Analysis of the transcriptional logic governing differential spatial expression in Hh target genes

This work provides theoretical tools to analyse the transcriptional effects of certain biochemical mechanisms (i.e. affinity and cooperativity) that have been proposed in previous literature to explain the proper spatial expression of Hedgehog target genes involved in Drosophila development. Specifically we have focused on the expression of decapentaplegic, wingless, stripe and patched. The transcription of these genes is believed to be controlled by enhancer modules able to interpret opposing gradients of the activator and repressor forms of the transcription factor Cubitus interruptus (Ci). This study is based on a thermodynamic approach, which provides expression rates for these genes. These expression rates are controlled by transcription factors which are competing and cooperating for common binding sites. We have made mathematical representations of the different expression rates which depend on multiple factors and variables. The expressions obtained with the model have been refined to produce simpler equivalent formulae which allow for their mathematical analysis. Thanks to this, we can evaluate the correlation between the different interactions involved in transcription and the biological features observed at tissular level. These mathematical models can be applied to other morphogenes to help understand the complex transcriptional logic of opposing activator and repressor gradients.

Hedgehog (Hh) is a morphogen, a signalling protein that induces several cellular 2 responses. It is involved in the development of different biological systems, for example 3 that of, the Drosophila melanogaster fly. In Drosophila's wing imaginal disc the 4 secretion of Hh from the Posterior compartment cells induces the expression of several 5 target genes inside the cells in the Anterior compartment. Among them are 6 decapentaplegic (dpp) and patched (ptc). Both give rise to the synthesis of their In [13], the spatial expression of both genes was related to the relatively higher 53 affinity between Cubitus proteins and dpp enhancers, than the affinity between Cubitus 54 proteins and ptc enhancers. Specifically, they confirm that, under moderate Hh signal, 55 low-affinity sites produce activation, whereas high-affinity sites produce repression. In 56 order to discriminate between the mechanisms that could give rise to a differential 57 spatial expression, they contrasted the experimental observations with fittings to a 58 thermodynamical model based on the ideas of Shea,Ackers and coworkers [2,19]. 59 Furthermore, by fitting a repressor cooperativity model in [13] they observed that CiR 60 plays a substantial role in the response to a moderate Hh signal. In [23] the authors also 61 proposed that the cooperativity between repressors may play an important role in the 62 change of the genetic expression along the imaginal disc, by using a mathematical model 63 of occupancy competition between repressors and activators. 64 The large amount of biochemical variables that are present in the system calls for 65 mathematical models that can shed some light on the origins of the differential spatial 66 expression in the target genes of Hh, among others. The thermodynamic model 67 proposed by Shea,Ackers and coworkers [2,19], also known as BEWARE [9] (Binding 68 Equilibrium Weighted Average Rate Expression), is a method frequently used in the 69 mathematical modelling of genetic transcription processes. See [3] or [8] for a general 70 discussion/comparison with other modelling approaches as for instance Boolean models. 71 However, this model gives rise to long and complex mathematical expressions even when 72 there are only a few transcription factors involved. For the analysis of independent and 73 specific binding sites and the analysis of two non competitive transcription factors, only 74 simple mathematical expressions have previously been proposed [4,7]. It is difficult to 75 decipher the biological effects in the model even if they are supported by numerical 76 tools [24] because the expressions inherently involve a great number of constants and 77 variables. 78 In this work we try to have a better understanding of the transcriptional logic of Hh 79 target genes from a theoretical point of view by using a thermodynamic model. Our effects. We conclude that differential expression of Hh target genes is due to the 86 combination of the differences in affinities between dpp and ptc binding sites and the 87 cooperativity between repressors. Confirming the results obtained in [23]. However, this 88 does not exclude that a different version of the transcriptional logic could be adopted by 89 some other biological systems. 90 In the case of a single activator gradient our analysis represents the well accepted 91 transcriptional logic which comes from the activator threshold model. This model 92 explains the role of certain biochemical factors involved in the signalling interpretation. 93 For example, differential affinities of activators for DNA elements [17] and cooperativity 94 between activators [10,13]. High-affinity binding sites and cooperativity between 95 activators benefit the binding of the activators to the enhancers, allowing the expression 96 of genes at low activator concentrations, so here we observe a broader response within 97 the activator gradient. In contrast, low-affinity sites and the absence of cooperativity 98 between activators restrict the gene expression to high activator concentration regions. 99 Although this rationale is well accepted in the single gradient scenario, it has not been 100 succesfully applied in combinatorial interactions as, for instance in opposing basal level [13], and hence there is no global activation/repression. The cellular 105 expression ranges (CERs) will not be determined only by signal intensity but also by 106 net activated/repressed cellular ranges. 107 Our analysis suggests that biochemical differences between genes can affect both, 108 signal modulation and changes in the net activated cellular ranges. That is, the 109 variation of CERs can be explained by analysing the combination of both aspects. 110 Another very interesting aspect is that using the same biochemical characteristics with 111 different cooperativity between activators and repressors will give drastically different 112 expression rates. Different types of cooperative interactions between transcription 113 factors will produce variations in the transcription logic. Another very interesting 114 aspect in our analysis is how the number of enhancers could modify the gene expression. 115 In [13] a transgenic fly line carrying a GFP reporter with a single high-affinity Ci site 116 was constructed in order to detect whether cooperativity between TFs played a 117 important role in the signalling process. Even in absence of cooperativity, this variation 118 of the number of enhancers could modify the gene expression, at least theoretically.

