Analysis of biochemical mechanisms provoking differential spatial expression in Hh target genes
aa r X i v : . [ q - b i o . M N ] S e p Analysis of biochemical mechanisms provokingdifferential spatial expression in Hh target genes
Manuel Camb ´on and ´Oscar S ´anchez University of Granada, Applied Mathematics Department, Granada, E18071, Spain * [email protected] + these authors contributed equally to this work ABSTRACT
This work seeks to analyse the transcriptional effects of some biochemical mechanisms proposed in previous literature whichattempts to explain the differential spatial expression of Hedgehog target genes involved in
Drosophila development. Specifi-cally, the expression of decapentaplegic and patched , genes whose transcription is believed to be controlled by the activatorand repressor forms of the transcription factor Cubitus interruptus (Ci). This study is based on a thermodynamic approachwhich provides binding equilibrium weighted average rate expressions for genes controlled by transcription factors competingand (possibly) cooperating for common binding sites, in the same way that Ci’s activator and repressor forms might do. Theseexpressions are refined to produce simpler equivalent formulae allowing their mathematical analysis. Thanks to this, we canevaluate the correlation between several molecular processes and biological features observed at tissular level. In particular,we will focus on how high/low/differential affinity and null/total/partial cooperation modify the activation/repression regions ofthe target genes or provoke signal modulation.
Introduction
Hedgehog (Hh) is a morphogen, signaling protein acting on cells directly to induce distinct cellular responses. It is involvedin several developmental systems such as the
Drosophila melanogaster embryo. In particular, in
Drosophila wing imaginaldisc, the secretion of Hh from cells in the posterior compartment induces the expression of several target genes in the anteriorcompartment cells causing the patterning of the central domain of the wing . Cubitus interruptus (Ci), a
Drosophila transcription factor (TF), controls the synthesis of the Hh target genes decapentaplegic ( dpp ) and patched ( ptc ). It has beenpreviously proposed that these genes are regulated by sets of a promoter and 3 binding sites (cis-regulatory elements, alsoknown as enhancers), and TFs of the Cubitus interruptus family in two opposite forms competing for the same genomicbinding sites . In the absence of Hh, Ci is cleaved to become a transcriptional repressor, CiR, but in the presence of Hh itis converted to the activator form, CiA. However, the same Hh signaling causes differential spatial expression of these targetgenes: ptc expression is restricted to the region of highest Hh signal concentration while dpp responds more broadly in a lowerzone of Hh signaling.In the work of Parker and coauthors the explanation of this fact was made considering several biochemical mechanismsin the binding of the TFs such as high-low, differential affinities and cooperativity. This discussion employs the fitting ofa thermodynamic model to discriminate between the different options. In , the same discussion is focussed mainly on thehigh-low affinity character of the enhancers from a experimental point of view, concluding that low-affinity binding sites arerequired for normal dpp activation in regions of relatively low signal. Another very interesting point in experiments shownin is the reinforcement of the Hh signaling when dpp low-affinity Ci/Gli sites are converted to high-affinity sites, in thesense that both repression and activation are generally stronger in the regions of respective net repression/activation.Here, we will reconsider the same questions performing the mathematical analysis of our own model building on the statis-tical thermodynamic method proposed by Shea, Ackers and coworkers , also known as the BEWARE method . BEWARE(Binding Equilibrium Weighted Average Rate Expression) is a well recognised method used frequently in mathematical mod-elling of gene transcription processes (see for instance for a general discussion). These weighted averages give rise to lengthymathematical expressions even for the case of only two TFs. This impedes the possibility of deciphering biological effectswithout the use of numerical tools . In addition, the averages involve a great amount of constants of diverse nature. Mak-ing approximation of these constants constitutes a dilemma in itself, and the effect of their modification in the biologicalsystem only has been tested by numerical and in vivo/vitro experiments (see for instance or ). Explicit simple analyticalexpressions have only been proposed in the literature for specific independent binding sites .In this work we formulated useful easily applicable expressions for the BEWARE operator in the Hh signaling pathwayhere the pair of transcription factors CiA and CiR, acting as activators and repressors, compete cooperatively for commonenhancers. Based on the experiments done in the literature, we apply these expressions performing a mathematical analy-sis of the behaviour of the system under variations of the biochemical mechanisms considered: binding affinity, interactionintensity and cooperativity. These expressions allow us to predict the effects of these mechanisms in presence of opposing ac-tivator/repressor TF gradients acting through the same cis-regulatory sites, a point not satisfactorily explained in any previoussystem . Results
Concise expressions of the BEWARE operator have been deduced for a family of two TFs competing for common bindingsites controlling transcription by the recruitment mechanism taking into consideration the following variables: number ofbinding sites, particular binding affinities for each TF species (activators-repressors), posible effects of cooperativity betweenboth TFs (total) or between the TFs of the same family (partial), and finally interaction intensity for activators and repressors.This operator determines the balance between opposing TFs that gives rise to equal gene activation rates. In particular, themathematical analysis performed to this model predicts:i) The existence of relations between the concentrations of the transcription factors determining a threshold with respectto the activation basal level (meanly, transcription rates in absence of TFs). This threshold defines two areas: ac-tivated/repressed region for activator and repressor concentrations inducing transcription rates greater/lower than thebasal. This threshold is linear in the case of null or total cooperativity and involves more entangled expressions in thecase of partial cooperativity.ii) The dependence of this threshold, and in consequence of the activated/repressed regions, on the relative affinity betweenactivators and repressors, TFs interaction intensity, and partial cooperation.iii) Variations of the intensity of the signal due to proportional changes in the TFs affinity constants or total coopera-tivity, where we will refer to straightened signaling effects to transcription rate increments/decrements in the acti-vated/repressed regions. However, we remark that these effects do not change the activation/repression threshold.We propose that the differential response of two genes in the same cell containing the same TFs concentrations could bejustified by the fact that the activation/repression regions are different for those genes. In consequence, the model proposesthat the proportional (in particular, equal) low-high affinity of the TFs, or total cooperativity can not explain solely the dif-ferential spatial expression of dpp and ptc although they justify perfectly the stronger activation/repression measured in theliterature for higher affinity modified binding sites. Methods:
