epistasis and pleiotropy as natural properties of transcriptional regulation

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mTheoretical Population Biology � 1256

theoretical population biology 49, 58�89 (1996)

Epistasis and Pleiotropy as Natural Propertiesof Transcriptional Regulation

Greg Gibson

Department of Biology, Natural Science Building, University of Michigan,Ann Arbor, Michigan 48109-1048

Received January 20, 1995

A statistical thermodynamic model of transcriptional regulation is employed toinvestigate the likely effects of genetic variation on the stabilization of gene expres-sion. The model is tailored to empirical data on the control of transcription of thehunchback gene by the morphogen Bicoid during Drosophila embryogenesis.Variable parameters include the number of binding sites for activator protein andthe DNA�protein and protein�protein cooperative binding energies. Recursions areperformed to derive transcriptional response curves over a range of concentrationsof activator. Sigmoidal responses are indicative of threshold-dependent activation ofgene expression, and the effects of variation of the parameters on the width andlocation of the threshold are considered. It is shown that there is a minimumthreshold width (maximum switch sensitivity) that is a function of the number ofbinding sites and the level of response desired, but independent of the bindingenergies. This places a constraint on the evolution of sensitive genetic switches thatgenerate discrete cell types. Inevitable trade-offs between threshold widths and loca-tions for multiple target genes of most transcriptional activators are found to occur.These naturally lead to epistatic and pleiotropic effects, and may favor the genera-tion of networks of compensatory mutations that together produce homeostaticdevelopmental pathways. � 1996 Academic Press, Inc.

Introduction

The elucidation of the molecular basis for quantitative genetic variationremains a major issue in contemporary biology. Classical populationgenetic theory was built on the assumption of primarily additive geneticvariance. While many of the important ideas have been confirmed and rein-terpreted in the light of empirical studies of metabolic control (e.g., Wright,1931; Kacser and Burns, 1973), they remain to be integrated with ourknowledge of morphological variation. In recent years, the identification ofthe key components of developmental genetic pathways has revolutionizedour understanding of how body plans are put together (Lawrence, 1992).Comparative analyses of the embryonic expression of such genes in diverse

article no. 0003

580040-5809�96 �18.00Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

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taxa are beginning to demonstrate how developmental mechanisms evolve(Patel, 1994). However, we remain a long way from being able to relatestanding levels of genetic variation in natural populations to themechanistic basis of homeostasis within species and the morphologicaldivergence between them. In addition to finding estimates for the numberof loci, the manner in which they interact, and the distribution of alleliceffects, there is a need to understand how variation at the molecular levelis translated into the observed phenotypic variation. Only then will it betruly possible to understand the balance of evolutionary forces thatproduce the morphological distributions observed in nature (Clark, 1991).

Clearly both empirical and experimental approaches are needed. Theadvent of rapid, and statistically sophisticated, methods of interval map-ping (Lander and Botstein, 1989; Zeng, 1994; Jansen and Stam, 1994) andfor the association of specific molecular polymorphisms with phenotypicvariation (Sing et al., 1992; Lai et al., 1994), has begun to allow theattribution of quantitative effects to defined genes (Tanksley, 1993). Inmany cases, these genes will be found to lie in genetically well-characterizedpathways of, for example, signal transduction and transcriptional regula-tion. The properties of regulatory pathways are just beginning to beexplored by theoreticians (McAdams and Shapiro, 1995; Ackers et al.,1982; Kerszberg and Changeux, 1994), and already major differences fromthe organization of substrate-modification enzyme pathways are indicated.The consequences of such differences for the expected distribution of quan-titative genetic variation have not been explicitly considered, beyond ageneral recognition that epistasis and pleiotropy may be common featuresof developmental genetic architecture (Barton and Turelli, 1989; Gavriletsand deJong, 1993). Here, I suggest an approach to addressing this deficitby considering the likely effects of genetic variation in the context of amodel for the setting of a transcriptional switch.

The particular genetic pathway considered is the threshold-dependentactivation of hunchback (hb) gene transcription by Bicoid (Bcd) protein inthe early Drosophila embryo. This genetic switch has been studied exten-sively studied at the molecular level (Driever et al., 1988a, 1988b, 1989;Struhl et al., 1989). It resembles in several respects the threshold-dependentactivation of repressor gene expression in the Escherichia coli lambda phage(Ptashne, 1992). After demonstrating that a mathematical model (Ackers etal., 1982) that successfully describes the operation of this lysogeny�lysisswitch can be adapted to describe the Bcd-hb interaction, I consider theeffect of subtly altering parameters of the model (protein�DNA bindingenergy; protein�protein cooperative interaction; number of binding sitesin the hb promoter; and Bcd concentration) in ways expected of pointmutations in the system. The analysis reveals that (i) the sharpness of athreshold-dependent switch is a function of the number of binding sites, but

59GENETIC VARIATION AND TRANSCRIPTION

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has a practical limit that can be considered by analogy with the steady-state level of flux predicted to evolve in metabolic pathways (Clark, 1991);(ii) slight variation in biophysical parameters can produce additive effectson the sharpness and location of the threshold, although the details of theresponse depend on the concentration of the activator protein; (iii) conse-quently, if multiple target genes are regulated at different concentrationsof the activator, the overall response of the system will be complex,demonstrating unavoidable pleiotropy and epistasis; and (iv) distinct com-binations of parameters that produce identical sharp responses when homo-zygous will produce different, more gradual responses when heterozygous,suggesting that underdominance may also be prevalent in transcriptionalregulation. These findings are considered in relation to the homeostasis ofdevelopmental systems.

The Model

Threshold-Dependent Activation of Transcription

A statistical thermodynamic model that accounts for many of thephysiological properties of the lysis�lysogeny switch of bacterial phagelambda has been described (Ackers et al., 1982) and is modified here toresemble a more complex eukaryotic switch. In the original model, thelambda repressor binds cooperatively to three DNA sites separating twodivergent promoters (Ptashne, 1992). Cooperativity refers to the increasedprobability of binding to a second (or third) site once the first (or second)site has been bound and is due to protein�protein interactions that reducethe free energy of binding in higher order complexes. The thermodynamicaspect of the model consists of mathematical expressions that define theprobability that the operator is in any particular configuration (from allsites free, to all sites bound) at a given concentration of repressor. Theinput parameters can in some cases be derived from genetic and biochemi-cal measurement of binding kinetics. The probability that either promoteris active is a simple summation of the predicted activity of the promotersfor each configuration multiplied by the probability of that configuration.Thus, when the right-hand-most binding site is occupied, transcriptionfrom the right-hand promoter is repressed; otherwise it is active. Conse-quently, the overall level of activity is the combined probability that theright-hand site is free. The model clearly demonstrates how cooperativebinding is necessary to achieve threshold-dependent switches in gene trans-cription as the concentration of repressor changes gradually (Ackers et al.,1982).