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This aspect, will play a key role in our argument in order to understand Hh target 120 genes. Although they seem to play a central role, affinity and cooperativity between 121 TFs are not the only biochemical factors involved in the interpretation of general 122 morphogen signalling. In the development of the chick embryo neural tube, another 123 paradigmatic morphogenetic patterning example, cells are differentiated in response to 124 Sonic Hedgehog (Shh) morphogenetic signals [22]. In this case, the Shh signal balances 125 the concentration of different versions of activators and repressors of the Gli family.

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These Gli TFs recognise target sequences which are very similar, however it has been 127 proposed that some TFs have more potency as activators or repressors [10] than others. 128 Thus, the different potency as activators and repressors could be another factor to be 129 taken into account by using differential TFs-RNAPs cooperativity.

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In this work we will study which of the previous factors can vary cellular expression 132 ranges by analysing their combined effect on both, the modulation of the signal and the 133 variation of the ranges of net activated/repressed cells. We begin with a modelling 134 exercise that will provide the mathematical representation (operators) of expression 135 levels to be later analysed. Firstly we apply the BEWARE method in order to obtain 136 expression rates of a gene controlled by two opposing general transcription factors: the 137 activator CiA and the repressor CiR. The "BEWARE operators" obtained include, for 138 each gen:

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• the transcription factors competing in order to bind one of n common  [CiR], will provide more or less gene expression than the basal. So, we define a 178 threshold separating concentrations of both TFs that would produce net activated cells 179 or net repressed cells using the BEWARE operators. Then, we can use the threshold 180 between net activation/repression concentrations to determine the limit between the 181 ranges of net activated or repressed cells. Once we have defined the ranges of activated 182 cells we can predict how biochemical differences will affect them as well as the signal 183 intensity. To do this analysis we need to assume that the opposing activator and 184 repressor gradients are monotone and do not change over time (see Fig.1 198 In [13], a transgenic fly line carrying a GFP reporter with a single high-affinity Ci site 199 (dppD-1xCi ptc ) was also constructed. In this case, the range of net activated cells was Net activated cellular (NAC) range described by a thermodynamic model. A) Schematic of the experiment for NAC range determination. The arrows represent all the possible interactions captured by a thermodynamic model determining the transcription rates: double-headed straight arrows show protein-DNA binding site affinities while single-headed black and red arrows are TFs-RNAP and TFs-TFs cooperativities respectively. The net activated cellular range of dppD3xCi ptc , a reporter gene with a version of the dpp enhancer with three high-affinity binding sites, is obtained by comparing its theoretical transcriptional activity with the activity of dppD3xCi KO , a gen containing different version of the dpp enhancer containing three null-affinity sites. Both cases are represented in the upper and lower schemes respectively. TFs binding sites are represented by rounded rectangles filled in green (high-affinity) or black (null affinity). B) Theoretical transcription rates predicted for both genes in cells of the Anterior compartment. This compartment occupies the 60% of the Drosophila imaginal disc and the Posterior compartment the rest (60% to 100%). The expression levels given by the BEWARE operators are between 0nM/min and 1nM/min being the basal level equal to 0.5nM/min. These reference expression levels have been chosen for a proper appreciation of signal modulation. Since dppD3xCi KO has been modelled independent of external factors it is expressed at basal level anywhere. Cells expressing dppD3xCi ptc more than the basal level are in the NAC range. The expression of both genes in the wing imaginal disk is also indicated by using coloured bars. The blue circle inside the bar, indicate the position of a cell expressing dppD3xCi ptc at the basal level. The color scale used in these bars is shown in C) black meaning no expression (0nM/min), and full color meaning high expression (1nM/min). The inset in B) depicts the activator/repressors (CiA/CiR) gradients generated by Hh signalling: activator concentrations are higher close to the Anterior/Posterior border.
TFs determines the effect of the binding sites reduction:

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• In the presence of total cooperativity or non-cooperativity between the TFs the 206 NAC range would essentially remain unaltered although a reduction of signal 207 intensity could be observed, that is, less repression/activation in the 208 repressed/activated cells.

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• If the activators CiA only cooperate between them a reduction in the NAC range 210 would be observed, so some net activated cells for dppD-3xCi ptc would change to 211 be net repressed for the gene dppD-1xCi ptc .

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• Finally, in the case of cooperativity only between repressors the NAC range would 213 be incremented, in concordance with the measurements for dppD-3xCi ptc and 214 dppD-1xCi ptc obtained in [13].

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These results (sumarised in Table 1 row 3)) are balancing a twofold consequence of 216 the reduction in the number of enhancers from 3 to 1. At one hand, the reduction 217 implies the vanishing of any possible kind of cooperativity between TFs. In the case of 218 total cooperativity between TFs these relations are symmetric for activators and 219 repressors, so their disappearance reduces the signalling, that is less transcription in 220 activated cells and more transcription in repressed cells, but it does not modify the 221 balance between net activated or repressed cellular ranges. In the case of asymmetric 222 cooperativity, that is, partial cooperativity either only between activators or repressors, 223 the cooperative specie is loosing that advantage. This would provoke a global reduction 224 of activation, in the case of activators cooperativity, and repression in the case of repressor cooperativity is abolished. The interpretation of these assertions on the 228 contrary allow us to affirm that one of the roles of asymmetric cooperativity between 229 activators/repressors is to increase/decrease the NAC range with respect to the NAC 230 range in the non cooperative case. A summary of these results can be found in Table 1 231 row 2). On the other hand, regardless of the cooperativity, the reduction in the number 232 of enhancers also implies that signalling has to be weakened. Both considerations 233 explains the theoretical transcriptional effects of the binding sites reduction.

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Under our modelling, the only transcriptional logic compatible with the experiments is 237 the one which occurs in presence of partial cooperativity between repressors.

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Nevertheless we can find more concordances in this direction by using other 239 experimental evidences. First column: schematic of the experiment 1: comparison of the expression ranges of reporter genes with 3 high-affinity sites (dppD-3xCi ptc ) and a single high-affinity Ci site (dppD-1xCi ptc ). A) corresponds to the non/total cooperativity case where, if cooperativity holds, all the TFs cooperate between them, C) to the activators cooperativity case, only activators cooperate, and finally E) to the repressors cooperativity case where only repressors cooperate. Second column shows the different transcriptional responses that can be theoretically described depending on the case of cooperativity considered. The schemes and plots employ the same keys explained in Fig. 1.
Experiment 2: Differential affinity effects. The second experiment we focus 241 on is the comparison of net transcriptional rates of reporter genes containing either three 242 high-affinity sites version of the ptc enhancer (dppD-3xCi ptc ) or three low-affinity dpp 243 sites (dppD-Ci wt ). It was observed that higher Ci affinity provides a reduction in the 244 net activated cellular region, that is a relevant intermediate region where net activated 245 cells for (dppD-Ci wt ) are net repressed for (dppD-3xCi ptc ). It was also observed that 246 increased affinity provides stronger activation in the region close to the A/P border as 247 well as a stronger repression in regions far from the same border (see Fig.2 D in [13]).