Deduction of the BEWARE operator
As a first step, we will apply the ideas of the statistical thermodynamic method to the genes dpp and ptc , controlled bythe transcription factors { CiA , CiR } . Thus, our goal in this point is to deduce expressions for the time evolution of theconcentration of protein P (either be Dpp or Ptc) in terms of the concentrations of the TFs, i.e., d [ P ] dt = BEWARE ([ CiA ] , [ CiR ]) (1)where ‘BEWARE()’ represents a mathematical function specifying the dependence with respect to the activation/repressionrole of the TFs, independently of other possible factors relevant for the protein evolution as for instance degradation orspace dispersion. In the model, the binding reactions of TFs and RNA polymerase (RNAP) in the enhancers and promoter,respectively, are much more faster than the synthesis of the protein P, hence it will be considered in thermodynamic equilibriumgiven by the Law of Mass Action. If B is a set of non occupied enhancers-promoter, the complexes BCiA , BCiR and
BRNAP have concentration at equilibrium given by [ BCiA ] = k ( )+ A k ( ) − A [ CiA ][ B ] : = [ CiA ] K ( ) A [ B ] , [ BCiR ] = k ( )+ R k ( ) − R [ CiR ][ B ] : = [ CiR ] K ( ) R [ B ] , [ BRNAP ] = k + RP k − RP [ RNAP ][ B ] : = [ RNAP ] K RP [ B ] , where K ( ) A , K ( ) R and K RP are dissociation constants of the activators, repressors and RNA polymerase, so the quotients [ CiA ] K ( ) A , [ CiR ] K ( ) R and [ RNAP ] K RP are dimensionless. The superscript ( ) stands for the dissociation constant of a reaction that takes place in bsence of another TF previously bound in other enhancer (note that, since the sets only have one promoter, the superscript isnot needed for the RNAP dissociation constant). The consecutive binding of more that one transcription factor is consideredas a sequential and competitive process, such that the reactions CiA + BCiA k ( )+ A −− ⇀↽ −− k ( ) − A BCiACiA or CiR + BCiA k ( )+ R −− ⇀↽ −− k ( ) − R BCiACiR will be given by equilibrium concentrations [ BCiACiA ] = [
CiA ][ CiA ] K ( ) A K ( ) A [ B ] and [ BCiACiR ] = [
CiA ][ CiR ] K ( ) A K ( ) R [ B ] , where now the superscript ( ) denotes the dissociation constant for a reaction of a TF that binds the operator with alreadyone TF in some other site. On the other hand, the competition is modelled such that the dissociation constant of the free sitesconfiguration does not depend on their position, but might depend on other existing bound TFs in the same set of enhancersby cooperativity or anticooperativity.We will denote by non cooperative TFs to all those proteins whose enhancer’s affinity is not modified by any previouslybound TFs, that is, they verify K ( ) A = K ( ) A and K ( ) R = K ( ) R . This assumption implies sequential independence of the equi-librium concentrations since [ BCiRCiA ] = [
BCiACiR ] . It is plausible to assume the same relation for later bindings, that is, K ( j ) A = K ( ) A and K ( j ) R = K ( ) R for j ≥ K A and K R skipping the superscript. Then, if all the TFs under consideration are non cooperative we easily deducethat the concentration at equilibrium of a configuration with j A activators and j R repressors bound is [ BCiA j A CiR j R ] = [ B ] (cid:18) [ CiA ] K A (cid:19) j A (cid:18) [ CiR ] K R (cid:19) j R (2)independently of the sequential order of binding and of the specific positions occupied for the TFs. Let us recall that Drosophila ’s cis-regulatory elements involve a total number of 3 binding sites, so we have a restriction for the possiblenumber of bound transcription factors. So, j A + j R ≤ j = − j A − j R ≥ i , h = A , R , that is: K ( ) i = K ( ) h / c where c is a positive constant bigger than 1 if proteins cooperate and smaller than the unity if anti cooperation occurs. Sincethe only difference between cooperativity and anticooperativity is a threshold value for c in the subsequent considerationsabout modelling we will refer to the constant c and not distinguish between both cases. If cooperation occurs it would benecessary to know which TFs are affected by others TFs since the equilibrium concentration will depend on these relations. Inthe literature total and partial cooperation have recently been proposed to play a very relevant role in the Hh/Shh target genesby means of the Ci/Gli TFs . Partial cooperation in the activators would occur when the existence of an activator modifyequally the affinity of any posterior activator binding, that is K ( j ) A = K ( ) A / c A for j ≥
2, and respectively for repressors. Totalcooperation would occur when the presence of a bound TF modify the affinity of any posterior binding in the same manner,i.e. K ( j ) A = K ( ) A / c and simultaneously K ( j ) R = K ( ) R / c for j ≥ ). In the subsequent we will denote by K A or K R the activator and repressor affinity constants, such that [ BCiA j A CiR j R ] = [ B ] c ( j A + j R − ) + (cid:18) [ CiA ] K A (cid:19) j A (cid:18) [ CiR ] K R (cid:19) j R (3)in the presence of total cooperativity while [ BCiA j A CiR j R ] = [ B ] c ( j A − ) + A c ( j R − ) + R (cid:18) [ CiA ] K A (cid:19) j A (cid:18) [ CiR ] K R (cid:19) j R (4)if partial cooperativity for TFs occurs. Here ( · ) + denotes the positive part function ( ( x ) + = x if x > {{ CiA , CiR } c } and {{ CiA } c A , { CiR } c R } the total and partial cooperativity respectively. Let s observe that this notation covers the case of non cooperativity since it would correspond to the case {{ CiA , CiR } } orequivalently {{ CiA } , { CiR } } .The binding sites are ordered spatially and, in general, there is not an unique spatial distribution for a configuration with j A activators, j R repressors and 3 − j A − j R free sites. For instance if we consider j A = j R = CiACiRO , CiRCiAO , CiAOCiR , CiROCiA , OCiACiR , OCiRCiA where O denotes theempty space). In our description spatial localization of bound particles is not considered, indeed for a concrete configurationwith j A activators, j R repressors and j free sites j ! j A ! j R ! spatial different configurations are plausible, where k ! denotes thefactorial of k .Regarding the promoter’s RNA polymerase binding process, the TFs work together trying to promote or repress thebinding process by a mechanism known as recruitment . Thus, we consider that the activators interact with RNAP with‘adhesive’ interaction that gives rise to a modification of the RNA polymerase binding affinity: K RP / a j A where where a isa constant bigger than 1, called for now on activator interaction with the RNA polymerase. On the other hand, the effect of j R repressors is modelled in terms of a ‘repulsive’ interaction that modifies the binding affinity K RP / r j R with a factor r < : Step 1: Construction of the sample space
