1 Introduction

Transcription

1 Publishe in IET Systems Biology eceive on 8th January 2010 evise on 13th July 2010 oi: /iet-syb Special issue on the Thir q-bio Conference on Cellular Information Processing Thermoynamic moels of combinatorial gene regulation by istant enhancers J. Narula O.. Igoshin Department of Bioengineering, ice University, 6500 Main Street, Houston, TX 77030, US ISSN bstract: The ynamical properties of istal an proximal gene regulatory elements are crucial to their functionality in gene regulatory networks. However, the multiplicity of regulatory interactions at trol elements makes their theoretical an experimental characterisation ifficult. Here a thermoynamic framework to escribe gene regulation by istant enhancers via a chromatin mechanism is evelope. In this mechanism transcription factors (TFs) moulate gene expression via shifts in the equilibrium between chromatin states. The esigns of ND, O, XO an NND two-input transcriptional gates for the chromatin mechanism are propose an compare to similar gates base on the irect physical interactions of TFs with the transcriptional machinery. n algorithm is evelope to estimate the thermoynamic parameters of chromatin mechanism gates from gene expression reporter ata an applie to characterise the response function for the Gata2-3 enhancer in hematopoietic stem cells. In aition waiting-time istributions for transcriptionally active states were analyse to expose the biophysical ifferences between the tact an chromatin mechanisms. These ifferences can be experimentally observe in single-cell experiments an therefore can serve as a signature of the gene regulation mechanism. Taken together these results inicate the iverse functionality an unique features of the chromatin mechanism of combinatorial gene regulation. 1 Introuction Differential regulation of gene expression is the key to cellular iversity in complex organisms. Its ynamical properties are trolle by unerlying gene regulatory networks (GNs) sisting of transcription factor (TF) genes an their cisregulatory elements that, together with basic transcriptional machinery, trol the expression levels of each gene [1]. The complexity of genetic regulation in higher organisms is relate to the complexity of the unerlying networks rather than the number of genes [2]. In particular, this complexity often manifests itself in combinatorial regulation of gene expression with multiple inputs verging on regulatory trol elements. Bining sites for transcriptional regulators are foun either in the immeiate vicinity of a transcription initiation site or in the enhancer sequences situate several kilobases upstream or ownstream [3]. The molecular mechanisms of gene regulation via istant enhancers are not very well unerstoo. The propose mechanisms of istal regulation of gene expression can be broaly characterise into two classes tact mechanisms an non-tact mechanisms. Contact mechanisms involve DN looping or packing that brings the enhancer-boun proteins close to the promoter (Fig. 1a) [4]. Non-tact mechanisms o not rely on irect physical tact of the enhancer-boun proteins an transcriptional machinery [5, 6]. Propose mechanisms of non-tact enhancer action inclue superhelical tension in negatively supercoile DN, nuclear localisation an nucleosome remoelling [3]. The nucleosome-remoelling hypothesis is particularly attractive as it explains why chromatin integration is often essential to observe any enhancer action [7, 8]. It also explains why enhancers in some single-cell measurements affect the probability of transcription rather than the rate of transcription [7, 9] an how in many cases enhancers regulate transcription in a manner that is inepenent of their orientation an istance relative to the transcription initiation site [10]. Dynamical moelling of GNs is often essential to unerstan their functionality as it provies information IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp oi: /iet-syb & The Institution of Engineering an Technology 2010

2 Figure 1 Two mechanisms of combinatorial gene regulation by istant enhancers a Contact mechanism: DN looping brings the istant enhancer-boun TFs an B close to the promoter-boun transcriptional machinery to allow protein protein interactions. TFs an B can activate transcription by stabilising promoter-boun transcriptional machinery or repress transcription by estabilising or sterically hinering the bining of the transcriptional machinery to the promoter b Chromatin mechanism: N polymerase bining sites are inaccessible in the close chromatin state where DN is tightly wrappe into nucleosomes. The bining sites become accessible when DN unwraps from nucleosomes an forms the open chromatin state. TFs an B (activators) bin to enhancer bining sites in the open chromatin state to shift