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If we accept that the cooperativity between TFs is working in the same way in the 249 binding to both, dpp and ptc, versions of the binding sites, our analysis suggests that 250 the effect of the reduction in affinity could again depend on the type of cooperativity 251 occurring between TFs:

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• In the presence of total cooperativity or non-cooperativity between the TFs the 253 NAC range would essentially remain unaltered although a reduction of signal 254 intensity could be observed, that is, less repression/activation in the net 255 repressed/activated cells.

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• If the activators CiA only cooperate between them the NAC range would be 257 reduced, because this range can be proved to be monotone increasing with affinity, 258 that is, the more affinity the wider the NAC range.

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• In the case of cooperativity only between repressors the NAC range would be 260 incremented, because this range is monotone decreasing with affinity, that is, the 261 more the affinity the narrower the NAC range.

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Again the repressor cooperative model is the only in concordance with the results 263 observed for dppD-3xCi ptc and dppD-3xCi wt in [13]. These theoretical behaviours can 264 be seen in Fig. 3. In all the previous analysis, summarised in Table 1  Thus, in the case of the Hh target genes, both results are only compatible with the 271 presence of cooperation between repressors. This conclusion coincides with the one 272 obtained in [13] by using numerical fittings to experimental data and with [16] where 273 the authors found that dpp requires low-affinity binding sites for normal activation in 274 regions of low Hh signalling. Indeed, our analysis allow us to interpret the roles of 275 differential affinity and repressor cooperativity for the Hh target genes in the following 276 terms:

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• The analysis of experiment 1 show us that repressors cooperativity reduces the 278 NAC ranges of both genes with respect to the non-cooperativity case.

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• Furthermore, the analysis of experiment 2 implies that this reduction of the NAC 280 region is less effective with low affinity binding sites of dpp than with high affinity 281 binding sites of ptc. ranges for dppD-1xCi ptc , dppD-Ci wt and dppD-3xCi ptc occupy from the 43%, 49% and 287 54% of the disc width respectively to the A/P border (which is around to the 60% of 288 the disc width). Although the results of our analysis have been before directly applied to the Hh target 292 genes, the analysis can be performed for any other genes controlled by opposing 293 transcription factors, A and R, activators and repressors respectively. We will adopt 294 from now on this general approach, and in the particular case of Hh target genes CiA 295 and CiR will play the role of A and R. As it was pointed out in the Introduction the 296 previous results are not compatible with the transcriptional logic of the activator 297 threshold model. So, in this section we will describe the versions of the transcriptional 298 logic that could be found depending on the cooperativity between the TFs: non/total 299 cooperativity, activators partial cooperativity and repressors partial cooperativity.

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The results of our analysis are summarised in Table 1 3) ↓ no. enhancers: gene1 has less binding sites than gene2, as for instance the genes 313 dppD-1xCi ptc and dppD-3xCi ptc , 314 4) ↓ A-RNAP coop.: the activator A is a weaker transcriptional activator for gene1 315 than for gene2, exhibiting weaker cooperativity between the activators and the 316 RNA polymerase, 317 5) ↓ R-RNAP coop.: the repressor R is a weaker transcriptional repressor for gene1 318 than for gene2, exhibiting weaker anti-cooperativity between the repressors and 319 the RNA polymerase.

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The results compiled in Table 1 also summarise the way in which the CER variations 321 happen. ↑ CER / ↓CER indicate gene1 has a broader/narrower CER than gene2

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The consequences of the differences in biochemical characteristics listed in Table 1 337 clearly justify the existence of several versions of the transcriptional logic in the 338 presence of opposing gradients depending on the kind of cooperativity between TFs.

