window Download PPT slide   Figure 8. Triploid intersex simulation. The solid curve corresponds to the trajectory of a mutant genotype with the dosage of dpn set equal to 3. The long-dashed and short-dashed curves correspond to trajectories with the number of dpn gene doses, respectively, equal to 3.5 and 4.0.

	DISCUSSION

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

In D. melanogaster, Sxl is the master regulatory gene that directs the processes of somatic sex determination and dosage compensation. Expression of Sxl protein is regulated in a two-step manner: it is first transcriptionally controlled by the X/A ratio signal during a short period around blastoderm stage and then post-transcriptionally controlled throughout the rest of development and adult life. The whole mechanism can be viewed in analogy to electric control circuits. It forms a binary switch that converts the X/A ratio of a cell into the all-or-none production of Sxl protein. The system regulatory logic is rather simple: it involves the transient triggering action of the X/A ratio signal (decision step) and the subsequent memorization of this action (maintenance step). For a long time, it has been understood that the positive autoregulatory loop controlling the splicing pattern of Sxl transcripts forms the switch "memory" device. Before being memorized, the state of the switch needs to be determined first. On the basis of current knowledge of the system, we used modeling to clarify how this decision is made—or more precisely at which molecular level the X/A ratio "digitization" (all-or-none response) occurs.

We developed a formal model for the sex-specific control of Sxl production. As any other model, the present one relies on simplifying assumptions and was constructed on the basis of the parsimony principle. To model the formation of the X/A signal we chose the most representative of each class of genes that form this polygenic signal. Although the X-linked genes run and sis-c also participate in the X/A signal, they were not included in the present model because they play a secondary role compared to sc and sis-a (CLINE 1988 ; TORRES and SANCHEZ 1989 , TORRES and SANCHEZ 1992 ; DUFFY and GERGEN 1991 ; SANCHEZ et al. 1994 , SANCHEZ et al. 1998 ; JINKS et al. 2000 ; SEFTON et al. 2000 ). The current knowledge suggests that sis-c and run operate as mechanisms that reinforce the effects of the two predominant X-linked genes. In addition, run operates in a region-specific manner: it has an effect restricted to the trunk region of the embryo whereas sc and sis-a are required for the activation of Sxl throughout the embryo. The product of sis-c is a secreted ligand that activates a JAK/STAT signal transduction cascade and indirectly promotes transcription from the establishment promoter (JINKS et al. 2000 ; SEFTON et al. 2000 ). Modeling the nonautonomous cellular aspect of the X/A signal assessment is beyond the scope of this work. Our model treats neither position-related effects (the case of run) nor cell-cell interactions (the case of sis-c) but it focuses on the description of one typical somatic cell.

Another aspect of Sxl regulation that was not considered in our model refers to the existence of a putative negative autoregulatory function of Sxl. The association of Sxl protein to the 3' untranslated region (UTR) of Sxl mRNA significantly decreases Sxl protein expression (YANOWITZ et al. 1999 ). It was speculated that this negative autoregulatory loop might play an important homeostatic role to maintain nontoxic levels of Sxl protein (MEISE et al. 1998 ; SACCONE et al. 1998 ; YANOWITZ et al. 1999 ). However, this negative loop appears to be not crucial for the decision-making process establishing the functional state of Sxl by the X/A signal. Its function might pertain to the later regulation of Sxl protein production after sex-specific splicing has been launched in females. This idea would be supported by the observation that the 3' UTRs of early Sxl mRNAs are shorter whereas those of the late Sxl mRNAs are larger (SALZ et al. 1989 ; SAMUELS et al. 1991 ). We surmise that this putative negative autoregulation of Sxl should not affect the model's conclusions as it would modify only the value of the threshold above which alternative splicing is launched.

The splicing mechanism exerted by Sxl proteins on primary transcripts originating at Sxl_Pm is oversimplified, although it includes the dimerization of Sxl—a key element of the process. Despite these simplifications, the model takes into account many of the known molecular details and features the hard core of the system's characteristics.