Let us consider [ CiA ] , [ CiR ] and [ RNAP ] concentrations of activators, repressors and RNA polymerases. Then, all thepossible ways of obtaining an equilibrium concentration with j A , j R and j P activators, repressors and RNA polymerases isgiven by the states Z ( ) ( j A , j R , j P = C ) = C ( C ) j ! j A ! j R ! [ B ] [ RNAP ] K RP (cid:18) a [ CiA ] K A (cid:19) j A (cid:18) r [ CiR ] K R (cid:19) j R , (5) Z ( ) ( j A , j R , j P = C ) = C ( C ) j ! j A ! j R ! [ B ] (cid:18) [ CiA ] K A (cid:19) j A (cid:18) [ CiR ] K R (cid:19) j R where j P = j P = j = − j A − j R ≥
0, andthe variable C describes the relation of cooperation between the TFs. More concretelly, by using (3) and (4), the cooperationfunction C takes the values C ( C = { CiA , CiR } c ) = c ( j A + j R − ) + (6)and C ( C = {{ CiA } c A , { CiR } c R } ) = c ( j A − ) + A c ( j R − ) + R . (7)This allow us to describe all the sample space, i.e. the space of all the possible configurations, as Ω = (cid:8) ( j A , j R , j P ) ; j A , j R ≥ , j A + j R ≤ , j P = , (cid:9) . Step 2: Definition of the probability
Once we have described all the possible configurations in terms of the concentrations of activator, repressor and RNApolymerase we easily obtain the probability of finding the promoter in a particular configuration of j P RNA polymerase and j A , j R TFs related by a cooperation relation C as P ( ) ( j A , j R , j P ; C ) = Z ( ) ( j A , j R , j P ; C ) ∑ { j ′ A , j ′ R , j ′ P }∈ Ω Z ( ) ( j ′ A , j ′ R , j ′ P ; C ) , (8)for all ( j A , j R , j P ) ∈ Ω . Step 3: Definition of the BEWARE operator
In this last step, the BEWARE operator is obtained in terms of the probabilities P ( ) . Following the work of Shea et al the synthesis of certain protein depends on the total probability of finding RNA polymerase in the promoter, more concretely,it is proportional to the marginal distribution of the case j P = . We will denote by the recruitment BEWARE operator tothe function BEWARE ([ CiA ] , [ CiR ] , [ RNAP ] ; C ) = C B j A + j R ≤ ∑ j A , j R ≥ P ( ) ( j A , j R , j P = C ) here in definition (8) expression (5) is assumed and C B is a constant of proportionality that could depend on other factorsdisregarded in this work. Splitting the denominator in two sums, depending on the existence of RNA polymerase bound to theconfiguration, this expression can be easily rewritten in terms of the regulation factor function, F reg :BEWARE ([ CiA ] , [ CiR ] , [ RNAP ] ; C ) = C B + ∑ j ′ A + j ′ R ≤ j ′ A , j ′ R ≥ Z ( ) ( j ′ A , j ′ R , j ′ P = C ) ∑ j ′ A + j ′ R ≤ j ′ A , j ′ R ≥ Z ( ) ( j ′ A , j ′ R , j ′ P = C ) = C B + K RP [ RNAP ] F reg ([ CiA ] , [ CiR ] ; C ) . (9)Doing some basic algebra (see Suplemental Material 1.1) this regulation factor can be transformed equivalently into simpleexpressions whose analysis can contribute to the understanding of the general process. These calculations, using a classicalstrategy employed for obtaining the derivation of the General Binding Equation more than a century ago , have not still beenapplied in this context up to the authors knowledge. Indeed, we can prove that the regulation factor can be equivalently writtenas F reg ([ CiA ] , [ CiR ] ; C ) = S ( ) (cid:0) a [ CiA ] K − A , r [ CiR ] K − R ; C (cid:1) S ( ) (cid:0) [ CiA ] K − A , [ CiR ] K − R ; C (cid:1) , (10)where the explicit expression of S ( ) ( x , y ; C ) depends on the kind of cooperativity presumed, that is S ( ) ( x , y ; {{ CiA , CiR } } ) = ( + x + y ) , (11) S ( ) ( x , y ; {{ CiA , CiR } c } ) = − c + c ( + cx + cy ) , (12) S ( ) ( x , y ; {{ CiA } c A , { CiR } c R } ) = ( + c A x + c R y ) c A c R + (cid:18) − c R (cid:19) ( + c A x ) c A + (cid:18) − c A (cid:19) ( + c R y ) c R + (cid:18) − c A (cid:19) (cid:18) − c R (cid:19) , (13)respectively for the non cooperative, total an partial cooperative cases. The first remark is that mathematical complexityin these expressions is mainly related with the assumed cooperativity. These ideas can be generalized in a multifunctionalframework and they can be applied in other different contexts as can be seen in . Determination of activation/repression regions
To cope with the problem treated in we are going to establish theoretical regions of activation/repression and posteriorlywe will put them in correspondence with biological observations. Using reporter genes in it was compared the activityof different versions of the dpp enhancer containing three low-affinity sites, three high-affinity sites or three null-affinity sites.The last one, the basal expression, collects the effects of all other factors than Ci on dpp since null-affinity sites interrupt Cubi-tus control. For instance, it takes into account that Engrailed prevents the transcription of dpp near the anterior/posterior (A/P)boundary and, in consequence, the basal expression depends on the distance of the cells to this boundary. More concretely,in the effects of Ci signaling with low- or high-affinity enhancers was measured comparing the gene activity versus thebasal at any cell. We propose to define activation/repression regions in the plane [ CiA ] − [ CiR ] separated by the basal state, thatis, we want to determine which concentrations [ CiA ] , [ CiR ] will provide more or less gene expression than the basal. Note thatthe basal state, determined by the absence of TFs, that is [ CiA ] = [
CiR ] =
0, corresponds to F reg = F reg >
1) or decrease (for F reg <
1) of the numberof RNAP molecules bound to the promoter with respect to the basal level as was pointed out in . The analysis of the influenceof the biochemical mechanisms (mainly related with affinities and cooperation) on the threshold between both regions willprovide us interesting information in order to understand the wide spreading of dpp .Let us first consider a BEWARE operator with non/total cooperativity ( c ≥
1) both for ptc and for dpp obeying the generalexpression (9) which depends on the regulation factor F reg ([ CiA ] , [ CiR ] ; { CiA , CiR } c ) : = − c + c (cid:16) + ac [ CiA ] K A + rc [ CiR ] K R (cid:17) − c + c (cid:16) + c [ CiA ] K A + c [ CiR ] K R (cid:17) . (14)It is quite easy to see that in the case of (14) the threshold (that is F reg =
1) is determined by the linear relation [ CiR ] K R = a − − r [ CiA ] K A (15) ividing the plane [ CiA ] − [ CiR ] into two parts that we can denominate activated region if [ CiR ] K R < a − − r [ CiA ] K A and repressedregion if on the contrary [ CiR ] K R > a − − r [ CiA ] K A . Let us observe that the threshold (15) is independent of the affinities values if equalaffinities are assumed, while differential affinities between activators and repressors at the common binding sites will modifythe slope of this linear threshold. See Fig. 1 (a) and (c) where some examples of these thresholds are depicted for differentvalues of the parameters. This trivial remark could give some insight into some experiments where the activation seems tooccur at small concentrations of activators even in the presence of a high gradient of repressors (see for instance ).Moreover, in the absence of cooperativity ( c =