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The region of net activated cells is highly relevant in the Non/Total cooperativity case 340 because the effects of some changes in biochemical characteristics only weaken the 341 signalling (↓ Sig) and in consequence they do not modify the NAC range ( \ NAC) (see 342 Key: ↑ increase, ↓ decrease, \ no change, ⇒ produces, NAC net activated cellular range, CER cellular expression range. Table 1. Transcriptional logics in the presence of opposing A/R gradients: This is a simplified comparison of the transcriptional response to different biochemical characteristics between two genes controlled by opposing activator/repressor gradients. The column headings are the kind of TF cooperativity analysed: non/total cooperativity (TFs can cooperate with any other TF), partial cooperativity only between activators or partial cooperativity only between repressors. The biochemical characteristics are: row 1): affinity of TFs for their binding sites, rows 2): cooperativity between TFS, row 3): number of enhancers, row 4) and 5): cooperativity between TFs and RNAP. The variation of affinity considered in row 1) is proportionally equivalent for both activators and repressors. The table shows whether the cellular expression ranges increase or decrease (↑ CER, ↓ CER) and how it works. Decreases in cooperativity between TFs and RNAP, rows 4) and 5), again produce globally higher/lower expression rates which cause the increase/decrease in the net activated cellular range (↑ NAC, ↓ NAC resp.) and CER. The response to differences in the other analysed biochemical characteristics varies depending on the kind of cooperation between TFs. If activators and repressor do not cooperate or cooperate globally the net activated cellular region remains unaltered ( \ NAC) and the signal is weakened (↓ Sig) provoking the decrease of activation in net activated cells but also repression in net repressed cells. On the other hand, if partial cooperation occurs between TFs the same biochemical characteristics can produce either increase or decrease of the net activated cellular region (↑↓ NAC) and in consequence broader or narrower expression ranges (↑↓ CER) depending on the type of cooperation.
NAC range is due to a global increment or decrement of expression rates as can be seen 347 in Fig. 5(B) and 6(B). Nevertheless, we can observe that the consequences resulting 348 from some others biochemical characteristics, as those described in Table 1 rows 4), 5), 349 are qualitatively the same, independently of cooperativity. Table 1 Table 1 rows 1), 2), 3). In these graphs thresholds and transcription 353 rates are represented in magenta for gene1 and green for gene2 in order to appreciate 354 relative differences.

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The same BEWARE operators can be used to find out the expression rates for a 356 single activator gradient simply by setting to zero the concentrations of repressors. In In this work we attempt to deepen the current understanding of genetic expression. By 362 using a theoretical approach we are able to isolate the biochemical mechanisms involved 363 in the expression of genes which operate through opposing activator and repressor TF 364 gradients, and the degree of their involvement. The BEWARE method is a well 365 accepted modelling tool which allows us to represent the delicate balance between 366 opposing signals using mathematical expressions and see how these proportions are 367 affected by the other biochemical characteristics involved. By reducing the previously 368 long BEWARE formulae into compact mathematical expressions we are able to deduce 369 the existence of several different forms of transcription logic, that is, several scenarios 370 where the same biochemical characteristics between genes produce absolutely different 371 consequences at a tissular level. The detailed description of the different scenarios and 372 the relationship between them allows us to contrast this theoretical framework with the 373 experimental evidence obtained in specific systems. This has been achieved in the case 374 of the Hedgehog Target genes dpp and ptc where we obtain conclusions analogous to 375 those obtained in previous work using other techniques.

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This work extends the applicability of the BEWARE method since the relevant 377 qualitative information can be extracted from the compact models. The fact that these 378 models could be applied in a similar way to other biological systems means there are 379 many interesting implications beyond the scope of this paper.
Here 'BEWARE()' represents a mathematical function specifying the dependence with 388 respect to the activation/repression role of the TFs. This is independent of other 389 possible factors relevant for the protein evolution as for instance degradation or spatial 390 dispersion. In the model, the binding reactions of TFs and RNAP in the enhancers and 391 promoter, respectively, are much faster than the synthesis of the protein G, hence it will 392 be considered in thermodynamic equilibrium given by the Law of Mass Action. If B is 393 an empty regulatory region, a set of non occupied enhancers-promoter, the complexes 394 BA, BR and BRNAP have concentration at equilibrium given by one transcription factor is considered as a sequential and competitive process, such that 406 the reactions are given by equilibrium concentrations A K (2) A

413
We denote as non cooperative TFs, all those proteins whose enhancer affinity is not 414 modified by any previously bound TFs, that is, they verify K independently of the sequential order of binding and of the specific positions occupied 423 by the TFs. Although Drosophila's wild type cis-regulatory elements involve a total 424 number of 3 binding sites we are going to compare with experiments where these 425 binding sites have been reduced to 1. Thus, we will consider in our model n ≥ 1 the 426 number of TFs binding sites. In all cases, we have a restriction for the possible number 427 of bound transcription factors. So, j A + j R ≤ n has to be verified, and in consequence 428 j 0 = n − j A − j R ≥ 0 denotes the number of free spaces in the configuration.