Our purpose has been to get deeper insight into the regulatory dynamics of the network. Model analysis together with numerical simulations has allowed us to untangle the roles of the system parts. In particular, the model clarifies the regulatory function of some of the actors of the primary signal and leads to four major conclusions:

1. The X/A ratio is signaled primarily by the relative amount of activator (Sc-Da and SisA-Da) and repressor (Dpn-Dpn) complexes. Activators and repressors of Sxl_Pe are not formed independently: their relative abundance is coupled through a titration scheme where the existence of sequestering complexes like SisA-Dpn lowers the availability of free proteins. The sequestration events involving the X/A signal components result in a first, but moderate, fivefold amplification of the difference between males and females. Our simulations corroborate the qualitative model proposed by PARKHURST and ISH-HOROWICZ (1992). This titration regulatory scheme, however, relies on the putative formation of the inactive complex SisA-Dpn. Though SisA and Dpn proteins were shown to interact (LIU and BELOTE 1995 ), further experiments are needed to support the formation of a stable heterodimer and confirm the regulatory importance of sequestration events.

2. The X/A signal is not at steady state when it controls early Sxl expression. X/A signal formation and activation of Sxl occur simultaneously and not sequentially. It is therefore important to keep a dynamical picture of these two entangled processes. This conclusion is experimentally supported by timed profiles of sis-a, sc, and Sxl transcription (ERICKSON and CLINE 1993 ).

3. Activation of the establishment promoter by the X/A signal does not depend on the absolute number of activator molecules or on the complete absence of repressor molecules; rather, it is a function of the relative amounts of activators and repressor. To ensure activation of Sxl_Pe, the X/A signal composition needs to be such that the relative concentration of repressor is "sufficiently" small and the relative concentration of activator is sufficiently high. The sex-specific composition of the X/A signal induces much more transcription at Sxl_Pe in females than in males. As a result, a 100-fold difference appears in the transient amount of early Sxl proteins expressed in the two sexes. Our simulation results are not compatible with the complete absence of early Sxl protein in males. This last point is particularly important as it conveys the idea that the sex determination decision is not made entirely through the action of the X/A signal on the establishment promoter, but also requires the action of the positive autoregulatory function of Sxl.

4. The RNA-splicing control of the late transcripts implements the switch behavior of the process. The establishment of Sxl autoregulation is determined by the amount of early Sxl proteins present before Sxl_Pm transcription is launched. When this amount is above a threshold value, autoregulation of Sxl is kick started. Consequently, males tolerate the production of early Sxl proteins up to a threshold concentration. Numerous experiments provided direct or indirect evidence for this requirement of a threshold concentration of early Sxl protein (CLINE 1980 ; KEYES et al. 1992 ; BERNSTEIN et al. 1995 ). Our results suggest that the positive feedback loop is crucial for both the establishment and the maintenance of the binary decision and that it exerts a triple function. It ensures the final conversion of the primary signal into an all-or-none response, confers robustness to the decision process (see the INTEGRATED MODEL section), and constitutes a memory device of the transient X/A ratio signal.

These conclusions emphasize that the binary decision results from the combined effects of three linked processes (Fig 7C): (i) formation of the X/A signal, (ii) activation of the establishment promoter as a function of the X/A signal composition, and (iii) determination of whether Sxl autoregulation is established on the basis of the amount of early Sxl proteins temporarily accumulated. The X/A ratio signal is transiently formed in both sexes but the activator/repressor ratio is higher in 2X;2A flies than in 1X;2A flies. Consequently, transcription is more induced in 2X;2A flies and accumulation of Sxl protein is important for these individuals only. This early burst of Sxl protein occurs around blastoderm stage. Afterward, when the maintenance promoter starts functioning constitutively, the residual abundance of Sxl in 2X;2A flies triggers the female-splicing pattern of the late transcripts, which ensures the permanent production of functional protein.

The model shows that the difference between females and males is amplified at each regulatory step. This picture reconciles the system complexity with the simple function it achieves, i.e., the conversion of a continuous signal (ratio of activators vs. repressors of the X/A signal) into a binary output (all-or-none production of Sxl protein from the late promoter). Given the existence of important random fluctuations in the concentration of molecular species, multiple steps where male-female differences are amplified might be an efficient strategy for securing robustness to decision-making systems.