1) the isolines of the regulation factor (contour lines determining values of CiRand CiA concentrations determining the same level of activation) are the straight lines in the [ CiA ] − [ CiR ] plane given by theformula: F reg = K ⇐⇒ [ CiR ] K R = K / − r − K / + K / − ar − K / [ CiA ] K A . (16)In the case of partial cooperativity, however, due to the more entangled expression obtained in (13) it is not so clear that theactivation threshold can be obtained explicitly as we did with (15) and we need to develop with a little more care the propermathematical analysis. Indeed, if we impose the threshold equation F reg ([ CiA ] , [ CiR ] ; {{ CiA } c A , { CiR } c R } ) = , it can be shown that this threshold is determined by an unique increasing function f , that divides the plane [CiA]-[CiR] in tworegions, activation ( [ CiR ] / K R < f ([ CiA ] / K A ) ) and repression ( [ CiR ] / K R > f ([ CiA ] / K A ) ) (see Supplemental Material 1.2 fordefinition and analysis of the function f ). Let us mention that the threshold of a BEWARE operator with partial cooperativityis not, in general, a straight line although it shows a linear asymptotic behaviour for large concentrations (see Fig. 1 (e) ). Biochemical mechanisms that modify genetic spatial expression
In this section we will describe the effect induced on the spatial gene expression by the molecular mechanisms: affinity,cooperativity and TFs interaction intensity. Since our main goal is to understand how these mechanims could modify theexpression of the Hh target genes we are going to assume transcription factors acting in the same way that Cubitus works inthe
Drosophila system. We recall that Hh is secreted from the posterior in the anterior compartment of the wing imaginaldisc, that results in opposing gradients of activator and repressor Ci. In order to model these concentration distributions weare going to adopt the time independent approach proposed in : [ CiA ] = he − x / √ D , (17)where x denotes the distance from the A/P boundary, h scales the activator concentration values and D is the steepness of thegradient. The A/P boundary is located around the 60% of the dorso-ventral (D/V) axis and the influence of Hh gradient can beappreciated in the middle of the anterior compartment, more concretely the cells located approximately between the 30% andthe 60% of the D/V axis. Furthermore, the description used in also considers the conservation of the total amount of Ci, i.e. [ CiR ] + [
CiA ] = h , (18)hence they will be restricted to a straight line in the [ CiA ] − [ CiR ] plane (see Fig. 1 (a) , (c) , (e) ). Inset in Fig. 1 (e) shows thedistributions of [ CiA ] , [ CiR ] under (17) and (18) that we consider in this work. The intersection point between the straight line(18) and the threshold, ([ CiA ] th , [ CiR ] th ) , will determine a boundary between genetically activated/repressed cells (representedby yellow circles in Fig. 1 (a) , (c) , (e) ). That is, repressed (resp. activated) cells will be those containing concentrations ([ CiA ] , [ CiR ]) verifying (18) such that [ CiA ] < [ CiA ] th (resp. [ CiA ] > [ CiA ] th ) and exhibiting, in consequence, transcriptionrates lower than the basal (resp. higher than the basal). Due to the monotone character of distribution (17) activated cellsare closer to the A/P boundary and the limit of the percentage of the wing imaginal disc occupied by activated cells will bedetermined by the distance x th verifying [ CiA ] th = he − x th / √ D represented by yellow circles in Fig. 1 (b) , (d) , (f) . From now we will refer to the space occupied by activated/repressed cellsas relative activated/repressed disc.By using previous considerations, we will analyse in next paragraphs the effect over the spatial dpp and ptc expressionrate due to the biochemical mechanisms:1. Equal Affinity ( K dppA = K dppR , K ptcA = K ptcR ). . Differential Affinity ( K dppA = K dppR , K ptcA = K ptcR ): In this case we have noticed that it will be relevant to distin-guish between Proportional Differential Affinity, where K dppR / K dppA = K ptcR / K ptcA and Independent Differential Affinity, K dppR / K dppA = K ptcR / K ptcA .3. Interaction intensity, where a dpp = a ptc , r dpp = r Ptc .4. Cooperativity: Global Cooperativity, where c dpp = c ptc , and Partial Differential Cooperativity, where c dppA = c dppR , c ptcA = c ptcR .Where the superscripts ptc and d pp stand for the parameters of the genes ptc and dpp , respectively. Let us remark that someof these mechanisms have been proposed in .Equation (15) shows a clear dependence of the activation-repression threshold (at least for the non-cooperative and totalcooperative case) on the activator-repressor affinity constants K A , K R , and activator-repressor interaction intensities a and r .Under Equal Affinity ( K dppR / K dppA = K ptcR / K ptcA =
1) the threshold doesn’t change. In the case of Differential Affinity thethreshold only will change if the affinities are not proportional. Since dpp shows a relative activated disc larger than ptc , andsupposing that this variation comes only from differential affinities, this effect can only be obtained by the non proportionalrelation K dppR / K dppA > K ptcR / K ptcA (see Figures 1.a and 2.a for graphical examples of these effects).On the other hand, the dependence with a and r in (15) clarifies the main contribution of the activator and repressorinteraction intensities to the model: the larger the slope (this is, the larger the value of a ), the larger will be the relativeactivation disc, and the larger value of r , the less the value of the slope and hence less relative activation disc. In particular,the wide spreading of dpp , for the same reason as before, could be motivated by ( a dpp − ) / ( − r dpp ) > ( a ptc − ) / ( − r ptc ) (see Figure 1.b for a graphical example of this effect). Note also the non-dependence in (15) with the cooperation constant c ,meaning that the activation threshold will remain the same if the TFs cooperate between them in a total manner, no matter thevalue of the coefficient c is.However, if the operator is under the partial cooperation hypothesis (13) the expression of the threshold in general doesnot follow the linear dependence (15) because the partial cooperation constants c A , c R play an important role in the definitionof the activated and repressed states. Indeed, in Suplemental material 1.2 we compare this threshold with the linear one (15)obtaining the following result in terms of the magnitudes¯ a = c A c R − ra − (cid:8) ( ar − )( − c A − c R ) + ( c A − c R )( a − r ) (cid:9) , (19)¯ a = c A c R − r ( a − ) (cid:8) ( + a + a )( − r ) c A + c A ( − r )( a r − ) − c R ( a − )( − ar ) − ( + r + r )( a − ) c R (cid:9) . Depending on the sign of these values, given by the sign of the terms between brackets, it can be proved that the inclusion ofpartial cooperativity provokes: • If ¯ a > a >
0: an increment in the activation range with respect to the non cooperative case. • If ¯ a < a <