the equilibrium towars the open state an increase the probability of gene transcription. TF C (repressor) bins to the enhancer in the close chromatin state an shifts the equilibrium away from the open state to ecrease the probability of transcription about network steay states an its responsiveness to physiologically important inputs an perturbations. Construction of such moels requires functional expressions that relate centrations of TFs to the rate of transcription of regulate genes. Common approaches to structing such input functions inclue the use of Boolean functions (such as logical gates), Hill functions an thermoynamic moels. Boolean moels an orinary ifferential equation (ODE) moels that use Hill functions provie useful qualitative information about the behaviour of regulatory networks. However, these approaches are base on phenomenological information about the networks rather than a specific biophysical mechanism of gene regulation [11 13]. In trast, thermoynamic treatment of transcriptional regulation provies a rigorous metho to translate hypotheses about the mechanism of transcriptional regulation into quantitative moels [14 16]. This approach has been extensively use to moel bacterial gene regulation but has not been wiely aopte for combinatorial regulation in higher organisms. We recently evelope a thermoynamic moel of istant enhancer activation via chromatin isruption an applie it to the ynamic moelling of the core network moule in haematopoiesis [7]. The moel assumes that the structure of chromatin in the gene neighbourhoo is in either an unstable open state that allows bining of the transcriptional machinery an gene transcription or a relatively stable close state that oes not allow transcription (see Fig. 1b). In the close chromatin state, the bining regions for the transcriptional machinery are wrappe in nucleosomes an are inaccessible, an no gene expression is possible from this state. The close chromatin state can spontaneously unwrap to an open state, where the bining sites become accessible an allow the transcriptional machinery to bin to the promoter an initiate transcription. 394 IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp & The Institution of Engineering an Technology 2010 oi: /iet-syb

3 This moel of chromatin structure ynamics is base upon experimental results that show that (i) the structure of chromatin, in particular nucleosomes, can impee transcriptional initiation [17, 18], (ii) chromatin exists in a ynamic equilibrium of open an close states [19] an (iii) TFs can isrupt chromatin structure by isplacing nucleosomes to trol gene transcription [20, 21]. The central iea of our moel is that by bining at the enhancer an moulating chromatin structure, TFs can trol the rate of gene expression without any physical interactions with the transcriptional machinery. In this paper, we further evelop a general thermoynamic framework to struct input functions of combinatorial gene regulation. We generalise this mechanism to inclue the possibility of negative regulation via stabilisation of the close chromatin formation. With that generalisation, we show that this chromatin mechanism is capable of generating the same logical input functions as the irect tact mechanism of transcriptional regulation [14, 15]. We further compare the sensitivities of the resulting input functions with respect to changes of the parameters an inicate important istinctions between tact an chromatin mechanisms. In aition, we escribe an approach that uses gene expression reporter measurements to estimate thermoynamic parameters an thereby characterise the complete response function for any enhancer esign an apply it to characterise the regulation of Gata2, an essential haematopoietic stem cell (HSC) gene, by a istant enhancer. Finally, we compare the ynamic properties of the two mechanisms with respect to waiting-time istributions of transcriptionally active an inactive states. Our results inicate that the chromatin mechanism of gene regulation can perform the same logic gate type input functions for transcriptional regulation as the tact mechanism. However, the ifferences in the biophysical mechanism (irect tact against chromatin) lea to ifferences in the esign of regulatory elements, in sensitivities to mutations an in ynamical properties. 