The Bicoid protein acts as a morphogen in early Drosophila embryo-genesis by activating transcription of a handful of target genes in a

60 GREG GIBSON

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threshold-dependent manner (Driever and Nu� sslein-Volhard, 1988b). Theconcentration of Bicoid declines logarithmically from a peak at the anteriorpole, due to diffusion from a localized source of mRNA translation(Driever and Nu� sslein-Volhard, 1988a). The best-characterized target geneis hunchback, which itself encodes a concentration-dependent transcriptionfactor responsible for dividing the embryo into segmental regions (seeHu� lskamp and Tautz, 1991, for a review). There are six known bindingsites for Bicoid in a segment of the hunchback promoter that allows foractivation of transcription of the gene over a relatively narrow change inBicoid concentration in the middle-region of the embryo (Driever et al.,1989). Increasing the dosage of the bicoid gene causes a posterior-shift inthe location of the hunchback threshold, as expected of an increase inBicoid protein (Driever and Nu� sslein-Volhard, 1988b). Targeted disruptionof binding sites as assayed in transgenic flies also disturbs the transcrip-tional response in several ways (Driever et al., 1989; Struhl et al., 1989): (i)it becomes less steep, reflecting loss of cooperativity; (ii) the overall level oftranscription is reduced, indicating that each bound Bicoid molecule con-tributes to transcriptional activation; and (iii) the location of the thresholdshifts anterior, since a higher concentration of Bicoid is required to achievesimilar levels of transcription.

Each of these effects can be mimicked by modifications of the lambdarepressor model. It should be emphasized that this study is not an attemptto precisely describe the Bicoid�hunchback interaction. First, the precedingproperties have only been determined semi-quantitatively, so that there isno way of deriving a meaningful ``goodness-of-fit'' measure for corre-spondence between the mathematical simulations and immunohistochemicaldata. Second, the wild-type hunchback promoter is undoubtedly more com-plex than the minimal derivative considered here: blocks of conservedsequence extend beyond the core region (Lukowitz et al., 1994) and higher-order protein�protein�DNA interactions have been implicated by recentstudies (Simpson-Brose et al., 1994). Third, the heuristic purpose of thisstudy is to develop a general framework for considering the populationgenetics of transcriptional switches that can be refined by later work,and�or adapted to describe similar switches (for example the Dorsal�snailinteraction that helps to establish the dorsal�ventral embryonic axis inDrosophila, Ip et al., 1992). The extent to which the conclusions presentedhere are limited by the simplifying assumptions of the model will only beclarified by further modeling in conjunction with more quantitative empiri-cal work.

A Modified Statistical Thermodynamic Model

There are three essentially independent steps in the modeling process asoutlined in the flow chart presented as Fig. 1: determining the binding free


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Fig. 1. Flow diagram of the modeling strategy. The transcription response from ahypothetical promoter with up to six binding sites, one of which may have different propertiesfrom the other, is calculated in three steps. DNA�protein and protein�protein binding freeenergies are entered into the fractional occupancy equations, which are calculated over arange of activator concentrations. This gives estimates of the proportion of time the promoterhas 0, 1, 2, 3, 4, 5, or 6 sites bound. The response curve is then calculated by summing thecontribution of each possible configuration by multiplying the number of ways each equiv-alent configuration occurs by the fractional response for that number of bound sites, andsumming. Output parameters such as threshold location and width, and response at a certainactivator concentration, can be read off the curve.

energy of each different promoter configuration, then estimating the frac-tional occupancy of each promoter configuration over a range of Bicoidconcentrations, and finally, relating fractional occupancies to the level oftranscription. The output is a response curve presenting the relative level oftarget gene transcription at a given concentration of activator as showndiagrammatically in Fig. 2. Alternatively, the response can be describedrelative to position along the anterior�posterior axis of the egg, to accountfor effects of change in dosage of the activator gene.

For a promoter with six equivalent binding sites, there are 64 possiblepromoter configurations. These can be reduced to just seven promoter``states'' if it is assumed that each configuration with the same number of

62 GREG GIBSON


Fig. 2. Model for the generation of a threshold response by cooperativeness. (a) Activatormonomers are able to bind to each of six sites with affinity u or v, with the binding of eachsite enhanced by protein�protein interactions between monomers of affinity w or W. Theassembled complex then interacts with the assembled transcription initiation complex (by asyet biochemically undefined mechanisms) to increase the rate of RNA polymerization. (b) Thetranscriptional response is scaled from 0 to 1, and arbitrary low and high thresholds aredefined (in this case, 0.25 and 0.75). The concentrations of activator that give these responsesare read off the curve, and their ratio determines the threshold width. In the text it is arguedthat the sharpness of the threshold, expressed as the minimal width, and its location, mayevolve to become relatively insensitive to genetic variation.


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occupied sites is thermodynamically and transcriptionally equivalent. Thestates ``no-sites occupied'' and ``6-sites occupied'' each have one configura-tion; ``1-'' and ``5-site'' states have 6 configurations; ``2-'' and ``4-site'' stateshave 15 configurations; and the ``3-sites occupied'' state has 20 possibleconfigurations as listed in Table I. The binding free energy for each con-figuration within a state is given by the expression:

2Gc=ks } u+as } w, (1)

where ks is the number of sites bound in the state�configuration, u is aparameter representing the DNA-binding affinity for each site consideredalone, and w is a parameter describing the protein�protein interaction(cooperative) affinity, which is multiplied by a modifier, as , that is specificfor each state. In the simplest case, as=ks&1 adds one unit of cooperativebinding for each additional site bound, but more realistically as increasesnon-additively as the promoter complex assembles, as discussed below.Also, in reality as would take on a slightly different value for each con-figuration within a state, due to effects of spacing and proximity to theTATA box on the interaction potential, but there is no obvious way to

TABLE I

Promoter Configuration States for Six Equivalent Binding Sites

Number of Number of ResponseState sites bound configurations Free energy value

ks ns 2Gs rs

0 0 1 0 0A 1 6 u 0.5C 2 15 2u+w 0.6E 3 20 3u+2w 0.7G 4 15 4u+4w 0.8I 5 6 5u+6w 0.9K 6 1 6u+8w 1.0

Note. The number of configurations, ns , is the number of ways in which ks of the six sitescan be bound to produce each state. The free energy is assumed to be a sum of DNA�proteinand protein�protein components and to be equivalent for each configuration within a state(but see discussion in the text). The response multiplier, rs , scales the output to give anincrease in the overall response as the number of binding sites increases. The modified recur-sions for varying the sixth site are indicated in Appendix 1. Note that there are undoubtedlynumerous combinations of parameters and scaler expressions (4) and (5) that will give asgood, if not better, fits to the data. No attempt was made to simultaneously optimize the func-tions. The response function was chosen first, then the cooperativity function. The set ofvalues indicated in the text simply establishes a base for comparison of the effects of subtlemodification of one or more parameters.

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incorporate such effects and to attempt to do so would introduce unne-cessary complexity.

The probability that a promoter exists in each configuration (that is, thefractional occupancy, fc) is determined by the expression

fc=[x]ks exp(&2Gc�RT )

�c [x]kc exp(&2Gc�RT ), (2)

where [x] is the concentration of activator protein, R and T are thefamiliar gas constant and temperature in degrees Kelvin, respectively, and2Gc is the free energy of binding per configuration from (1). The expressionin the numerator is the statistical thermodynamic derivation of promoteroccupancy for a configuration (Hill, 1960; Ackers et al., 1982) and takes ona value of unity for ``no-sites bound'' (k0=0 implies [x]0 e0=1). The sum-mation in the denominator is over all 64 configurations and is performedsimply by multiplying the expression for each state by the number of con-figurations in the state and summing.

The overall transcriptional response at a given concentration of activatoris estimated by multiplying each fractional occupancy by a response valueand summing over each configuration:

response=:c

rs } fc , (3)

where rs is a scaler that defines the relationship between the promoteroccupancy and the level of transcription. In the simplest case, rs increaseslinearly with the number of sites bound, up to a maximum value of 1,so that the total response scales over the range 0 to 1. The summationis again over each of the 64 configurations, with the simplifying assump-tion that each configuration within a state is equivalent with respect totranscriptional response; hence rs replaces rc . (Note that our modeldeparts from the lambda case where the response function representedrepression through occupancy of a single site, not activation throughmultiple sites.)