429
On the other hand, cooperativity occurs when the existence of other previously 430 bound proteins affects the affinity of the new binding protein of type i, k = A, R, that 431 is: k /c where c is a positive constant greater than 1 if proteins cooperate, and less than 1 if 433 anti-cooperativity occurs. Since the only difference between cooperativity and 434 anti-cooperativity is a threshold value for c, in the subsequent modelling we will refer to 435 Hh/Shh target genes by means of the Ci/Gli TFs [10,13,23]. Partial cooperativity of 440 the activators would occur when the existence of a bound activator modifies equally the 441 affinity of any posterior activator binding, that is K (j) A /c A for j ≥ 2. The same 442 applies for repressors. Total cooperativity would occur when the presence of a bound TF 443 modifies the affinity of any posterior binding in the same manner, i.e. K (j) R /c for j ≥ 2 (see for instance [11]). From now on, we 445 will denote the activator and repressor dissociation constants as K A or K R , such that in the presence of total cooperativity, while if partial cooperativity for TFs occurs. Here, (·) + denotes the positive part function . This is needed because the cooperativity will 449 not take place unless two or more cooperative TFs are present in the configuration. In 450 the rest of this paper, we will designate the cases when the The binding sites are ordered spatially and, in general, there is not a unique spatial 455 distribution for a configuration with j A activators, j R repressors and n − j A − j R free

462
Regarding the promoter's RNA polymerase binding process, the TFs work together 463 trying to promote or repress the binding process [12] by a mechanism known as 464 recruitment [14,15]. Thus, we consider that the activators interact with RNAP with 465 'adhesive' interaction [4] that gives rise to a modification of the RNA polymerase binding 466 affinity: K RP /a j A where a is a cooperativity constant greater than 1. In contrast, the 467 effect of j R repressors is modelled in terms of a 'repulsive' interaction that modifies the 468 binding affinity K RP /r j R with an anti-cooperativity factor r < 1 (repressor interaction). 469 We will refer to these parameters as TF transcriptional activation/repression intensity. 470 By using the previous guidelines we will now describe the concentrations of all 471 possible configurations as was done in [2, 19]:

472
Step 1: Construction of the sample space 473 All the possible ways of obtaining an equilibrium concentration with j A , j R and j P 474 activators, repressors and RNA polymerases is given by the states where j P = 1 means there is a bound RNA polymerase and j P = 0 there is none, 476 j 0 = n − j A − j R ≥ 0, and the variable C describes the relation of cooperativity between 477 the TFs. Specifically, by using (3) and (4), the cooperativity function C takes the values 478 and This allows us to describe the entire sample space, i.e. the space of all the possible 480 configurations, by 481 Ω = (j A , j R , j P ) ; j A , j R ≥ 0, j A + j R ≤ n, j P = 0, 1 .
Step 2: Definition of the probability 482 Once we have described all the possible configurations in terms of the concentrations 483 of activator, repressor and RNA polymerase, we easily obtain the probability of finding 484 the promoter in a particular configuration of j P RNA polymerase and j A , j R TFs 485 related by a cooperativity relation C as for all (j A , j R , j P ) ∈ Ω.

487
Step 3: Definition of the BEWARE operator

488
In this last step, the BEWARE operator is obtained in terms of the probabilities 489 P (n) . Following the work of Shea et al [19] the synthesis of a certain protein depends on 490 the total probability of finding RNA polymerase in the promoter, specifically, the 491 synthesis is proportional to the marginal distribution of the case j P = 1 [4,7,13]. This 492 justifies the definition of the BEWARE operator as where in definition (8) expression (5) is assumed and C B is a proportionality constant 494 that could depend on other factors not considered in this work. Splitting the 495 denominator in two sums, when RNA polymerase is bound or not bound to the 496 configuration, this expression can be rewritten in terms of the regulation factor function, 497 F reg : Doing some basic algebra, this regulation factor can be reduced to facilitate the 499 understanding of the general process (see Supplementary Material 1.1). This has been 500 done by using a classic strategy employed for obtaining the General Binding Equation 501 more than a century ago [5]. This, have not yet been applied, in this context, to the 502 authors knowledge. In fact, we can prove that the regulation factor can be equivalently 503 written as where the explicit expression of S (n) (x, y; C) depends on the kind of cooperativity 505 presumed, that is for the non cooperative, total and partial cooperative cases respectively. Note that the 507 mathematical complexity in these expressions is mainly related to the assumed 508 cooperativity.