0: a decrement of the activation range with respect to the non cooperative case. • Otherwise: an increment or decrement depending on the total amount of Ci ( h ) considered. A detailed explanation canbe found in Suplemental material 1.2.See for instance Fig. 1 (e) and (f) where the values adopted verify ¯ a < a < Biochemical mechanisms that modulate signaling
Now we are going to prove very easily the next qualitative property: proportional binding affinities and total cooperation mod-ulate the activation/repression intensity. However, as we mentioned in the previous section, these biochemical mechanisms donot change the activation range.Indeed, by using electrophoretic mobility shift assays in vitro, it was measured in that Ci ptc sites affinities are consid-erably higher than affinities of Ci dpp sites which in terms of dissociation constants could be interpreted as K ptcR << K dppR and K ptcA << K dppA . Previously, we described how independent differential affinities were able to change the activation regions,and now we are going to see what could be the effect over the transcription levels if proportional differential affinities areassumed, that is, K ptcR = δ K dppR and K ptcA = δ K dppA being δ << . (20) [ C i R ] ( n M ) [ CiA ]( nM ) Independent Differential Affinity thresholds K dppA = . × K dppR = . × K ptcA = . K ptcR = . [ CiR ] = h − [ CiA ] (a) . . . . B E W A R E ( n M m i n − ) % Disc widthIndependent Differential Affinity K dppA = . × nM K dppR = . × nMK ptcA = . nM K ptcR = . nM (b) [ C i R ] ( n M ) [ CiA ]( nM ) Interaction Intensity thresholds a dpp = . r dpp = . × − a ptc = . r ptc = . [ CiR ] = h − [ CiA ] (c) . . . . B E W A R E ( n M m i n − ) % Disc widthInteraction Intensity a dpp = . r dpp = × − a ptc = . r ptc = . (d) [ C i R ] ( n M ) [ CiA ]( nM ) Partial Differential Cooperativity thresholds c dppA = . c dppR = . c ptcA = . c ptcR = . [ CiR ] = h − [ CiA ] (e) . . . . [ CiA ][ CiR ] B E W A R E ( n M m i n − ) % Disc widthPartial Differential Cooperativity c dppA = . c dppR = . c ptcA = . c ptcR = . (f) Figure 1.
Biochemical mechanisms changing activation/repression regions and spatial genetic expression. Blue lines areobtained from a non cooperative BEWARE operator ( (9), (10), (11)), under Cubitus distributions ((17),(18)) determined byparameters in table 1. Red dashed lines correspond to BEWARE operator involving new biochemical mechanism:Differential Independent Affinity (figures (a) and (b) ), Differential Control Intensity effects ( (c) and (d) ) and DifferentialPartial Cooperativity ( (e) and (f) ) with partial cooperative BEWARE operator (13). Inset in (f) shows the opposing gradientsof activator and repressor Ci considered in this work. Yellow circles in (a) , (c) and (e) are the intersection points ([ CiA ] th , [ CiR ] th ) while in (b) , (d) , (f) are the determined by distances x th .Since both genes are controlled by the common Ci signaling we can compare the corresponding protein production by com- aring the regulation factors for both genes, that is, F dppreg = − c + c (cid:18) + ac [ CiA ] K dppA + rc [ CiR ] K dppR (cid:19) − c + c (cid:18) + c [ CiA ] K dppA + c [ CiR ] K dppR (cid:19) versus F ptcreg = − c + c (cid:18) + ac [ CiA ] K ptcA + rc [ CiR ] K ptcR (cid:19) − c + c (cid:18) + c [ CiA ] K ptcA + c [ CiR ] K ptcR (cid:19) = − c + c (cid:18) + ac [ CiA ] δ K dppA + rc [ CiR ] δ K dppR (cid:19) − c + c (cid:18) + c [ CiA ] δ K dppA + c [ CiR ] δ K dppR (cid:19) where relation (20) has been replaced. Let us remark that this can be done thanks to the fact that the BEWARE operator (9)is monotone with respect to the Regulation Factor. It can be shown, by some monotonicity properties of these operators (seeSupplemental Material 1.3), that F ptcreg > F dppreg ( > ) , if [ CiA ] and [ CiR ] belong to the activation regionand F ptcreg < F dppreg ( < ) , if [ CiA ] and [ CiR ] belong to the repression region . This result can be interpreted in the following terms: binding affinity reduction (increment in δ ) provokes less activation in theactivation region and less repression in the repression region. That is, for low-affinity binding sites the signaling is attenuated.This effect can be clearly observed in previous literature, more concretely in Figure S6 (C) where a fit of a biophysical modelof BEWARE recruitment type considering non cooperativity ( c =
1) and equal affinity ( K R = K A ) were fitted to high-affinity(3 × Ci ptc ) and low-affinity (3 × Ci wt ) data. In this figure the threshold between activation/repression coincide in both fittingsand the relative expression of the low-affinity fitting is attenuated with respect to the high-affinity one.On the other hand, in the previous section we discussed that the total cooperation does not modify the threshold either. Infact we can prove (see Supplemental Material 1.3) that cooperativity strengthens the signaling in the sense that, for 1 ≤ c ≤ c , F reg ([ CiA ] , [ CiR ] ; { CiA , CiR } c ) ≤ F reg ([ CiA ] , [ CiR ] ; { CiA , CiR } c ) when [ CiA ] , [ CiR ] lay in the activation region while F reg ([ CiA ] , [ CiR ] ; { CiA , CiR } c ) ≥ F reg ([ CiA ] , [ CiR ] ; { CiA , CiR } c ) in the opposite case. That is, the increment on the value of total cooperativity provokes more activation/repression in theircorresponding regions. See Figure 2 for two examples of both attenuation (Fig 2 ( a ) and reinforcement (Fig 2 ( b ) effects. Discussion and conclusion
The modelling proposed in this paper provides mathematically treatable BEWARE modules enclosing two well accepted the-oretical approaches in the gene transcription modelling framework: the statistical thermodynamic method and the recruitmentmechanism. These expressions are susceptible of being tested in deep detail by using mathematical analysis. So they are use-ful tools for unravelling the complex balances that control the transcription process in gene expression. This is a methodologythat has been fruitfully applied in many other aspects of quantitative biology.The particular application of these ideas to the problem of differential spatial activation of the Hh target genes predictsthat: • some a priory hypotheses (proportional low-high affinity and total cooperation) should be disregarded as unique respon-sible of the change of spatial expression. • other a priory hypotheses (differential affinity and partial cooperation) are available alternatives to be tested, • new hypothesis (interaction intensity of the TFs ) could be taken into consideration, although the biological interpreta-tions of this point should be evaluated.The contrast with biological evidences, probably from a different point of view, will be necessary to improve the theoreticalunderstanding of this particular problem. From a more broader point of view, the repercussion of the analysis of thesefunctionals may be of deeper bearing as soon as this would be performed in a model focus on the mechanisms controlling thebalance between the transcription factors . . . . . B E W A R E ( n M m i n − ) % Disc widthProportional Differential Affinity K A = . × nM K R = . × nM ˜ K A = K A ×
10 ˜ K R = K R × (a) . . . . B E W A R E ( n M m i n − ) % Disc widthTotal Cooperativity c = c = (b) Figure 2.