Basic Properties of the Model

Using these three sets of equations, a simple program was written in theC language (Think CTM software; Symantec) and run on a Macintoshcomputer to derive transcriptional response curves over a range of Bicoidconcentrations in the early embryo. The concentration of activator, [x],was varied over the physiologically reasonable range of 10&12 to 10&10 M(cf. Krause et al., 1988), while u and w took on values of 4.0_104 to5.4_104 and 0 to 1.4_104 J�mol, respectively. These values were derived



from binding energies determined by in vitro measurements for Anten-napedia-class homeodomains (equilibrium binding constant KD=10&9�10&10 M: Affolter et al., 1990; Goutte and Johnson, 1993). Appropriateadjustment of the parameters, as well as running similar recursions for justthree, four, or five binding sites, creates response curves that reproduceeach of the qualitative observations of Driever et al. (1989) and Struhl etal. (1989) concerning the effect of Bicoid concentration and hunchbackpromoter sequence arrangements on the transcriptional response thresholdas shown in Fig. 3. Since the real binding sites do not actually have equiv-alent affinities and are to some extent context-dependent with regard totheir effect on transcription, the empirical data cannot be directly com-pared with the simulation. Of most importance, though, is the result thatremoval of one strong and one weak binding site shifts the location of thethreshold by 100 egg-length (comparing constructs pThb1 and pThb2 inTable I and Fig. 2 of Driever et al., 1989, with five and six strong bindingsites in Fig. 3 here), reduces the overall transcription up to 500 and hasa perceptible effect on the width of the threshold.

Fig. 3. Haploid response curves for different promoter configurations. Simulatedresponse curves are drawn for promoters with 3, 4, 5, or 6 equivalent binding sites, withu=5_104 J�mol, and w=1_104 J�mol. With these values the 6-site threshold places at 500egg length at 25 %C for the indicated activator concentration range. Values for the functionsrs and as are from Table I. The shape of the activator (Bicoid) gradient is indicated by thedotted line, which Driever and Nu� sslein-Volhard (1988a) showed to be log-linear��althoughwe note that they did not estimate the concentration of Bicoid protein in moles. As thenumber of binding sites increases, the threshold shifts to the right and sharpens, while themaximum response observed at the anterior pole increases.

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The overall decrease in transcription observed as the number of bindingsites is reduced, is most easily accounted for in our model by adjustmentof the vector of rs values that relates response to the number of sites boundin each configuration of a state. An additive relationship between thenumber of sites bound and the transcriptional response can be representedby the function:

rs(0)=0(4)

rs(ks)=:+; } (ks&1).

Parameter values :=0.5 and ;=0.1 produce the rs values listed in Table I,which in turn provide the curves shown in Fig. 3. Since there is currentlyno way of measuring these parameters, they provide a potentially largesource of error in the analysis. Consequently, a number of simulations wererun with a range of functions that include multiplicative and additiveincreases in transcription as occupancy increases (data not shown). It wasfound that the form of this vector has remarkably little effect on the loca-tion and width of the transcription threshold, both of which are dominatedby the level of cooperativity. Actually, perhaps counterintuitively, thenarrowest thresholds are achieved when the maximum transcriptional out-put is independent of occupancy, that is, when having one site boundactivates transcription as strongly as when six sites are bound.

By contrast, effects on the width of the threshold are highly dependenton the increase in cooperativity due to each additional activator molecule,as represented by the distribution of as values in Eq. (1). In those few caseswhere quantitative measurements have been made, the free energy ofprotein�protein interactions has been of the order of one quarter of the freeenergy of protein�DNA interactions (Ackers et al., 1982; Ptashne, 1992;Vershon and Johnson, 1993). With that constraint, threshold widths asnarrow as that produced by Bicoid's activation of hunchback, cannot beproduced if cooperative interactions merely increase linearly with eachextra site bound. That is, a three-fold increase in hunchback transcriptionover just 50 egg length (a 1.5-fold increase in Bicoid concentration,Driever and Nu� sslein-Volhard, 1988a) requires that cooperativity increaseat a greater than additive rate, as modeled by the functions:

as(0)=0(5)

as(ks)=#+=ks".

A linear component could be built into expression (5), but it was notrequired here; arbitrarily adjusting the parameter values to #=&1; ==0.7,and "=1.42 provides as=(0, 0, 1, 2, 4, 6, 8) for ks=(0, 1, 2, 3, 4, 5, 6) and


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produces thresholds of the desired width. This particular function producesa nearly bilinear increase in cooperativity, with greater per-moleculecooperativity being provided by complexes with more than three boundactivator molecules.

A thermodynamic justification for this adjustment may be sought in theconsideration that cooperativity can be provided by protein�protein inter-actions mediated by a domain of each activator molecule that is distinctfrom the DNA-binding domain, and is thus available to interact with freemolecules in solution. Transient interactions would increase the local con-centration of free activator, increasing the likelihood of a favorable encoun-ter of the free molecules with vacant DNA sites. In other words, once anucleating complex above a critical size has been formed, protein�proteininteractions may have an increased effect on further growth of the trans-cription complex. Alternatively, cooperativity alone may not account forthe sharpness of the hunchback threshold, and as yet unidentified geneticinteractions, including chromatin effects and mRNA sequestration, couldcontribute to the unexpected sharpness of the threshold. Clearly moresophisticated modeling combined with experimental manipulation mightelucidate the details of threshold function. The conclusions that followshould be judged with this caveat on the assumptions of the model in mind.

Analytical Determinants of Threshold Width

In a developing embryo, one of the most crucial parameters of a geneticswitch is the ``threshold width,'' which can be defined as the ratio of thetwo concentrations of activator required to produce a specified activationof a target gene. The analytical proof given in Appendix 2 gives the sur-prising result that the minimum threshold width (that is, the most sensitiveswitch) is determined simply by the number of binding sites for activator,irrespective of cooperative interaction energies. Simulations confirm thatafter a certain point, greater cooperativity merely shifts the location of thethreshold without making it any more narrow. More precisely,

mimimum threshold width= k- (tmax �tmin), (6)

where k is the number of binding sites, and tmax and tmin are values satis-fying the relation ``fractional response=(t�(1+t))'' for the upper and lowerlimits of the target gene activation. For example, for a gene activated from250 to 750 of maximal activity, tmin=1�3 (since 0.25=0.33�(1+0.33))and tmax=3 (since 0.75=3�(1+3)). In this example, a target gene with twobinding sites would require at least a three-fold (3=2

- (3�0.33)) increase inactivator concentration, whereas one with six binding sites would onlyrequire a 1.44-fold increase in activator (1.44=6

- (3�0.33)). Notably,

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increasing the number of binding sites still further has little effect ondecreasing the threshold width, as shown in Fig. 4a. Thus, the sharpesttranscriptional responses produced by cooperative DNA-binding are likelyto involve at least five or six binding sites for activator proteins, with addi-tional binding sites either contributing redundantly to the narrowness ofthe threshold, or providing a different aspect of regulation (for example,input from an alternate regulatory pathway). The generality of this findingwill depend to some extent on the assumptions of the model, in particularthe equivalence of DNA�protein binding affinities for all of the sites on anygiven promoter.