To set transcription in motion at the establishment promoter, two "bolts" need to be simultaneously unfastened. These bolts represent requirements on the relative concentration of transcription activators and repressor (see above). Mutant analyses have pointed out the advantage of this dual-locking control. Although intuition tells us that male cells lacking repressor should be committed into the female phenotype, mutant males homozygous for dpn display only patchy patterns of Sxl expression (YOUNGER-SHEPHERD et al. 1992 ), which indicates that not all the embryo cells express Sxl protein. The absence of repressor is thus not sufficient to activate transcription in every cell. Our model accounts for this striking result: it shows that even though one transcription bolt is released when the amount of repressor is nil, the second bolt remains partly fastened as the amount of activators is not sufficiently high. As a result, Sxl transcription is partially impaired. On the other hand, mosaicism reveals that the amount of early Sxl proteins accumulated in some cells is high enough to switch on the autoregulatory loop on Sxl production. This might be due to intracellular fluctuations in the concentration of activators. The absence of repressor probably amplifies the effect of such fluctuations on transcription. Since mosaic flies are lethal, the use of activators alone would clearly not be an efficient strategy to control the production of early Sxl protein.

We speculate that the cooption of a repressor as a component of the X/A signal has been favored by selection to prevent accidental activation of the establishment promoter in males. The use of a dual control mechanism on transcription enables us to buffer fluctuations in the amount of repressor and activators without affecting wild-type male and wild-type female viability; its advantage is in providing robustness to the decision process, a property further enhanced by the coupling of the transcription bolts through the existence of sequestering complexes and competitive binding for regulatory sites at the establishment promoter. These conclusions provide a putative explanation for the coevolution of the molecular organization of the establishment promoter and the X/A signal. Strikingly, interspecies comparisons have revealed that the activator-binding sequences (E-boxes) and the repressor-binding sequences (D-box) as well as their locations in the establishment promoter are conserved in drosophilids (PENALVA et al. 1996 ; YANG et al. 2001 ). Significant conservation has been also demonstrated for the structure and function of the products of the X-linked genes sc (BOTELLA et al. 1996 ) and sis-a (ERICKSON and CLINE 1998 ).

The model developed herein gives an integrated and dynamic picture of Sxl protein production, the key process triggering sex determination and dosage compensation. It provides a theoretical framework that accounts for important experimental observations and corroborates most of the working hypotheses formulated so far. The model points out that certain types of behavior are incompatible with the structure of the system (e.g., the absence of activators in males and/or the absence of repressors in females and the establishment of the positive autoregulatory function of Sxl by a small number of early Sxl protein molecules). It also highlights the different roles of the positive feedback loop that is involved in the alternative splicing process.

The aim of this study has been to capture the key aspects of the regulation of Sxl expression with a minimum number of equations and parameters. Hence, only the core mechanisms of the process have been considered here. More specific aspects of the regulation of Sxl need to be addressed in further work, for instance, by taking into account the numerator genes run and sis-c when more data become available. Exploring whether the conclusions drawn for the regulation of Sxl are relevant to other developmental switches that involve the conversion of small quantitative differences in a signal into the all-or-none production of a gene product represents a challenging issue.

	FOOTNOTES

² Present address: Institute of Biotechnology, University of Helsinki, Helsinki, FIN-00014 Finland.

	ACKNOWLEDGMENTS

M.L. thanks Michael Ashburner, Benjamin Audit, Daniel Bopp, Dennis Bray, Jose Pereira-Leal, Denis Thieffry, and Juan Valcarcel for their support and precious help at various stages of this work. We also thank the referees for their helpful comments. L.S. is supported by grant PB98-0466 from Dirección General de Investigación Cientifica y Técnica, Ministerio de Educacion y Cienca. M.K. acknowledges financial support by the European Space Agency under contract no. 90042.

Manuscript received March 14, 2003; Accepted for publication August 5, 2003.

	APPENDIX A

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

FORMATION OF THE X/A SIGNAL
Kinetic equation of the model: The reaction scheme (1) is described by the following ordinary differential equation system,

(A1)

where the dot above variables denotes the time derivative. The steady-state solution of (A1) verifies the following conditions:

A little algebra leads to the steady-state values:

Initial conditions:
For Emc and Da, the initial conditions (ICs) are fixed so that the proteins remain in nonlimiting concentrations during the time the X/A signal is present. The production of Dpn, Sc, and SisA is supposed to start simultaneously at the same time (t = 0 sec in our simulations).

Parameter set:
For convenience, the system components are described in numbers of molecules enclosed in the nucleus instead of concentrations. Notwithstanding, on- and off-rate constants are given in molar units (M) to facilitate interpretation. The two unit systems are obviously equivalent but differ by a conversion factor (denoted as ).