Biochemical mechanisms that only attenuate or reinforce signaling. Blue lines are obtained from a noncooperative BEWARE operator ( (9), (10), (11)), under Cubitus distributions ((17),(18)) determined by parameters in table 1.Red dashed lines correspond to BEWARE operator involving the biochemical mechanism (tilda parameters): DifferentialTotal Affinity ( (a) ) and Differential Total Cooperativity effects ( (b) ) with total cooperative BEWARE operator (12).
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Acknowledgements
O. S. would like to thanks professor J. Garc´ıa-Ojalvo for pointing out reference . This work has been partially supported bythe MINECO-Feder (Spain) research grant number MTM2014-53406-R, the Junta de Andaluc´ıa (Spain) Project FQM 954,and the MINECO (Spain) research grant FPI2015/074837 (M.C.). SUPPLEMENTAL MATERIAL
In this subsection we present the different procedures used in order to rewrite the regulation factor used in the BEWAREoperator (9).We will work with two transcription factors: [ CiA ] and [ CiR ] , with activation and repression interaction intensityconstants a and r , and binding affinities K A and K R . The number of enhancers in the promoter will be denoted by n . In the next subsection we will show a procedure for obtaining the sums: S ( n ) x ( x A , x R ; C ) = j A + j R ≤ n ∑ j A , j R ≥ C ( C ) n ! j ! j A ! j R ! x j x j A A x j R R being j = n − j A − j R (21)appearing in regulation factor definition (10) by using the Multinomial theorem: ( x + x A + x R ) n = ∑ j + j A + j R = n n ! j ! j A ! j R ! x j x j A A x j R R where j , j A , j R ≥ . (22)Indeed, recalling expression (9), the BEWARE operator is written in terms of the Regulator Factor F reg ([ CiA ] , [ CiR ] ; C ) = j A + j R ≤ n ∑ j A , j R ≥ C ( C ) n ! j ! j A ! j R ! (cid:16) a [ CiA ] K A (cid:17) j A (cid:16) r [ CiR ] K R (cid:17) j R j A + j R ≤ n ∑ j A , j R ≥ C ( C ) n ! j ! j A ! j R ! (cid:16) [ CiA ] K A (cid:17) j A (cid:16) [ CiR ] K R (cid:17) j R = S ( n ) ( aK − A [ CiA ] , rK − R [ CiR ] ; C ) S ( n ) ( K − A [ CiA ] , K − R [ CiR ] ; C ) . We will get several expression for the regulation factor depending on the cooperation between TFs, i.e., non-cooperative(11), total-cooperative (12) and partial-cooperative (13), represented by C ( C ) . Next lines show the computations for thenumerators of the regulation factors, for each cooperation hypothesis, and the denominators will follow the same deductionimposing a = r = • Regulation factor for non-cooperative species
In this case the C ( C ) = S ( n ) ( aK − A [ CiA ] , rK − R [ CiR ] ; { CiA , CiR } )= j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! (cid:18) a [ CiA ] K A (cid:19) j A (cid:18) r [ CiR ] K R (cid:19) j R = ∑ j + j A + j R = n n ! j ! j A ! j R ! 1 j (cid:18) a [ CiA ] K A (cid:19) j A (cid:18) r [ CiR ] K R (cid:19) j R = ( + aK − A [ CiA ] + rK − R [ CiR ]) n . • Regulation factor for total-cooperative species
If the transcription factors cooperate between all of them, then the cooperation function is described by (6). Thecooperation function makes a bit more difficult the calculus of the polynomial (22), and first we need to get rid of thecooperation in both the numerator and denominator of the regulation factor. This can be easily achieved by splitting thesum S ( n ) ( aK − A [ CiA ] , rK − R [ CiR ] ; { CiA , CiR } c ) = j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! c ( j A + j R − ) + (cid:18) a [ CiA ] K A (cid:19) j A (cid:18) r [ CiR ] K R (cid:19) j R = + j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! c ( j A + j R − ) + (cid:18) a [ CiA ] K A (cid:19) j A (cid:18) r [ CiR ] K R (cid:19) j R = + c j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! (cid:18) ca [ CiA ] K A (cid:19) j A (cid:18) cr [ CiR ] K R (cid:19) j R = − c + c j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! (cid:18) ca [ CiA ] K A (cid:19) j A (cid:18) cr [ CiR ] K R (cid:19) j R , nd using (22) as before S ( n ) ( aK − A [ CiA ] , rK − R [ CiR ] ; { CiA , CiR } c ) = − c + c ( + caK − A [ CiA ] + crK − R [ CiR ]) n . • Regulation factor for partial-cooperative species
If the TFs cooperate independently (eq. (7)), we can split the sum twice in the same way as in the total-cooperationcase, i.e., S ( n ) ( aK − A [ CiA ] , rK − R [ CiR ] ; {{ CiA } c A , { CiR } c R } ) = j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! c ( j A − ) + A c ( j R − ) + R (cid:18) a [ CiA ] K A (cid:19) j A (cid:18) r [ CiR ] K R (cid:19) j R = j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! c ( j A − ) + A c ( j R − ) + R (cid:18) a [ CiA ] K A (cid:19) j A (cid:18) r [ CiR ] K R (cid:19) j R + j R ≤ n ∑ j A ≡ j R ≥ n ! j ! j R ! c ( j R − ) + R (cid:18) r [ CiR ] K R (cid:19) j R + j A ≤ n ∑ j A ≥ j R ≡ n ! j ! j A ! c ( j A − ) + A (cid:18) a [ CiA ] K A (cid:19) j A + = c A c R j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! (cid:18) c A a [ CiA ] K A (cid:19) j A (cid:18) c R r [ CiR ] K R (cid:19) j R + c R j R ≤ n ∑ j A ≡ j R ≥ n ! j ! j R ! (cid:18) c R r [ CiR ] K R (cid:19) j R + c A j A ≤ n ∑ j A ≥ j R ≡ n ! j ! j A ! (cid:18) c A a [ CiA ] K A (cid:19) j A + = c A c R j