A corollary of the formulation is that the threshold width is also a func-tion of the concentration range over which the response is measured,relative to the maximal activation. Consider, for example, a three-foldincrease from 300 to 900 gene activity. In this case, (tmax �tmin)=9�0.43=21, which requires a 1.66-fold increase in activator concentrationfor a promoter with six sites, more than for the threshold at a lower levelof activation. Alternatively, for an increase from 2.50 to 7.50 maximalactivity, (tmax�tmin)=0.081�0.025=3.24, requiring just a 1.22-fold change inactivator concentration for the same promoter. These effects are sum-marized in Fig. 4b, a plot of the minimum threshold width obtainable forpromoters with four and six binding sites, for two-fold and three-foldincreases in transcription of the target gene. The minimum theoreticalwidth is smallest at low concentrations of activator protein, suggesting thatthe sharpest genetic switches might actually be produced at barely detec-table levels of activator protein. Intriguingly, genetic analysis of earlyembryogenesis in Drosophila indicates that this is often the case; Bicoiditself activates hunchback at the lower end of the concentration gradient(Driever and Nu� sslein-Volhard, 1988a), while the Hunchback protein hasalso been shown to perform important genetic regulatory functions inregions of expression below the limit of detection by immunohistochemicalstaining (Hu� lskamp et al., 1990).

This result also has important implications for the organization ofregulatory cascades where the target of one activator is itself the activatorof a further set of genes (as is the case for Hunchback and many otherdevelopmental genetic pathways). If two genes are both activated by three-fold increases in Hunchback activity, but over different concentrations (onenear the peak of expression, the other in a region of low expression), thenthey will in fact be responding to markedly different changes in concentra-tion of the activator of hunchback, namely Bicoid. Consequently, if varia-tion exists for the level of expression of the primary activator, then differentdownstream target genes will experience markedly different effects on theirexpression. This in turn raises the question of how developmentalhomeostasis is maintained in complex systems.



Fig. 4. Determinants of the minimum threshold width. (A) Minimum threshold width asa function of the number of binding sites. For any given response, the addition of extrabinding sites has a decreasing effect on the minimum threshold width. Shown is the curve fora response from 250 to 750 of maximal transcriptional response. (B) Minimum thresholdwidth as a function of activator concentration. The four curves show the calculated minimumthreshold width (assuming maximal cooperativity) for two-fold (lower curves) and three-fold(upper curves) responses, with four (heavy lines) and six (light lines) binding sites. At loweractivator concentrations (on an arbitrary scale), the overall response will be lower, but theincrease in activator concentration required to produce the same relative increase in outputis smaller, so that the sharpest thresholds can actually be produced with barely detectableamounts of protein, as is observed with Bicoid and Hunchback.

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Genetic Variation in a Haploid Model of Transcription

The effects of point mutations on a genetic switch can be modeled bysystematic variation of the values of the DNA-binding and protein�proteininteraction parameters, u and w. This was done by supposing that five ofthe six binding sites are equivalent and by introducing two newparameters: v is the DNA-binding affinity for a protein bound to the sixthsite, and W is the cooperativity associated with binding to this site. Thisrequires an extra five recursion equations representing five new configura-tion states as shown in Appendix 1, which were iterated over a range ofactivator concentrations as before. Figure 5 shows plots of response curvesfor a range of values of v, w, and W, against decreasing activator con-centration along the abscissa. The precise shapes of the response curvesis of less interest than their robustness to parameter changes of the orderthat might be expected of single nucleotide (and amino acid) changessuch as are constantly being generated in natural populations. Twofeatures of the response curves are considered here: the location and thethreshold width.

As expected, increasing the affinity of the DNA�protein interactions bymodifying the parameter v has the effect of shifting the location of thethreshold (but not its width) relative to the Bicoid gradient. It is thoughtthat such a mechanism may be used in many developmental geneticsystems to allow a single activator to activate multiple target genes at dif-ferent concentrations (Lawrence, 1992). Thus, Bicoid activates severalother ``gap'' genes anterior to the hunchback threshold (Hu� lskamp andTautz, 1991), while the Dorsal protein almost certainly defines the dorsal�ventral axis in a concentration-dependent manner also via differentialtarget�gene affinity (Ip et al., 1992).

The most striking observation is that variation of each of the threeparameters over a physiologically relevant range produces nearly additiveeffects on the response curves. Variation of v over the range 4.0_104 to5.4_104 J�mol (that is, from nonspecific DNA-binding to the highestobserved protein�DNA affinity; Ptashne, 1992) shifts the threshold loca-tion gradually over 200 egg length, where 50 egg-length corresponds toan approximately 1.5-fold change in activator concentration (Driever andNu� sslein-Volhard, 1988a). Note, though, that as the binding affinityapproaches nonspecificity, the effect on the response curve becomes negli-gible, as expected. A similar result can be achieved by changing theparameter u, perhaps representing an amino acid substitution that affectsthe protein's specific affinity for each of the binding sites (not shown). If theassumptions of the model are correct, these results suggest that the majoreffect of subtle differences in promoter sequences or protein�DNA bindingaffinities may be to regulate the precise location of a threshold response.



Similarly, variation in the level of cooperativity over physiologicallymeaningful ranges can produce observable differences in the location of thethreshold. For the case of overall cooperativity (Fig. 5B), the change inthreshold location is nearly additive over a reasonable range of values.However, for increases in the cooperativity of a single site (perhaps onethat binds a different activator protein) there is a multiplicative componentto the variance, best seen by consideration of the effects of increasing the

Fig. 5. Effects of variable input parameters on response curves. The simplified 7-statemodel was modified to include one variable binding site, as explained in Appendix 1 and inthe text. Functions rs and as still take on values from Table I. Unless indicated, u=5_104,v=5_104, w=W=1_104 J�mol. (a) Increments of the sixth-site DNA-protein affinity, v,result in shifts of the response curve towards the posterior pole, without affecting the widthof the threshold. The effect is multiplicative as the affinity increases from nonspecific(104 J�mol), but quickly becomes additive in the physiological range (105 J�mol). Responsecurves are drawn for v=1_104, 2_104, 4_104, 6_104, 8_104, 1_105, 1.2_105, and1.4_105 J�mol. (b) Increasing the overall protein�protein cooperativity results in additiveshifts in the location of the threshold, with dramatic effects over a narrow range of parameterincrease. There is also a gradual increase in width of the threshold as the cooperativityincreases. From left to right, w=8_103, 8.5_103, 9_103, 9.5_103, 1_104, 1.05_104,1.1_104, 1.15_104 J�mol. (c) The effect of increasing the level of cooperativity associated withjust the sixth binding site is similar to that of increasing the overall DNA-binding affinity,over an order of magnitude smaller range of increments: W=0, 2_103, 4_103, 6_103,8_103, 1_104, 1.2_104, and 1.4_104 J�mol. However, still greater increases in W produceever-greater posterior shifts in the location of the curve (not shown). See Table II for effectson threshold widths.

72 GREG GIBSON


Fig. 5��Continued.


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parameter W. As W approaches 0, the effect on location of the threholdbecomes negligible, but as it increases, the threshold shifts an ever-increas-ing distance along the body axis (Fig. 5C). This result implies the existenceof constraints on the evolution of interactions between activators with dif-ferent cooperative effects, since they would tend to disrupt the concentra-tion range over which a switch is activated. Whether or not such con-straints exist has not yet been investigated empirically.