509
Versions of transcription logic in the presence of opposing 510 gradients 511 In this section we are going to describe what the transcriptional reaction of genes, 512 controlled by the same opposing TFs, would be when there are biochemical differences 513 between them. As we explained in the section Results the consequences of such 514 differences will depend on the type of cooperativity occurring between the TFs. The 515 analysis of the case of single gradients can be found in Supplementary Material 1.4.

516
Transcriptional logic in the case of opposing gradients and non/total 517 cooperativity between TFs

518
We observe that expression (12) coincides with (11) when c = 1 which allows us to use 519 the same mathematical expression for both cases, non cooperativity and total 520 cooperativity. So, in both cases the transcription rates are given by (9) with being the regulation factor, where c = 1 if there is no cooperativity between TFs and 522 c > 1 total cooperativity occurs. In fact, we prove that the transcription logic will be 523 basically the same in both cases.

525
Thanks to the BEWARE operator we can theoretically describe which 526 concentrations of activators and repressors will cause higher or lower gene expressions 527 than the basal level, that is, we can describe in great detail the effect of the balance of 528 both signals. Note that the basal state, determined by the absence of TFs, that is

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[A] = [R] = 0, corresponds to F reg = 1 in expression (9). Thus, the regulation factor 530 describes an effective increase (for F reg > 1) or decrease (for F reg < 1) of the number of 531 May 23, 2018 15/21 RNAP molecules bound to the promoter, with respect to the basal level, as was 532 stablished in [4]. 533 We can see that in the case of (14) the threshold between activation/repression 534 concentrations (that is F reg = 1) is determined by the linear relation  K R /K A , it increases with respect to a and decreases with respect to r (since r < 1).

542
This justifies the behaviour of the net activated cellular ranges in the case of non/total 543 cooperativity stated in Table 1 as we will now explain.

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We can take this information and by considering appropriate gradients of activators 545 and repressors we can define the tissular regions of net activated and net repressed cells, 546 which are, regions of cells expressing more or less than the basal expression level (see 547 Fig. 1 for a detailed explanation). For the sake of clarity, and taking into account that 548 our main goal is to understand how these mechanisms could modify the expression of 549 the Hh target genes, we are going to assume that transcription factors act in the same 550 way as Cubitus works in the Drosophila system. Hh secreted from the posterior into the 551 anterior compartment of the wing imaginal disc results in opposing gradients of 552 activator and repressor Ci. The A/P boundary is located at around 60% of the Hence the concentrations will be restricted to a straight line in the imaginal disc occupied by activated cells will be determined by the distance x th given by 577 This limit is represented by blue circles in Fig. 4  Now, by using previous considerations, we want to justify the transcription logic in 581 presence of total cooperativity or in absence of any cooperativity, results collected in 582 Table 1, column a) and represented in Suplemental Material Fig. 4. In this case it is 583 quite easy to see the behaviour of the net activated cellular range. Equation (15), which 584 determines the threshold between activation/repression concentrations, does not depend 585 on the number of enhancers n or the cooperativity c and depends on the TFs affinities 586 in terms of the ratio K R /K A . Thus, these regions do not change ( \ NAC) for genes 587 gene1 and gene2 such that: 588 1) The affinity for the binding sites of gene1 is smaller than the affinity for the 589 binding sites of gene2 in a proportional manner. In terms of the dissociation 590 constants, this would be expressed as K g2 R = δK g1 R and K g2 A = δK g1 A being 591 0 < δ < 1 and this occurs because we are considering proportional change of 592 affinity for activator and repressors, K g1 R /K g1 2) The TFs cooperate in less intense manner for gene1 than for gene2, that is 594 c g1 < c g2 .