Kinetic constants:
As mentioned in the Introduction, the model developed herein does not integrate the distinction between cell nucleus and cytoplasm; i.e., no transport between these two cell compartments was considered. Therefore, the system is assimilated as a cell nucleus whose shape just after cellularization can be approximated as an ellipsoid with long and short axis equal to 12 and 4–5 µm, respectively (LECUIT and WIESCHAUS 2000 ; T. LECUIT, personal communication). The system volume is chosen equal to 2.5 x 10^-13 liter. Let us denote this volume as V and the on-rates of a dimerization reaction as k^M_i when expressed in (mol·sec)^-1 and k^m_i when expressed in molecule per second. Dimensional analysis readily shows that , where N_A represents the Avogadro number. Given the system volume, we deduce that the conversion factor is equal to 1.5 x 10¹¹. The same relation exists for the production rates. In Table A11, the parameters used in the numerical simulations are given in molar and seconds.

Since no forward and backward rate constant has been measured for the protein complexes forming the X/A signal (except for Dpn), the parameter choice was based on the following considerations: (i) dimerization of proteins is characterized by on-rates that are typically in the range of 0.5–5 x 10⁶ M^-1 sec^-1 (NORTHRUP and ERICKSON 1992 ); (ii) real-time kinetic measurements for dimerization of the transcription factor NF-B p65 and the basal transcription factor TBP have shown an on-rate constant of 2.3 x 10⁶ M^-1 sec^-1, whereas the off-rate has been estimated to be 8 x 10^-4 sec^-1 (PAAL et al. 1997 ); and (iii) C-Jun and C-Fos (bZIP proteins) heterodimerize with a dissociation constant K_d of 1 nM (HEUER et al. 1996 ). Thereon, we assume that typical dissociation constants for bHLH and bZIP proteins lie in the range of 0.1–10 nM. It should be emphasized that the conclusions drawn in the text remain valid when most of the parameter values are considerably varied (results not shown). Further information about parameter values is given as comments in Table A11.

Production fluxes:
Production rates of Sc, SisA, and Dpn proteins are important parameters of the ODE system. To our knowledge, no measure has been done for these parameters. Fixing their values is not simple as many processes, such as transcription and translation, passive and active transport from cytoplasm to nucleus, RNA degradation, etc., condition import of proteins into the nucleus. As mentioned above, the model does not integrate the distinction between cell nucleus and cytoplasm. Here, we show how the influx of the protein Sc can be estimated in the oversimplifying situation where transcription and translation would occur in the same compartment. The main assumption is that the amount of proteins is proportional to the amount of mRNAs. Genes have generally several elongating polymerases on them at the same time. In eukaryotes, the center-to-center spacing of polymerases is typically 100 nucleotides (KENNELL and RIEZMAN 1977 ; ALBERTS et al. 1994 ). Given that the transcription unit of gene sc is 1200 bp long, 12 polymerases are assumed to transcribe a single gene simultaneously. Transcription rates are typically in the order of 50 bp sec^-1 in eukaryotes (HARGROVE et al. 1991 ; JACKSON et al. 2000 ). On average, the polymerase machinery is expected to complete a transcription round of sc in 24 sec. Therefore, a rough but fairly high estimation of sc mRNA production rate is 30 molecules min^-1 gene^-1. Let us denote it as F_ms. If we assume that sc mRNA is degraded following a first-order kinetics with a fast decay rate d_ms of 0.002 sec^-1 (which corresponds to a half-life of 5 min), the steady-state number of sc mRNAs (denoted as ms^st) is estimated to be F_ms/d_ms = 250. In eukaryotes, the translation machinery is made up of ribosomes spaced by ±100 amino acids (aa) on the mRNA (KENNELL and RIEZMAN 1977 ; ALBERTS et al. 1994 ); the average translation rate is 2 aa/sec (ALBERTS et al. 1994 ; LEWIN 1997 ; JACKSON et al. 2000 ). Since Sc protein is 350 aa long, three ribosomes are expected to process a single mRNA molecule at the same time and the apparent translation rate v_tsl is estimated to be 1 molecule min^-1 mRNA^-1. It can be thus inferred that the production flux of Sc proteins is approximately ms^st x v_tsl = 4–5 proteins sec^-1. Production of Sc and SisA proteins is modeled as follows: they are rapidly switched on to a constant value, remain unchanged for 40 min, and are then switched off according to the observed expression window (see Table A22). The same scheme is applied to Dpn except that the production rate remains on for 1 hr before it is switched off.