Modification of the input parameters affects threshold widths in twomajor ways that differ from the effects on threshold location, as indicatedin Table II. First, alteration of the DNA-binding affinity parameters haslittle effect on threshold width; the shape of the response curves is essen-tially independent of u over physiological ranges of site-specifc binding insimulations (not shown). Second, modification of the cooperativityparameters has clearly nonadditive effects on threshold widths, with a clear``optimum'' (that is, minimum width�maximum sensitivity) for w=W. Asthe cooperativity of the sixth site interaction increases from 0, the thresholdwidth decreases as expected, but only to the point where it is of equivalentstrength to the mean cooperative interactions of the other sites. Thereafter,increasing the sixth site cooperativity actually results in a slightlybroader threshold. Similarly, increasing the parameter v has subtle effectson threshold widths, most likely indirectly as a result of increasedcooperative interactions, as the sixth site is occupied at a lower concentra-tion of the activator. It follows from this result that protein�protein inter-actions among transcription factors might be expected to maintain a fairly

TABLE II

Threshold Widths at Different Parameter Values

v tw W tw w tw

4.0 1.63 0.0 1.63 0.80 1.584.2 1.58 0.2 1.60 0.85 1.564.4 1.55 0.4 1.55 0.90 1.534.6 1.52 0.6 1.52 0.95 1.504.8 1.50 0.8 1.50 1.00 1.505.0 1.50 1.0 1.50 1.05 1.495.2 1.50 1.2 1.50 1.10 1.475.4 1.50 1.4 1.50 1.15 1.475.6 1.53 1.6 1.52 1.20 1.47

Note. Threshold widths (tw) represent the ratio ofactivator concentrations that produce transcriptional respon-ses of 750 and 250 of the maximum, over the indicatedrange of variable parameter values (_104 J�mol). Values ofthe function rs and as were taken from Table I. Unlessindicated, u=5_104, v=5_104, w=W=1_104 J�mol.

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constant value relative to the DNA-binding affinity of their site-specificDNA interactions. Further, given the earlier finding that increasingcooperativeness fails to sharpen transcription thresholds beyond a certainvalue, it is hard to imagine how selection pressures could favor the evolu-tion of very high protein�protein interaction energies among DNA-bindingproteins. This conclusion does not obviously extend, however, to interac-tions between non-DNA binding transcription activation factors (TAFs)and DNA-binding proteins.

Compensatory Mutations

The similarity of shape, and additivity of location, of response curvesraises the possibility that the same response might be produced by differentcombinations of input parameters. This is clearly the case, indicating thatthe effects of one mutation can be compensated for by another mutationaffecting a different parameter. Such compensatory mutations could arise inthe same locus, or at a second locus; and combinations of multiple muta-tions could interact to produce the same response curves. Thus, the samegenetic output can be readily produced by numerous genotypic combina-tions.

An example is provided in Table III, in which the response of a promoterwith a low DNA-protein but high cooperative affinity sixth-binding site iscompared with ones with high DNA�protein affinity but low cooperativity.The response curves are essentially identical. Suppose a mutation arises in

TABLE III

Nearly Equivalent Haploid Parameter Combinations

Threshold Thresholdv W w width location

4.0 1.0 1.00 1.63 1.005.0 0.0 1.00 1.63 1.005.0 0.5 0.96 1.53 1.005.0 0.8 0.92 1.52 1.00

Note. v, W, and w in J�mol_104, with u a constant=5.0_104 J�mol. Values of functions rs and as from Table I.Threshold widths were calculated over the response from250 to 750 of maximum. Threshold location, in this caseresponse=0.5 which is approximately the midpoint of thesigmoidal inflexion, is expressed in moles _10&11 ofactivator protein. The four parameter combinations shownproduce very similar response curves, indicating the poten-tial for extensive networks of compensatory mutations.


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the promoter that increases the binding affinity at one site. The width ofthe threshold might be slightly increased, but this effect would be offset byan anterior-shift in the location of the threshold. A second mutation,perhaps one that decreases the protein�protein interactions between trans-cription factor molecules, could then arise in or invade the population,compensating for the first mutation.

The possibilities for compensatory mutation would seem to be especiallyrich in multicomponent regulatory pathways. Consider the biochemicalmeaning of the parameters under consideration. The DNA-bindingaffinities can be altered by point mutations in either the promoter of thegene, or the coding sequence of the genes encoding the protein (orproteins) that bind to it. The cooperativity parameters similarly can bemodified by mutations affecting the protein sequences, or the promoter(spacing and base-content can affect the looping�bending potential of theDNA), or even the accessory factors that bind the proteins independentlyof the DNA. The location of the threshold can also be affected by variationin the level of expression of the activator, or the efficiency of translation ofthe target gene. For a regulatory switch consisting of tens of interactinggenetic components, all exposed to ongoing mutation pressure, it is con-ceivable that a network of compensatory effects will emerge. The majorityof individuals would have almost identical threshold settings, at least as faras selection is concerned, and yet the population will harbor a large poolof hidden genetic variation. Without any change in the visible regulatoryoutput, the genetic architecture of the developmental switches could becontinuously turning over in a dynamic, yet canalized, equilibrium.

Diploidy and Regulatory Underdominance

Additive haploid genetic variance is generally expected to generateadditive diploid genetic variance, so that the shape of a response curveproduced in a heterozygous individual would be intermediate between thatof the alternate homozygotes. This expectation is not met for the modeldeveloped here, where heterozygosity for two ``genotypes'' that individuallyproduce identical response curves is considered. Simulations were per-formed by running two sets of recursions simultaneously, one for each ofthe two promoters, and summing the individual outputs to get the overallresponse curve. Five binding sites were kept constant (u=5_104 J�mol),the sixth-site parameters v and W were covaried (assumed to both be dueto a single mutation in the promoter), and the general cooperativityparameter w was independently varied (assumed to be due to an unlinkedmutation in the gene encoding the activator protein). Further, the generalcooperativity was averaged (w=[wi+wj]�2) prior to running the recur-sions since the two protein isoforms would likely mix in the nucleus. Notethat asymmetric cooperative interactions (w=g(wi , wj), where g is a complex

76 GREG GIBSON


function) could also be considered. The restricted averaging produces thedepartures from additivity, since when both sets of parameters are averagedand run in a single recursion, the response curves for all genotypes areequivalent.

The curves in Fig. 6 illustrate the effect of heterozygosity. For diploidhomozygotes vWiiwii and vWjjwjj the calculated response curves are identi-cal to those seen in haploids, when the response is plotted as a percentageof the maximum output. In organisms homozygous for the alternate alleliccombinations, vWiiwjj and vWjjwii , the width of the threshold is unaltered,but the curves are displaced considerably. In organisms heterozygous forthe vWi and vWj alleles, the threshold is distinctly broader and displaced

Fig. 6. Increase in threshold width due to heterozygosity. The 12-state model of Fig. 5and Appendix 1 were further modified by consideration of diploidy as described in the text.A two-locus, two-allele model is considered for which one allele affects both the DNA-bindingaffinity, v, and cooperativeness, W, of the sixth site (perhaps due to a small insertion thatincreases the affinity but decreases the cooperativity by an effect on spacing), while the otherallele affects the overall cooperativeness w (perhaps due to a point mutation that alters thesequence of the activator protein). Two double homozygote combinations (dotted lines) werechosen to have almost identical response curves (see Table III). The alternate phasehomozygote combination response curves (dashed lines) have similar slopes, but markedlydisplaced locations. Response curves when the first locus is heterozygous are intermediate inlocation (solid lines) but have significantly less sharp thresholds. The double heterozygote(not shown) similarly has a wider threshold, but it would otherwise overlay the doublehomozygote curves.