2 esults 2.1 Thermoynamic formalism to moel combinatorial gene regulation via chromatin mechanism Quantitative characterisation of gene regulation requires a mathematical expression relating the rate of gene transcription to the centrations of TFs that regulate its expression. Because the initiation of transcription is usually the rate-limiting step in gene expression [22], at thermoynamic equilibrium the rate of gene expression I is given by the prouct of the bining probability p B of transcriptional machinery to the promoter an the rate of N polymerase isomerisation I 0 I = I 0 p B (1) We assume that in both tact an chromatin mechanisms of enhancer action, TFs at the enhancer moulate the transcriptional rate via probability p B. Bining of TFs an the transcriptional machinery at DN bining sites in the regulatory region generates multiple protein-boun DN figurations or microstates. t thermoynamic equilibrium, the probability p(a) of each microstate a is given by a Boltzmann istribution p(a) = e G a Z Here Z = S a e G a is a partition function that represents the sum of the Boltzmann weights of all possible figurations an G a is the imensionless free energy of each figuration in the units of kt. In orer to compute p B, we then simply sum up probabilities of all figurations where the transcriptional machinery is boun to the promoter (we enote this set by a T ) p B = p(a) (3) a[a T Using (2) an (3) we obtain the following expression for the cumulative probability for transcriptional machinery boun to the promoter p B = (2) Z ON Z ON + Z OFF (4) where we split the partition function Z into two parts Z ON an Z OFF, corresponing to transcriptional machinery boun an not boun states, respectively Z ON = e G a ; Z OFF = e G a (5) a[a T aóa T The free energies epen on the bining affinities, cooperative interaction energies an centrations of all boun proteins in that figuration. Therefore TFs can activate gene transcription by increasing Z ON or repress the transcription rate by increasing Z OFF. Concentrations of TFs an N polymerase/transcriptional machinery enter (4) an (5) via entropic tributions to free energies of each boun figuration G a = G 0 a i log([c i ]) (6) Here [C i ] stans for the centration of the ith protein an summation is over all the boun protein monomers in the figuration an G 0 a is the stanar free energy of the figuration at unit TF centration. Thus far the formalism is very general an can be applie to both mechanisms of enhancer action epicte in Fig. 1. The chromatin mechanism (Fig. 1b) inclues microstates corresponing to both open an close chromatin IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp oi: /iet-syb & The Institution of Engineering an Technology 2010

4 figurations. Transcription activators that attach to DN bining sites in the open chromatin state shift the equilibrium towars the open state an increase the probability of gene transcription (p B ). Similarly repressors bin to an stabilise the DN in the close chromatin state, thereby ecreasing p B. This can occur for instance if the bining site sists of the sequence motifs that are only brought in close physical tact by DN packaging (see Fig. 1b) [23, 24]. saresult,uner the chromatin mechanism TFs moulate the rate of transcription even without irect physical interaction with the transcriptional machinery. We first sier open an close chromatin states in the absence of TF or transcriptional machinery bining an efine their respective free energies as G 0 an G 1. Thereafter we set G 0 ¼ 0 an measure all the other free energies from this reference state. We efine an equilibrium stant of spontaneous DN opening (equilibrium stant for transitions between open an close chromatin in the absence of TF boun) as e G 1 = K (7) In most cases, the probability of spontaneous opening is low resulting in a large equilibrium stant, K 1 [19]. The bining of the transcriptional machinery occurs only in an open state an therefore only these states tribute to Z ON. On the other han, Z OFF inclues tributions from open chromatin enhancer figurations (a [ a Open ) an close chromatin enhancer figurations (a [ a Close ) an is represente as follows where Z OFF = Z Close + Z Open (8) Z Open = 1 + Z Close = K + a[a Open e G a an a[a Close e G a We assume that enhancer-boun TFs o not interact irectly with the promoter-boun transcriptional machinery. Therefore for each open chromatin microstate of the enhancer, the bining free energy for transcriptional machinery is the same value enote G T. s a result, the partition function Z ON is factorise as (9) Z ON = e G T Z Open (10) These general equations can be use to moel any gene regulatory element that functions via the chromatin mechanism. We will use these equations to moel the istant regulatory elements shown in Fig. 2b that implement input functions corresponing to various logic gates. 