A + j R ≤ n ∑ j A , j R ≥ n ! j ! j A ! j R ! (cid:18) c A a [ CiA ] K A (cid:19) j A (cid:18) c R r [ CiR ] K R (cid:19) j R + (cid:18) − c A (cid:19) c R j R ≤ n ∑ j A ≡ j R ≥ n ! j ! j R ! (cid:18) c R r [ CiR ] K R (cid:19) j R + (cid:18) − c R (cid:19) c A j A ≤ n ∑ j A ≥ j R ≡ n ! j ! j A ! (cid:18) c A a [ CiA ] K A (cid:19) j A + − c A c R where, by the previous deduction, S ( n ) ( aK − A [ CiA ] , rK − R [ CiR ] ; {{ CiA } c A , { CiR } c R } ) (23) = c A c R ( + c A aK − A [ CiA ] + c R rK − R [ CiR ]) n + (cid:18) − c R (cid:19) ( + c A aK − A [ CiA ]) n c A + (cid:18) − c A (cid:19) ( + c R rK − R [ CiR ]) n c R + (cid:18) − c A (cid:19) (cid:18) − c R (cid:19) . In this subsection, as we announced at the end of Section Methods, we develop the mathematical analysis of the function f defining the threshold determined by F reg ([ CiA ] , [ CiR ] ; {{ CiA } c A , { CiR } c R } ) = . More concretly we are going to: • prove that the threshold is a regular increasing curve in the plane [ CiA ] − [ CiR ] by using the implicit function theorem. • describe the behaviour of this threshold. mposing the previous threshold condition we end up with the equivalent polynomial equation of the form G (cid:18) [ CiA ] K A , [ CiR ] K R (cid:19) = G (cid:0) ˜ A , ˜ R (cid:1) = (cid:0) + c A a ˜ A + c R r ˜ R (cid:1) − (cid:0) + c A ˜ A + c R ˜ R (cid:1) +( c R − ) h(cid:0) + c A a ˜ A (cid:1) − (cid:0) + c A ˜ A (cid:1) i + ( c A − ) h(cid:0) + c R r ˜ R (cid:1) − (cid:0) + c R ˜ R (cid:1) i . Please note that, by definition, G takes negative values in the repression region and positive values in the activation region.We are going to prove that G fulfils the hypothesis of the implicit function theorem. With some basic calculations werewrite the function G as a polynomial in the ˜ R = [ CiR ] K R repression variable, and ˜ A = [ CiA ] K A dependent coefficients, G ( ˜ A , ˜ R ) ≡ P ( ˜ R ) = a ( ˜ A ) + a ( ˜ A ) ˜ R + a ( ˜ A ) ˜ R + a ( ˜ A ) ˜ R (25)with ( a ( ˜ A ) = c R (cid:2) ( + ac A ˜ A ) − ( + c A ˜ A ) (cid:3) a i ( ˜ A ) = ( − i ) ! i ! c Ri (cid:2) r i ( + ac A ˜ A ) − i − ( + c A ˜ A ) − i + ( c A − )( r i − ) (cid:3) ∀ i = , , . Here we state some lemmas allowing us to employ the implicit function theorem.
Lemma 1.1
Let a > , r < and c A , c R ≥ . Then, for any positive value ˜ A, P ( ˜ R ) has an unique positive root, ˜ R ∗ , andP ′ ( ˜ R ∗ ) = ∂ G ∂ ˜ R ( ˜ A , ˜ R ∗ ) < . Proof.
First, note that a > a = c R c A ( r − ) <
0, due to the hypothesis on the parameters a > r <
1, and˜ A >
0. Then, it is clear that lim ˜ R → P ( ˜ R ) = a > ˜ R → ∞ P ( ˜ R ) = − ∞ , and hence there exist at least one positive root of P ( ˜ R ) . Note also that, if P has no real extrema, then the result is triviallyverified. If there exist real extrema of P , then their sign will provide information about the number of roots. In the casesof pairs of positive-negative and negative-negative extrema, it can be easily checked the existence of a unique positive root,verifying the result. In the remaining case, the existence of two positive extrema would imply the existence of three positiveroots. We are going to prove that this case cannot be achieved with the conditions of the parameters that the polynomial workswith.The hypothesis of two real possitive extrema would imply ˜ R (+) > ⇐⇒ − a + q a − a a < , ˜ R ( − ) > ⇐⇒ − a − q a − a a < , being ˜ R ± = − a ± q a − a a a , (26)due to a < a − a a ≥
0. For the same reason ˜ R ( − ) ≥ ˜ R (+) and in consequence condition (26) can beequivalently written as q a − a a < a . (27)From (27) we easily deduce that a > a < re necessary conditions because a <
0. Let us prove that both conditions, (28) and (29), are not compatible and in conse-quence (26) can not be verified.(28) can be written equivalently as ( r a − ) ˜ A + r − > r a > A > − r r a − ( ra − ) c A ˜ A + ( ra − ) ˜ A + r − < . In particular, when r a >
1, this inequality requieres 2 ( ra − ) ˜ A + r − < A < − r ( ra − ) . (31)Now, let us observe that necessary conditions (30) and (31), respectively for (28) and (29), are not compatible at all since r < r a > − r r a − > − r ( r a − ) > − r ( ra − ) . (cid:3) In an absolutely symmetric manner we can prove the analogous result fixing the variable ˜ R and the corresponding polyno-mial ¯ P ( ˜ A ) = G ( ˜ A , ˜ R ) . Lemma 1.2
Let a > , r < and c A , c R ≥ . Then, for any positive value ˜ R, ¯ P ( ˜ A ) has an unique positive root, ˜ A ∗ , and ¯ P ′ ( ˜ A ∗ ) = ∂ G ∂ ˜ A ( ˜ A ∗ , ˜ R ) > . Both results allow us to define a bijective function such that f ( ˜ A ) = ˜ R ∗ and f − ( ˜ R ) = ˜ A ∗ because of the uniqueness of theroots of P ( ˜ R ) and ¯ P ( ˜ A ) . The implicit function theorem gives that f is regular and increasing, since, G ( ˜ A , f ( ˜ A )) = A > = ∂ G ∂ ˜ A (cid:0) ˜ A , f ( ˜ A ) (cid:1) + ∂ G ∂ ˜ R (cid:0) ˜ A , f ( ˜ A ) (cid:1) f ′ ( ˜ A ) = ¯ P ′ ( ˜ A ) + P ′ ( f ( ˜ A )) f ′ ( ˜ A ) = ⇒ f ′ ( ˜ A ) = − ¯ P ′ ( ˜ A ) P ′ ( f ( ˜ A )) > . Indeed, the threshold could be computed explicitly by applying the classical Tartaglia-Cardano’s method (see Secc. 3.8.).However, we can show without using these explicit expressions that f tends