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according to the w genotype. In other words, heterozygosity decreases thesharpness of the threshold, especially over the response range 0.25 to 0.75.The magnitude of effect of heterozygosity on the slope of each responsecurve is actually a function of the portion of the sigmoidal responses underconsideration. One consequence is presented in the next section, but athorough treatment is beyond the scope of this paper. Since in the develop-ing organism it is generally desirable to produce sharp genetic switches sothat cell types may be discretely determined (see, for example, Lawrence etal., 1987), this effect could reduce the fitness of a heterozygous organism.This would produce the effect of underdominance, but to my knowledge,the possibility has not yet been specifically addressed in any empiricalsystem.

Diploidy and Regulatory Pleiotropy

Finally, consider the consequences for diploidy in a regulatory cascade,where the primary target gene (modeled here as hunchback) encodes aprotein that itself regulates three different secondary target genes in aconcentration-dependent manner. Three different traits are considered: thelocation of the threshold inflexion point; the width of the threshold; andthe level of response at a given egg-length. Optimality for these traits isassumed to reflect the needs to maintain the relative spacing of the expres-sion of the three secondary target genes; to establish sharp boundaries ofexpression; and to coordinate the output of the cascade relative to inde-pendent cascades (such as a morphogen gradient issuing from the posteriorpole). In reality, embryogenesis is sufficiently homeostatic to account forconsiderable variation without measurable effects on survival, so that itwould be impossible to actually quantitate fitness effects of the sort dis-cussed. However, the point of the discussion is to illustrate how complexpleiotropy readily emerges from this very simple model of transcriptionalregulation.

The results presented in Table IV summarize the responses for each ofthe nine possible genotypes produced by the combination of two loci withtwo alleles considered in the previous section. Each matrix presents traitvalues for the nine genotypes. Columns of matrices refer to the response atdifferent points along the egg, to the width of the threshold (fold increasein Bicoid concentration required to achieve the indicated increase inhunchback transcription), and to the location of the threshold (in this case,concentration of Bicoid at which the lower threshold is obtained), asdescribed in the legend. In each matrix, it can be seen that the doublehomozygotes at the upper left and bottom right corners of the matrix haveessentially equivalent trait values for all three traits (the parameters weredeliberately chosen to produce this circumstance), while the remaining

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TABLE IV

Effects of Heterozygosity on Shape and Location of Response Curves

Responsea Widthb Locationc

vW: ii ij jj ii ij jj ii ij jj

at 150 egg length from 0.8�0.95 for response of 0.8

wii 0.94 0.97 0.99 1.36 1.38 1.31 0.8 0.7 0.6wij 0.93 0.96 0.98 1.38 1.38 1.32 1.0 0.8 0.6wjj 0.84 0.90 0.97 1.38 1.39 1.34 1.2 0.9 0.8


wii 0.65 0.79 0.93 1.63 1.71 1.50 1.0 0.8 0.7wij 0.44 0.64 0.85 1.63 1.72 1.52 1.2 1.0 0.8wjj 0.25 0.46 0.68 1.65 1.75 1.52 1.4 1.1 1.0


wii 0.04 0.09 0.13 1.35 1.31 1.30 5.3 4.4 4.0wij 0.02 0.04 0.06 1.38 1.32 1.32 6.2 5.2 4.7wjj 0.02 0.02 0.03 1.42 1.35 1.35 7.2 6.0 5.4

Note. vi=4_104 J�mol, vj=5_104 J�mol, Wi=1_104 J�mol, Wj=0.8_104 J�mol, wi=1.00_104 J�mol, wj=0.92_104 J�mol. Values of functions rs and as are from Table I.

a The left set of three matrices show the calculated transcriptional response (max=1.0) at150, 350, and 550 egg length (from anterior pole; top to bottom, respectively) for theindicated two-locus genotype combinations. At 150 EL (high concentration of activator),weak negative epistasis is indicated; at 350 EL the trait is nearly additive; at 550 EL weakadditive-by-multiplicative epistasis is apparent.

b The middle three matrices show the calculated threshold width for responses from 0.8 to0.95 (top), 0.25 to 0.75 (middle), and 0.025 to 0.075 (bottom). In the high range, the vWj alleleis recessive; in the middle range, skewed underdominance is observed, and in the low range,the vWj and wj haplotypes are both dominant.

c The right set of three matrices show the location at which a response of 0.8 (top), 0.25(middle), or 0.025 (bottom) is obtained, as a multiple of the concentration of activatorrequired to give a response of 0.25 for either of the double homozygote genotypes. In all threecases, the trait is nearly additive.

genotypes produce deviations that depend on the location of the thresholdand type of response considered.

All three target genes show essentially additive responses with respect tothreshold location, as implied by the asymmetry of the matrices about thetop left�bottom right axis. The effect on threshold width, though, is quitedifferent for each target gene: the high target shows slight dominance forone promoter allele, the intermediate target is clearly underdominant, andthe low target is relatively unaffected. The effect on response at various egglengths is different again: targets at anterior and posterior positions wouldshow multiplicative epistasis (the values at one corner are markedly


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different), while the intermediate gene is strongly additive. Regulatorypleiotropy and epistasis thus emerge as intrinsic properties of the hierarchi-cal organization of a cascade of transcriptional switches.

Discussion

Does Cooperativity Explain Threshold-Generation on Complex Promoters?

Transcription in multicellular organisms must be regulated in such amanner as to achieve a balance between efficient buffering of developmen-tal genetic switches and generation of sharp thresholds that enable discretechanges in cell fate determination in adjacent cells. The problem is par-ticularly acute in relation to morphogenetic gradients, where one activatorprotein sets the response levels for multiple target genes. At the biochemicallevel, we need to understand how sigmoidal kinetics can be generated whileat the same time allowing for buffering of genetic polymorphism and har-boring of redundant signals. Solution of this problem will require betterempirical measurement and quantitative modeling of experimentallymanipulable genetic switches. While recognizing that the model developedhere is incomplete at some levels, the following robust conclusions cannevertheless be drawn from the analysis:

(i) in a promoter optimized to achieve a sharp threshold response,no more than five or six binding sites for proteins are likely to contributeto the cooperative narrowing of the threshold width, so that the regulationcontributed by additional sites will be either redundant or affecting thelocation (or perhaps timing) of the response;

(ii) in a promoter optimized to achieve a sharp threshold response,the energy of protein�protein interactions has no effect (above a certainlevel) on the threshold width, and this level (about 250 of theprotein�DNA binding energy) is close to that observed for many knowntranscription factors;

(iii) minor variation in individual parameters can be balanced byvariation in other parameters, allowing for the same threshold to be set bya number of different genotypic combinations, upon which evolutionaryforces can act.