2.2 Implementation of cis-regulatory logic gate functions with chromatin mechanism Buchler et al. [15] showe that ND, O, NND an XO logic type cis-regulatory functions can be implemente with the tact moel. In this section, we show that enhancer-boun TFs can prouce similar logic gate input functions without TF transcriptional machinery interactions base on the thermoynamic formalism given by (4), (5) an (8) (10). The parameter values for each gate were numerically etermine to minimise the meansquare ifference from the corresponing gate function of the tact mechanism ([15] an Fig. S1). Our implementation of the ND gate type input function is schematically shown in Fig. 2a. Both TFs are activators an we assume that they only bin to DN in the open chromatin state an thereby increase p B. The probability of transcription for the enhancer element is calculate by using the following expressions for Z ON an Z OFF in (4) Z ON = e G T (1 + []e G + [B]e G B + [][B]e G G B G B ) Z OFF = K []e G + [B]e G B + [][B]e G G B G B (11) Clearly, the probability of transcription for an ND gate is maximum at saturating TF centrations, that is [], [B] 1 [cf. (4) an (11)]. We calculate the transcription rate normalise to this maximum level of expression an the results are shown in Fig. 3a. Note that because promoterboun transcriptional machinery an enhancer-boun TFs o not interact, the response function value at saturating levels of is always the same as the value at saturating levels of B. This is not generally true for the tact moel as the response function value at saturating centrations epens on the free energy of the TF transcriptional machinery interaction. However, for an appropriate choice of free energies this saturation effect will not be observe an the resulting input functions are very similar. The implementation of the O logic input function is shown in Fig. 2b. The esign of the O logic gate is similar to the ND gate in that both TFs increase the rate of gene expression by bining to the enhancer. The expressions for Z ON an Z OFF for the O logic enhancer are the same as the ones specifie in (11) for the ND gate. However, crucial ifferences between the ND logic an O logic gate esigns are in the parameters characterising the strength of TF-enhancer bining. In the case of ND gate, TFs bin to the enhancer weakly an the TF TF interactions stabilise the TF DN complex. On the other han, for O gate each TF bins very strongly to its bining site in the enhancer. We substitute the expressions from (11) into (4) to calculate the transcription rate for the O gate normalise relative to the maximum rate at [], [B] 1. Fig. 3b shows the transcriptional 396 IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp & The Institution of Engineering an Technology 2010 oi: /iet-syb

5 Figure 2 Designs of istant enhancers that exhibit a logic gate response a ND gate response: the bining sites for the TFs in the istant enhancer are weak an cooperative interaction between TFs is strong b O gate response: the TF bining sites in the enhancer are strong an there is no cooperativity c NND gate response: the TFs bin weakly to the enhancer sites in the close chromatin figuration XO gate response: only one TF can bin to the enhancer in the open chromatin state because the bining sites overlap. Bining of the TFs to the close chromatin state is weak but highly cooperative. Dashe lines inicate irect physical interaction. Weaker bining sites are hatche In the chromatin mechanism, activators increase the probability of transcription by bining to the enhancer in the open chromatin state an repressors ecrease gene expression by bining to enhancers in the close chromatin state input function of the O gate. The O gate type logic of the input function shown here an the tact moel input function are very similar (cf. Fig. 3b an Fig. S1b). TFs an B act as repressors of transcription in the NND gate input function. Therefore we assume that both TFs bin to the enhancer in the close chromatin state (Fig. 2b) an ecrease the probability of transcription. This NND gate is implemente with bining sites in the enhancer (silencer), unlike the implementation for the tact moel NND gate where the TF bining sites must overlap with the bining site of the transcriptional machinery [15]. p B epens on the following Z ON an Z OFF Z ON = e G T Z OFF = K []e G + [B]e G B + [][B]e G G B (12) The resulting analytical expression of the chromatin mechanism NND is ientical to that of the tact mechanism because of the lack of TF transcriptional machinery interaction energies in either mechanism. Therefore we can analytically fin parameter values for the chromatin mechanism such that its normalise transcription rate shown in Fig. 3c is ientical to the normalise transcription rate of the tact mechanism [see (12) an (13) in Supplementary information an Fig. S1c]. The XO gate input function is obtaine with the chromatin mechanism as shown in Fig. 2. In this esign, TFs an B both have two bining sites in the enhancer. One pair of bining sites is only accessible to the TFs in the open chromatin state (bining affinities G 1, G 1 B) an the two sites overlap such that only one TF can be boun at a time. The other pair of bining sites is weak an only accessible to TFs for bining in the close chromatin state (bining affinities G 2, G 2 B). The two TFs bin to the close chromatin sites cooperatively (high free energy of interaction G 2 B). Z ON an Z OFF in this case are given by the following equations Z ON = e G T (1 + []e G1 + [B]e G1 B) Z OFF = K + []e G2 + [B]e G2 B + [][B]e G2 G2 B G2 B []e G1 + [B]e G1 B (13) We use the above equations with numerically estimate parameters (see the Methos section) to calculate the normalise transcription rate relative to the maximum IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp oi: /iet-syb & The Institution of Engineering an Technology 2010