Protein degradation rates:
Degradation of the components that form the X/A signal is assumed to be appropriately described by a first-order reaction. It has been shown that the amounts of sc and sis-a mRNAs and proteins abruptly increase during nuclear cycle 12 and subsequently decrease so that by early cycle 14, very little mRNAs of both can be detected (ERICKSON and CLINE 1993 , ERICKSON and CLINE 1998 ; DESHPANDE et al. 1995 and Table A22). Similarly, gene dpn has been shown to be substantially expressed from cycle 12 to cycle 14; after that it undergoes quick turnover (BIER et al. 1992 ; EMERY and BIER 1995 ). Numerical simulations show that this phenomenology cannot be accounted for by means of constant degradation rates. They suggest that the degradation processes are biphasic. Hence, degradation processes are assumed to be moderately slow initially (i.e., until cycle 14) and significantly increased during a subsequent period (i.e., after cycle 14). This hypothesis is supported by recent experimental studies that have pointed out the existence of two machineries responsible for RNA degradation in Drosophila. One degradation machinery is maternal and the other zygotic (BASHIRULLAH et al. 1999 ). In these studies, slow degradation has been observed during the first 2 hr after fertilization when only the maternal degradation pathway is functioning. Degradation then becomes significantly faster when the zygotic pathway acts cooperatively with the maternal pathway. Therefore, it has been assumed that faster degradation for the Sc and SisA proteins started after 2400 sec and for Dpn protein after 3600 sec.

	APPENDIX B

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

ACTIVATION OF Sxl BY THE X/A SIGNAL
Competitive binding for the D-boxes and E-box 1 (downstream domain):
On the basis of the model assumptions, the probability transition matrix of the master Equation 4 can be constructed,

where A and R denote the numbers of activator and repressor molecules, respectively. Since the formation of the X/A signal evolves on a timescale slower than the dynamics of the interactions between the promoter and the regulatory factors (see text body), the time dependence of A and R is omitted in this Appendix A (though A and R change over the time). The steady-state distribution vector ^st of (4) verifies the following relation:

(B1)

As seen in (B1), ^st is the normalized eigenvector of corresponding to the eigenvalue zero. Some algebraic manipulations lead to

Multiple E-box domain:
The subsystem E-box 2–7 is made up of six independent E-boxes. The master equation for this domain is

wherein the transition probability vector is the seven-component vector,

with C_i referring to the ith class of equivalence defined in the text. is the (7 x 7) probability transition matrix

As the steady-state probability distribution vector is the eigenvector of corresponding to the eigenvalue zero, ^st can be calculated as

(B3)

where the ⁿC_i denote binomial coefficients.

Determination of parameter k_t:
In the absence of experimental measures, the numerical value of the parameter k_t in Equation 8 is adjusted so that the function _rx(A, R) remains in the order of magnitude of realistic transcription rate. We denote as F_h the number of Sxl transcripts constitutively produced by the maintenance promoter. By consistency, k_t is chosen so that

Numerical simulations drove our choice of setting k_t = 0.75.

	APPENDIX C

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

ESTABLISHMENT OF Sxl AUTOREGULATION
Kinetic equations of the model:
From the reaction scheme (9a–9d), the following differential equation system can be derived by application of the law of mass action:

(C1)

It is assumed that degradation of the protein x, the mRNA rx, and the primary transcripts h follows first-order reactions. In contrast, the degradation of h.Sxl and h.Sxl₂ is not considered. The steady-state solution of this ODE system verifies the following conditions:

(C2)

Algebraic manipulations of (C2) yield the third-degree polynomial in variable x:

(C3)

The steady-state values of Sxl are given by the solutions of

The existence of solution x₀ is guaranteed for any parameter values. To ensure that both x₊ and x_- are real positive, the parameter set of system (C1) must verify the following necessary and sufficient condition: (_x.F_h/d_rx · _tsl/d_x -d_h/k₆) 2. The steady states defined by x₀, x_-, and x₊ are points in a five-dimensional space that can be unequivocally computed from system (C2). As indicated in the text body, these states are denoted as z₀, z_-, and z₊:

A linearized stability analysis shows that z₀ and z₊ correspond to stable steady states while z_- is an unstable steady state.

Parameter set:
The parameter set used to numerically integrate system (C1) is presented in Table C1. Parameter values are extrapolated from semiquantitative data found in the literature as explained in Table C1 comments.