The utility of any theoretical model must be judged by the closeness offit between theoretical prediction and empirical measurement. Significantdifferences may imply the operation of novel mechanisms that contribute tothe phenomenon under investigation, in this case the generation ofsigmoidal transcriptional responses. Here I adapted a model that success-fully describes a well-studied genetic switch in bacteria to a more complex

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eukaryotic promoter and asked the question whether threshold widths asnarrow as those seen in early embryos can be derived. A disparitybetween calculable and actual threshold widths was encountered, someof which can be explained by incomplete specification of the parametersof the model. In particular, two arbitrary functions were used to producea ``reasonable fit'' into which variation was introduced. A nonlinearincrease in the value of the multiplier as in Eq. (1) was found to benecessary to produce sharp response curves and was justified on theassumption of an increase in per-molecule cooperativity as the numberof molecules in the transcription complex increases. Since for technicaland theoretical reasons cooperativity parameters where more than threeproteins interact with the DNA cannot be measured (see Ackers et al.,1982), the validity of this aspect of the model cannot be tested directly.The nature of the ``response function,'' relating configuration of thepromoter to transcriptional activity, is also arbitrary in the model,although simulations indicate that it will not dominate the output underconditions that ensure near-linear increase in transcription as the numberof binding sites increases (as found empirically). This function cannot yetbe modelled thermodynamically, since the details of how transcriptionfactors promote transcription are unclear (Herschlag and Johnson, 1993).Given these uncertainties, there is little point in attempting furtheroptimization of the known parameters.

Even so, the observed threshold is actually slightly sharper than the min-imum predicted by the model, suggesting that Bicoid cooperativity may notact alone to produce the transcriptional threshold. A similar conclusionwas reached on the basis of quite different mathematical simulations bytwo other groups (Reinitz et al., 1994; Kerszberg and Changeux, 1994).Further, a recent report (Simpson-Brose et al., 1994) provides empiricalevidence that Bicoid requires cooperative interaction with a second proteinto activate the hunchback promoter, as well as a number of other targetpromoters. Intriguingly, the Hunchback protein itself, which is initiallyprovided by maternal expression, is this accessory factor. Preliminarysimulations (unpublished data) indicate that the conditions under whichtwo unequal gradients will interact to produce an even sharper gradient ofactivity that may act as the true morphogen are surprisingly restricted. Itwill be interesting to learn whether feedback autoregulation supports thissharpening process and whether it is a common feature of threshold-dependent eukaryotic genetic switches (see also Ip et al., 1992). In addition,a series of alternate mechanisms, such as chromatin effects acting to main-tain transcriptionally active promoters (cf. Lukowitz et al., 1994) mayoperate quite indepenently of the described protein�protein interactions toenhance threshold function. Arguments can be brought for and againstsuch mechanisms, whose relevance will likely only be appreciated once


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more quantitative models are applied to the description of thresholdresponses. However, it should be emphasized that cooperativity isincreasingly implicated in regulatory switches in a host of developmentalprocesses and is almost certain to be a major mechanism by which bothspatial and temporal discrete switches in gene expression are achieved.

Genetic Variance and Transcriptional Regulation

A major objective of evolutionary quantitative genetics is to movefrom statistical descriptions of genetic variation based on measurementof phenotypes, to mathematical models based on Mendelian principles.Frustration arises when it is recognized that so many underlying geneticparameters concerning the number of alleles and their modes of interac-tion are unknown and, in fact, generally cannot be derived frommorphological analysis alone. Metabolic control theory represents oneattempt to incorporate explicit knowledge of the underlying moleculargenetics into the mathematical description (Kacser and Burns, 1973).Attempts have been made to reconcile the theory of mutation�selectionbalance with a thermodynamic model of metabolic variation (Clark,1991), the spirit of which motivated this study of how genetic variationmight affect developmental regulation. Unfortunately, the following con-siderations raise doubts as to whether it will ever be possible to con-struct a neutral description of the expected distribution of developmentalgenetic variation.

First, as discussed above, we are a long way from a complete biochemi-cal description of transcriptional regulation, let alone signal transductionand intercellular communication. Most isolated steps in a regulatoryhierarchy involve physical interaction between multiple components, asopposed to the activity of single enzymes, and so they may be too complexto model using a handful of kinetic parameters. Second, althoughindividual genetic switches may operate on common principles, theorganization of each genetic pathway may be so different (involvingparallel processes and complex branching and feedback interactions) as topreclude generalization. Third, the relationship between fitness and geneexpression is almost impossible to describe. That is to say, even given agood mechanistic description of the regulatory pathway, there is noobvious function relating it to selection pressures. With metabolic controltheory, outputs expected to relate directly to fitness, such as flux, efficiency,and intermediate metabolite concentrations can be measured (Kacser andBurns, 1973; Watt, 1986; Clark, 1991). What are the developmentalanalogs of these outputs? Possibilities include the sharpness of sigmoidalresponses, the stability of threshold locations within a field of cells, andhomeostatic buffering against environmental fluctuation. Yet it is known

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that dramatic changes in the fate map of early development can be com-pensated for by later developmental processes (Driever and Nu� sslein-Volhard, 1988b); the regulative capacity of embryos really precludes reduc-tionist isolation of segments of genetic hierarchies with respect to fitnesseffects. Certainly selection must act in some way to stabilize ontogeneticprograms, but biochemical analysis does not obviously help us to under-stand how.

Recognizing that a sophisticated prediction of the distribution of geneticvariation in regulatory pathways is not currently feasible, the followingconclusions are nevertheless worth considering:

(i) The distribution of allelic variance affecting a genetic switch atequilibrium will almost certainly be asymmetric, with cis- and trans-actingcomponents contributing differentially to each variable aspect of the switch.This conclusion follows from a comparison of the effects of modification ofthe DNA-binding and cooperativity parameters in the model. DNA-binding affinity can be modified over a large range with near-additiveeffects on the location of the threshold (and no effect on its width), whileprotein�protein affinities have multiplicative effects on the threshold loca-tion when high, and no effect when low, and a large effect on the thresholdwidth (but whether it is additive is highly dependent on the values of theother parameters). If minimization of the threshold width is the majortarget of selection, the existence of a biophysical constraint upon this widthsuggests that transcriptional switches may evolve towards a state ofdominance similar to that postulated to occur for enzymes that maintainmetabolic flux (Kacser and Burns, 1981).

(ii) The potential for epistasis and antagonistic pleiotropy is builtinto the architecture of regulatory pathways. This conclusion follows fromcomparison of the responses of downstream target genes to low, inter-mediate, and high concentrations of the initially activated gene product.Genetic variance affecting the first switch in a pathway (e.g., Bicoid activa-tion of hunchback) will tend to affect multiple switches at the second level(e.g., targets of Hunchback protein) unequally. This will be a generalproperty of gradient-response mechanisms for gene regulation, irrespectiveof the role of cooperativity in the establishment of the threshold. To thisquantitative derivation can be added a host of qualitative arguments estab-lishing the potential for nonadditive interactions between regulatorymolecules.

(iii) It follows that it is not possible to make any general observa-tions about covariation between regulatory components. The ability to doso depends upon the identification of a measure intrinsic to the geneticpathway that can be related to fitness (such as flux or metabolite concen-trations in linear substrate-modification pathways). Since the later


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ontogenetic consequences of variation at a single switch is wholly aproperty of the relation of the switch to the downstream target genes andsince different factors can reverse their interactions at different switches,depending on the structure of the target promoter, covariance is almostimpossible to derive.

(iv) An explanation for the evolution of hybrid incompatibility ispresented. It was shown that multiple genotypic combinations can readilygenerate equivalent threshold responses. Different combinations couldevolve in a stepwise fashion through the gradual replacement of singlealleles of small effect (as in Orr, 1994). Since heterozygosity is predicted toreduce the sharpness of threshold responses, extrapolated over manyswitches, hybrids could display significantly reduced developmentalcanalization.