6 these ifferences, we calculate the logarithmic sensitivity as follows [25] S Gi = log(p B) G i (14) Here p B is the probability of transcription as given by (4) an (11), an the inex i ¼, B, B inicates a specific free energy. These free energies are easily affecte by mutations in the DN TF bining sites. Therefore the sensitivities are important inicators of the evolutionary robustness an aaptability of the transcriptional response. Figure 3 Gene expression response functions for the logic gates of the chromatin mechanism a For the ND gate response, the normalise rate of transcription is calculate relative to the transcription rate at high TF centrations [] ¼ [B] ¼ 10 3 (G ¼ G B ¼ 5.26, G B ¼ 20.70, K ¼ 39.4, e G T = 1.16) b For the O gate type response, the normalise rate of transcription is calculate using (11) relative to the transcription rate at [] ¼ [B] ¼ 10 3 (G ¼ G B ¼ 3.38, G B ¼ 1.4, K ¼ 24.14, e G T = 2.15) c Normalise rate of gene transcription for the NND response function relative to the transcription rate at [] ¼ [B] ¼ 0 (G ¼ G B ¼ 1.61, G B ¼ 0, K ¼ 19, e G T = 2000) Normalise rate of transcription from (13) relative to the rate of transcription at [] ¼ 10 3, [B] ¼ 0 shows XO logic (G 1 ¼ G 1 B ¼ 24.52, G 2 ¼ G 2 B ¼ 20.10, G 2 B ¼ 26.22, K ¼ 7.51, e G T = 1.15) Parameters were etermine numerically so that the response function for each logic gate is as close as possible to the corresponing response function for the respective logic gate of the tact mechanism transcription rate for the case [] 1, [B] ¼ 0 (or equivalently [] ¼ 0, [B] 1). Despite our attempt to match the response functions of the tact mechanism XO gates (single promoter moel), the shape of the resulting response functions is slightly ifferent (cf. Fig. 3 an Fig. S1). However, we argue that the chromatin mechanism s esign mimics an XO gate better than the tact mechanism s esign. In fact, the response function of the XO gate of the chromatin mechanism is similar to the response of the tact mechanism XO gate that involves two promoters (cf. [15]). 2.3 Sensitivities of logic input functions to free energy values The logic gate response functions iscusse above epen on the values of free energies G i of TF bining an interactions. Even though the chromatin mechanism is capable of matching the response function gates, it still may possess ifferent sensitivities to parameter variation. To quantify The ND gate response is most sensitive to the free energies G, G B an G B at high centrations of TFs an B, respectively. The sensitivity of the ND gate response to these free energies is similar for the chromatin mechanism an the irect tact mechanism (see Figs. S2a ). The sensitivities of the O gate response to free energies G, G B an G B were calculate using (11). The chromatin mechanism O gate response is sensitive to G an G B in a larger range of TF centrations than the irect tact mechanism (see Figs. 4a an b). However, the irect tact mechanism shows more sensitivity to G B near saturating centrations of TFs an B (see Figs. 4c an ). The sensitivities of the NND gate response to variations in the TF bining an interaction energies are ientical for the two mechanisms. This is expecte because the moels for the two systems are exactly the same as shown above [see (12) an (13) in the Supplementary information an Figs. S2e h]. The sensitivities of the XO gates to various free energies iffer significantly between the two mechanisms (see Figs. S2i l ). However, these ifferences in sensitivity are mainly because of the issimilarity of the response functions themselves. In summary, we foun that the sensitivities to free energies for ND an NND logic gate responses of the chromatin mechanism o not iffer significantly from the corresponing sensitivities of the tact mechanism. However, the chromatin mechanism O gate response is more sensitive to free energies of TF-enhancer bining. This suggests that the O gate response of the chromatin mechanism is more sensitive to mutations in the TF bining sites. 2.4 Parameter estimation for chromatin moel from experiments Statistical thermoynamic moels can be use to preict the transcriptional response combinatorial cis-regulatory enhancers have over a range of TF centrations an quantitatively characterise ifferent esigns of gene regulation as shown above. But these moels usually have a 398 IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp & The Institution of Engineering an Technology 2010 oi: /iet-syb