asymptotically to a straight line as the concen-tration of the TFs increases. This can be easily shown by simply evaluating the threshold condition (24) on ˜ R = f ( ˜ A ) anddividing by ˜ A the whole equation. Then, tending the activators concentration to infinity leads to the equationlim ˜ A → ∞ (cid:18) c A a + c R r f ( ˜ A ) ˜ A (cid:19) − (cid:18) c A + c R f ( ˜ A ) ˜ A (cid:19) + ( c R − ) h ( c A a ) − c A i + ( c A − ) "(cid:18) c R r f ( ˜ A ) ˜ A (cid:19) − (cid:18) c R f ( ˜ A ) ˜ A (cid:19) = ( c A a + c R r α ) − ( c A + c R α ) + ( c R − ) h ( c A a ) − c A i + ( c A − ) h ( c R r α ) − ( c R α ) i = , where α = lim ˜ A → ∞ f ( ˜ A ) ˜ A is the slope that needs to be finite in the limit in order to fullfill the equation, and hence in the limit f ′ ( ˜ A ) → α .Indeed, one can compare this threshold with (15) by simply evaluating G in the straight line (15) since G (cid:18) ˜ A , − ar − A (cid:19) = ¯ P ( ˜ A ) = ( ¯ a + ¯ a ˜ A ) ˜ A , where the coefficients ¯ a and ¯ a are given by (19). This convoluted relation between the parameters defines the activation-repression range compared to the linear threshold, where depending on the sign of the coefficients we get: If ¯ a > a >
0, then the threshold for the BEWARE operator with partial cooperativity is over the threshold (15).In consequence, the activation range will increase due to partial cooperativity. • If ¯ a < a <
0, then the threshold for the BEWARE operator with partial cooperativity is under the threshold (15).In consequence, the activation range will decrease due to partial cooperativity. • If ¯ a < a >
0, the threshold for the the BEWARE operator with partial cooperativity is over the threshold (15) if˜ A = [ CiA ] K A < − ¯ a ¯ a and is under (15) otherwise. In this case, the change in the activation range will be determined by thetotal level of Ci protein in the system, h , by eq.(18). Let us consider [ CiA ] lth = h a − − r K R K A + [ CiA ] th the intersection point between the threshold for the BEWAREoperator with partial cooperativity.Then, if [ CiA ] lth < − ¯ a ¯ a it can be easily checked that [ CiA ] th < [ CiA ] lth . On the other hand, when, [ CiA ] lth > − ¯ a ¯ a the reverseinequality holds [ CiA ] th > [ CiA ] lth . That is, the activation range is larger or shorter with partial cooperativity dependingon h . • If ¯ a > a <
0, the situation is exactly opposite to the previous one. Now, if [ CiA ] lth > − ¯ a ¯ a then [ CiA ] th verifies − ¯ a ¯ a < [ CiA ] th < [ CiA ] lth and the activation range will increase. Furthermore, if [ CiA ] lth < − ¯ a ¯ a then the activation rangewill decrease because − ¯ a ¯ a > [ CiA ] th > [ CiA ] lth holds. In the main work we have stated that the regulation factor is monotonous decreasing and increasing with respect the affinityconstant in the activated and repressed zones, i.e.,
Lemma 1.3
Let us consider the functionG ( δ ) = (cid:0) − c (cid:1) + c (cid:16) + a c [ CiA ] δ K A + r c [ CiR ] δ K R (cid:17) n (cid:0) − c (cid:1) + c (cid:16) + c [ CiA ] δ K A + c [ CiR ] δ K R (cid:17) n = (cid:0) − c (cid:1) δ n + c (cid:16) δ + ac [ CiA ] K A + rc [ CiR ] K R (cid:17) n (cid:0) − c (cid:1) δ n + c (cid:16) δ + c [ CiA ] K A + c [ CiR ] K R (cid:17) n where [ CiA ] , [ CiR ] , K A , K R , a , r , δ are positive real numbers, n ≥ is a natural exponent and c is a real constant bigger orequal to . Then G is decreasing with respect to δ if and only if [ CiA ] and [ CiR ] verifya [ CiA ] δ K A + r [ CiR ] δ K R > [ CiA ] δ K A + [ CiR ] δ K R or equivalently G ( δ ) > , (32) and increasing otherwise. Proof.
This assertion can be easily verified computing ∂ G ∂δ = (cid:16)(cid:0) − c (cid:1) n δ n − c (cid:0) h ( β ) − h ( α ) (cid:1) + nc α n − β n − ( β − α ) (cid:17)(cid:16) (cid:0) − c (cid:1) δ n + c β n (cid:17) where α = δ + ac [ CiA ] K A + rc [ CiR ] K R , β = δ + c [ CiA ] K A + c [ CiR ] K R and h ( x ) = x n − ( x − δ ) . In the case of cooperativity ( c ≥
1) the signof this expression depends only on the differences h ( β ) − h ( α ) and β − α . Indeed both differences take always the samesign since α , β > δ and h ( x ) is an strictly increasing function for x ≥ δ . Condition (32) is equivalent to β < α implying thenegative character of ∂ G ∂δ while in the opposite case, β > α , ∂ G ∂δ is positive by the previous considerations. (cid:3) On the other hand, the regulation factor is monotonous increasing and decreasing with respect the total cooperation con-stant in the activated and repressed zones, i.e., emma 1.4
Let us consider the functionH ( c ) = (cid:0) − c (cid:1) + c (cid:16) + ca [ CiA ] K A + cr [ CiR ] δ K R (cid:17) n (cid:0) − c (cid:1) + c (cid:16) + c [ CiA ] K A + c [ CiR ] K R (cid:17) n = c − + (cid:16) + ca [ CiA ] K A + cr [ CiR ] K R (cid:17) n c − + (cid:16) + c [ CiA ] K A + c [ CiR ] K R (cid:17) n where [ CiA ] , [ CiR ] , K A , K R , a , r , are positive real numbers, n ≥ is a natural exponent and c is a real constant bigger orequal to . Then H is increasing with respect to c if and only if [ CiA ] and [ CiR ] verifya [ CiA ] K A + r [ CiR ] K R > [ CiA ] K A + [ CiR ] K R or equivalently H ( c ) > , (33) and decreasing otherwise. Proof.