(v) Certain combinations of small recombination and selection coef-ficients in conjunction with underdominance have been shown analyticallyto allow the maintenance of polymorphism accompanied by linkage dis-equilibrium between two loci (Hastings, 1982). In the absence of a theoryconcerning the strength of stabilizing selection acting upon regulatoryswitches, it is unclear what magnitude of linkage disequilibrium could bemaintained and whether it is detectable in natural populations. Scatterednonrandom associations between polymorphisms separated by severalkilobases in the promoters of Drosophila genes are usually considered to betransient, or artifacts of sampling error (for example Lai et al., 1994;Zapata and Alvarez, 1992). Although it seems unlikely that between-locuseffects would be strong enough to maintain variation, the model discussedhere suggests that within-promoter linkage disequilibrium could undersome circumstances be functionally significant.

Genetically well-defined switches, such as the Bicoid�hunchback interac-tion, the regulation of homeotic gene expression, bristle determination, andsignal transduction in the Drosophila eye provide ideal opportunities forstudying the molecular basis of quantitative genetic variation. The detailedmechanisms for the maintenance of regulatory variation are expected todiffer from those found in the context of metabolic control. Pleiotropy andepistasis are likely to be common elements of the genetic architecture ofregulatory traits. Understanding how these effects are generated at themolecular level is an important component of efforts to model the joint dis-tribution of phenotypic and genotypic effects (Gavrilets and de Jong, 1993;Caballero and Keightley, 1994) and, hence, to clarify the relative roles ofstabilizing and directional selection and of drift in the evolution of develop-mental programs.

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APPENDIX 1: Variable Sixth-Site Recursions

Response curves over a range of activator concentration were calculatedusing the following recursions for the haploid state. Each expression ``a''through ``k'' represents the expected occupancy for the configuration of onestate (see Table I). z=1�RT=4.0326_10&4 at 298 K (25 %C). Parametersu and v, the DNA�protein binding free energies, range from 4.0_104 to5.4_104 J, while w and W, the protein�protein interaction free energies,range from 0 to 1.4_104 J. The concentration of activator, x, ranged from10&12 to 10&10 M, increasing by a multiplier of 1.01 in each successiverecursion:

a=x } exp(u } z)

b=x } exp(v } z)

c=x2 } exp((2u+w) } z)

d=x2 } exp((2u+W) } z)

e=x3 } exp((3u+2w) } z)

f=x3 } exp((2u+v+w+W ) } z)

g=x4 } exp((4u+4w) } z)

h=x4 } exp((3u+v+3w+W ) } z)

i=x5 } exp((5u+6w) } z)

j=x5 } exp((4u+v+5w+W ) } z)

k=x6 } exp((5u+v+7w+W ) } z).

Each of these expressions is multiplied by the number of different con-figurations that give rise to the state (total=64 configurations, 7 states)and summed to give the denominator of the response equation (1). (Theunity term represents the vacant promoter state.) Fractional occupanciesare then multiplied by the response scaler for the state, in all cases (unlessstated otherwise) taking the values (0, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0) for (0, 1,2, 3, 4, 5, 6) sites bound:

response=\0.5(5a+b)+0.6(10c+5d )+7(e+f )

+0.8(5g+10h)+0.9(i+5j)+k +1+b+i+k+5(a+d+g+j)+10(c+e+f+h)

.

For the diploid case, the contributions of heterozygous trans-acting factorsto cooperativity were simply averaged by calculating w=(wi+wj)�2, andsubstituting this value into separate recursions for the cis-acting allelesvi Wi and vjWj (keeping u constant), summing each pair of output terms toproduce the diploid response curve.


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APPENDIX 2: Derivation of the Theoretical Minimum

Threshold Width

By definition, the threshold width��which we seek to minimize��is theratio of activator concentrations that result in a given change in responseof the target gene:

threshold width=xmax �xmin , (7a)

where xmax and xmin give the maximum and minimum (upper and lower)activator concentrations, respectively.

First consider the case of a promoter with a single binding site. Then thetranscriptional response is equal to the fractional occupancy of the site:

response=xe(&2G�RT )

1+xe(&2G�RT )=A

1+A, (8)

where

A=xe(&2G�RT ),

whence

x=A�e(&2G�RT ).

Substituting above eliminates the term e(&2G�RT ), such that

threshold width=Amax �Amin . (7b)

That is to say, the threshold width is independent of temperature andDNA-binding affinity, but it is a direct function of the range of responserequired. For example, to increase transcription of the target gene from0.25 to 0.75 units of activity requires a nine-fold increase in activator con-centration (Amax=3, Amin=1�3); or to increase it from 0.1 to 0.3 unitsrequires a four-fold increase in activator concentration (Amax=0.43,Amin=0.11). Notice how much smaller increases in activator concentrationare required to produce the same relative increase in target gene expressionat lower concentrations. A practical lower limit of the threshold width willbe set by the background level of transcription from an unoccupiedpromoter, as well as the concentration below which the target gene-producthas no activity.

Now consider the case of a complex promoter with k equivalent bindingsites, and for simplicity let the response be equal to the sum of the frac-tional occupancies,

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response=\ke(&u�RT )x+ne(&(2u+w)�RT )x2+ } } }

+exp(&(ku+mw)�RT ) xk +\1+ke(&u�RT )x+ne(&(2u+w)�RT )x2+ } } }

+exp(&(ku+mw)�RT ) xk +=

kA+nA2e(&w�RT )+ } } } +Ak(e(&w�RT ))m

1+kA+nA2e(&w�RT )+ } } } +Ak(e(&w�RT ))m

=kA+nA2C+ } } } +AkCm

1+kA+nA2C+ } } } +AkC m=t

1+t, (9)

where n is the number of configurations in the relevant promoter state, mstands for the maximal value of the cooperativity function as (but makesno assumptions about the form of this function), and C=e(&w�RT ). Put theratio tH�tL=., where the subscripts denote high and low responses; then

kAH+nA2H C+ } } } +Ak

HCm=.(kAL+nA2LC+ } } } +Ak

LCm) (10a)

and rearranging:

Cm(AkH&.Ak

L)=k(.AL&AH)+nC(.A2L&A2

H)+ } } } . (10b)

As C increases (practically, wmax=1.4_104 J, implies Cmax=283) and forlarge m (>k), the LHS increases at a greater rate than the RHS, so for thisequality to hold requires that (Ak

max&.Akmin) tends towards zero, that is,

Akmax � .Ak

min , Amax�Amin � k- ..

But (Amax �Amin) remains equivalent to (xmax�xmin), the threshold width;hence in the limit,

the minimum threshold width=k- (tmax �tmin) (6)

is defined by the k th root of the ratio of a parameter determined by thehigher and lower responses, namely the ratio of numbers that satisfy theupper and lower relations ``response=t�(1+t).'' With each successive bind-ing site, the decrease in threshold width becomes less and less. Further, thislimit is essentially unaffected by any reasonable function relating fractionaloccupancy to the response for each configuration, as the terms ``n'' on theRHS of the equality are negligible. The requirement that m be greater thank would only be violated if the cooperativity function as decreases substan-tially as the number of binding sites increases. Note, though, that theobserved threshold widths suggest that this function actually increases in agreater-than-linear manner.


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Acknowledgments

I particularly thank Professors David Hogness, Ward Watt, and Mark Feldman forencouragement during the inception of this work, and Julian Adams during its completion.Thanks, too, to David Pollock and Magnus Nordborg for helpful discussions along the way,and the anonymous reviewers for helpful comments. This work was initiated while I wassupported by a Post-Doctoral Fellowship from the Helen Hay Whitney Foundation, in theDepartment of Developmental Biology at Stanford University.

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epistasis and pleiotropy as natural properties of transcriptional regulation

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