7 for the Gata2 enhancer reporter structs have recently become available [29]. s shown in Fig. 5, Gata2 bins to an enhancer 3 kb upstream (Gata2 3) along with another TF Fli1 to upregulate its own transcription [29]. Both TFs enhance Gata2 gene expression [29]; therefore we assume that they bin to the Gata2 3 enhancer only in the open chromatin state. The effect of the Gata2 3 enhancer on gene expression was recently measure experimentally an reporte in [29]. The authors clone the Gata2 3 enhancer upstream of a SV40 promoter trolling a LacZ reporter gene [7, 29]. Thereafter, this struct was integrate into the genome of haematopoietic progenitor cells that show high centrations of Gata2 an Fli1. The Figure 4 Sensitivity of O gate response to variations of free energies values a an b Sensitivity of the transcription probability to the free energy of TF bining, G for the tact mechanism an chromatin mechanism, respectively. The chromatin mechanism has a larger region of high sensitivity than the response of the tact mechanism c an Sensitivity of the transcriptional probability to the interaction energy between two TFs G B for the tact mechanism an chromatin mechanism, respectively. For both mechanisms, the response is sensitive to G B only at high TF centrations. In this region, the response for the tact mechanism is more sensitive large number of inepenent parameters the free energies of all the figurations. Direct measurement of these parameters can be very cumbersome an without the parameter values it is ifficult to relate results from these moels to experimental information about gene expression. This problem greatly limits the utility of thermoynamic moels. In this section, we outline an approach that reuces the imensions of the unknown parameter space for the chromatin mechanism using experimental measurements of gene expression from enhancer-reporter structs. s a result, a hanful of reporter measurements allow us to quantitatively restruct the full transcriptional response function. To illustrate our approach for parameter estimation, we evelop a thermoynamic moel of the regulation of Gata2, a gene that regulates the specification an ifferentiation of HSCs [26 30]. Enforce over-expression an knockout experiments have shown that that the trol of Gata2 expression has major implications for HSC function [29 32]. Gata2 gene expression is an ieal example for the illustration of our parameter estimation approach because its regulation is epenent on the presence of multiple TFs as well as the chromatin organisation of istant upstream regulatory regions [29, 31, 32]. Moreover, experimental gene expression measurements Figure 5 pplication of the parameter estimation metho to the Gata2 3 enhancer a Schematic representation of Gata2 3 an mutant enhancer reporter structs. The wil-type (wt) enhancer tains both Gata2 an Fli1 bining sites, Enhancer 1 (E1) tains only a Gata2 bining site an Enhancer 2 (E2) tains only Fli1 bining sites. The numbers show the fol expression enhancement relative to the expression from Enhancer 3 (E3), which oes not have any TF bining sites (ata taken from [7, 29]). These measurements are use in (20) (22) to calculate the parameters of the Gata2 response function b Gata2 enhancer response function shows ND type logic for a chromatin equilibrium stant of K ¼ 300. We have normalise the Gata2 an Fli1 centrations with the respective wil-type centrations an the white lines emarcate this physiologically relevant range of TF centrations. Note that the transcription rates are normalise relative to the minimum transcription rate at [Gata2] ¼ [Fli1] ¼ 0. The fol change uner over-expression of Gata2 an Fli1 is K c Sensitivity of the Gata2 response to the value of the chromatin equilibrium stant K. The Gata2 response function is not sensitive to the value of K within the range of wil-type centrations of TFs (emarcate by white lines). However, the response is sensitive to the value of the chromatin equilibrium stant when TFs are overexpresse IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp oi: /iet-syb & The Institution of Engineering an Technology 2010