Table C1. Set of parameter values used in numerical simulations. Establishment of Sxl autoregulation Kinetic constant Value Comments k6 l6 k7 l7 6.66 x 10-7 molecule-1 sec-1 10-2 sec-1 1.33 x 10-6 molecule-1 sec-1 10-3 sec-1 The dissociation constants corresponding to the binding of Sxl monomers to Sxl transcripts and the binding of Sxl monomers to the complex Sxl::transcript are K6 = l6/k6 = 10-7 M and K7 = 5 x 10-9 M, respectively. These values were based on the following experimental facts: 1. Very tight binding between ribonucleoproteins and RNA is characterized by dissociation constants that are >10-9 M. For weak binding, dissociation constants are typically in the order of 10-6 M ( LAIRD-OFFRINGA and BELASCO 1995 ; VARANI and NAGAI 1998 ; KATSAMBA et al. 2001 ). 2. Experiments of Wang and Bell have shown that a 15-fold difference exists between the binding of Sxl to a single RNA-binding site and two RNA-binding sites ( WANG and BELL 1994 ). This difference strongly suggests the existence of cooperative effects. 3. Dissociation constants for the binding of Sxl-derived proteins with polypyrimidine RNA sequences are in the order of 10-8–10-10 M ( KANAAR et al. 1995 ). 4. The off-rates li of the complexes hx and hx2 are apparently larger than 15 min-1 (J. VALCARCEL, personal communication). By combining points 1, 2, and 4, we infer that the on-rates have an order of magnitude of 10-5–10-6 (M sec)-1. This qualitatively agrees with the rates measured for the spliceosomal protein U1A ( LAIRD-OFFRINGA and BELASCO 1995 ; KATSAMBA et al. 2001 ). The 20-fold difference between K6 and K7 is justified by consideration no. 2. k8 1 sec-1 In absence of experimental data, the splicing rate was tentatively set equal to 1 event per hx2 complex and per second. The value of this parameter poorly influences the system dynamics. tsl 0.03 molecule-1 sec-1 RNA-1 This estimation of the translation rate tsl is based on the fact that (i) Sxl protein sequence is 350 aa long; (ii) average translation rate is 2 aa/sec; and (iii) simultaneous translation rounds are achieved by ribosomes spaced by 80 aa ( ALBERTS et al. 1994 ; LEWIN 1997 ). Fh 0.06 molecule-1 sec-1 Influx per X chromosome corresponding to the production of 4 transcript molecules/min and per Sxl gene. This estimation is based on the following data and assumptions: (i) Sxl transcription unit length is 14,000 bp; (ii) transcription rate/chain elongation rate of the polymerase machinery is 50 bp sec-1; (iii) one gene is processed by 40 polymerase complexes at one time ( ALBERTS et al. 1994 ; LEWIN 1997 ). x = 1 for males = 2 for females No. of doses of Sxl gene. drx 5 x 10-4 sec-1 rx1/2 25 min In higher eukaryotes the half-life of mRNAs can vary from minutes to hours ( ALBERTS et al. 1994 ; LEWIN 1997 ). The quick turnover of Sxl mRNA that occurs 30 min after its production necessitates its half-life to be short. The mRNAs of C-Fos, C-Myc, and Pgk1 are subject to fairly fast degradation and have an estimated half-life of 30 min ( HARGROVE and SCHMIDT 1989 ; CAO and PARKER 2001 ). We fixed the value of rx1/2 by assuming that these half-lives are representative for Sxl mRNA. dx 10-4 sec-1 x1/2 120 min mRNA molecules are usually more labile than proteins. Here, Sxl protein is supposed to be four to five times more stable than its mRNAs. This value ensures that the amount of Sxl proteins in females before SxlPm constitutive transcription is sufficient to catalyze female splicing whereas early Sxl mRNAs are virtually absent. dh 5 x 10-3 sec-1 h1/2 2.5 min It is thought that very little of the primary transcripts leaves the nucleus for the cytoplasm and half of them are potentially degraded before producing mRNA ( ALBERTS et al. 1994 ). Therefore, the pre-RNAs are supposed to undergo a fast nuclear degradation. We (arbitrarily) fixed Sxl pre-mRNA half-life such that it is 10 times smaller than its RNA half-life.

	LITERATURE CITED

TOP ABSTRACT MODELS INTEGRATED MODEL DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

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