596
Here, the superscripts g1 and g2 stand for the parameters of the genes gene1 and gene2, 597 respectively. However, in the case of differential affinities, where the proportionality is 598 not verified, the net activated cellular range would change. For instance, if 599 K g2 R /K g2 A > K g1 R /K g1 A then the net activated cellular range for gene1 would be narrower 600 than for gene2.

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The rest of the assertions in Table 1, column a) requiere some simple monotonicity 602 properties that have been checked with Lemma 1. regulation factor F reg which allow us to extrapolate these estimates to expression rates. 608 These three results have been interpreted as a signal weakening. Biochemical differences 609 1), 2) and 3) can not modify the character of net activation/repression but are able to 610 make signalling less efficient. That is, they do not change the NAC range but they 611 cause less activation in the activated region and less repression in the repressed region 612 (↓ Sig). In consequence, we can say that in situations 1), 2) and 3) the expression rates 613 will decrease in the net activated cellular range and increase in the repressed cells, 614 which will attenuate the cellular expression range. See Fig. 4 in Supplementary Material 615 where these variations have been depicted.

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In contrast, assertion 3) in Lemma 1.2 implies that 617 • If A is a weaker transcriptional activator in gene1 than in gene2, exhibiting lower 618 cooperativity between the activators and the RNA polymerase (a g1 < a g2 ), then 619 expression of gene1 will be smaller than in gene2 because Obviously, this involves globally lower transcription rates (↓ Act) and more 621 restricted net activated cellular ranges (↓ NAC) for gene1 than for gene2.

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• If R is a weaker transcriptional repressor in gene1 than in gene2, exhibiting lower 623 anti-cooperativity between the repressors and the RNA polymerase 624 (r g2 < r g1 < 1), then expression rate of gene1 will be higher than gene2 expression 625 because . Obviously, this involves globally higher transcription rates (↓ Rep) and wider net 627 activated cellular ranges (↑ NAC) for gene1 than for gene2.

628
Transcription logic in the case of partial cooperativity between TFs

629
In the case of partial cooperativity, the expression rates are given by (9) where the 630 regulation factor is defined by: Compared to (14), in this complex representation of the expression rates the activation 632 threshold is not as clear. So we need to do a little more delicate mathematical analysis. 633 In fact, if we impose the threshold equation it can be shown that, if n = 3 and c A , c R ≥ 1, this threshold is determined by an 635 unique increasing function f , verifying  Material). Note also that the same analysis implies that the function f is independent 643 of the TF affinities: K A and K R .

644
We can understand the effects of partial cooperativity clearly in certain limit regimes 645 such as:

646
• Cooperativity only between repressors, that is c A = 1 and c R > 1. The expression 647 rates have been proved to be monotonic decreasing with respect to repressors 648 cooperativity, that is, the more cooperativity the less expression because 649 cooperativity increases repression effectivity (see Lemma 1.3 2)). Then, obviously, 650 less cooperativity implies less repression provoking wider NAC and CER.
• Cooperativity only between activators, that is c A > 1 and c R = 1. The

652
counterpart results in this other case shows that expression rates are increasing 653 with cooperativity between activators since it increases activation effectivity (see 654 Lemma 1. 3 1)).Then a reduction in cooperativity between activators reduces NAC 655 and CER. 656 These result have been summarised in Table1 cells 2b) and 2c) and represented in
Depending on the positive or negative sign of these values it can be proven that the 662 change from partial cooperativity to non cooperativity causes:

663
• Ifā 2 > 0 ,ā 3 > 0: we see an decrement in the net activated cellular range with 664 respect to the non cooperative case.

665
• Ifā 2 < 0 ,ā 3 < 0: we see a increment in the net activated cellular range with 666 respect to the non cooperative case.

667
• In all other cases: An increase or decrease of the net activated cellular range can 668 occur depending on the total amount of TFs considered (h), the binding affinities, 669 K A , K R as well as the activation/repression intensities. A detailed explanation 670 can be found in Supplementary Material 1.3.1.

671
Another interesting aspect is how these thresholds depend on the affinities of the 672 TFs in presence of partial cooperativity. Let us consider again gene2 with high affinity 673 binding sites and gene1 with proportionally low affinity binding sites, that is, 674 K g2 R = δK g1 R and K g2 A = δK g1 A being 0 < δ < 1 .
where the conservation of TF concentration, (17)