8 cells were then isrupte an analyse for b-galactosiase activity. ssuming that the reporter protein is stable, the level of b-galactosiase activity in cells with the enhancer reporter struct is irectly proportional to the rate of reporter transcription in these cells. The measure rate of transcription of the reporter I G is proportional to the probability p B that the promoter is boun by the transcriptional machinery [see (4)]. Since N polymerase bins typical core promoters very weakly [14, 33 35] we fin from (8) (10) that Z ON Z OFF (15) ccoringly, we keep only Z OFF in the enominator of (4) for p B to obtain absence of enhancers. Note in (16) that the factors e G T will cancel as ratios of transcription rates are compute. We use the equations above to relate the free energies of ifferent figurations to the fol enhancement of gene expression. The enhancer E1 can only bin Gata2. ccoringly, the ratio of transcription rates I G E1/I G E3 epens only on the free energy G G of TF Gata2 an the equilibrium stant K G I G E1 IE3 G = pe1 B p E3 = (e G G + 1)/(K G + e G G + 1) B 1/(K G + 1) (17) Similarly, only Fli1 can bin to the enhancer E2 an the rate of gene transcription from this enhancer relative to the expression rate from E3 only epens upon G F an K G (see (19)). p B = e G T (1 + e G G + e G F + e G FG ) K s e G G + e G F + e G FG (16) I G E2 IE3 G = pe2 B p E3 = (e G F + 1)/(K G + e G F + 1) B 1/(K G + 1) (18) Here G G, G F an G FG represent the free energies of the Gata2-boun, Fli1 imer-boun an Gata2 Fli1 imerboun enhancer figurations, respectively. G FG inclues the bining affinities G G, G F as well as the free energy of the Gata2 Fli1 protein protein interaction. These free energies follow the efinition in (6) an inclue entropic tributions from wil-type centrations of Gata2 an Fli1 an the centrations of Gata2 an Fli1 are normalise with these wil-type centrations. K G represents the equilibrium stant for transitions between open an close chromatin. The general iea behin the approach is that if the bining site of a TF is mutate or elete, then the bining of that TF to the mutate enhancer becomes energetically unfavourable an the corresponing terms are exclue from both the numerator an enominator in the expression for p B. This allows us to compute one of the remaining free energies from the ratio of transcription rates of reporters with wil-type an mutate enhancers. Fig. 5a shows the fol expression enhancement for the reporter struct in the presence of the wil-type (wt) Gata2 3 enhancer an three reuce versions of this enhancer: Enhancer 1 (E1) Fli1 bining sites elete, Enhancer 2 (E2) Gata2 bining site elete, Enhancer 3 (E3) all bining sites elete. ll the experimental ata have been abstracte from [7, 29]. The fol change in gene expression for ifferent enhancer reporter structs was calculate by normalising the level of b-galactosiase activity of cells with enhancer reporter structs with the b-galactosiase activity levels of cells with enhancerless reporter structs. Therefore fol enhancements of gene expression reflect the ratio of p B in the presence an The ratio of transcription rates from the wil-type Gata2 3 enhancer an the enhancer E3 is easily structe using (16) an this ratio epens on the free energies G G, G F an G FG (see (19)). Equations (17) an (18) are solve analytically for G G an G F in terms of the equilibrium stant K G. ( ) G G = log (1 I E1/I G E3)(K G G + 1) (K G + 1 IE1 G /I E3 G ) ( ) G F = log (1 I E2/I G E3)(K G G + 1) (K G + 1 IE1 G /I E3 G ) (20) (21) These solutions are use in (19) to solve for G FG as a function of only K G. ( G FG = log (1 I wt/i G E3)(K G G + 1) (K G + 1 Iwt/I G E3 G ) ) (1 I E1/I G E3)(K G G + 1) (K G + 1 IE1 G /I E3 G ) (1 I E2/I G E3)(K G G + 1) (K G + 1 IE2 G /I E3 G ) (22) Thus, using experimental ata from [7, 29] for fol enhancement of gene expression in (20) (22), we reuce the imensions of the parameter space to one. If we know K G we can uniquely etermine the free energies G G, G F an G FG. The unknown parameter K G can only be experimentally etermine through overexpression of one of the TFs but these ata are currently unavailable. We assume an appropriate value for K G to calculate the free energies an I G wt IE3 G = pwt B p E3 = (e G G + e G F + e G FG + 1)/(K G + e G G + e G F + e G FG + 1) B 1/(K G + 1) (19) 400 IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp & The Institution of Engineering an Technology 2010 oi: /iet-syb

9 the response function of the Gata2 3 enhancer. The response function is shown in Fig. 5b an inicates that the Gata2 3 enhancer functions as an asymmetric ND gate. Cooperative bining of Fli1 an Gata2 preicte from our estimations ensures that a high expression level is achieve only in the vicinity of maximal centrations of both TFs. s the exact value of K G is unknown, we explore the sensitivity of the Gata2 response to the value of the chromatin equilibrium stant. The logarithmic sensitivity was calculate with (14) an is shown in Fig. 5c. Wefoun that the ND logic property of the response function is not sensitive to the value of the equilibrium stant K G (see Fig. 5c an Fig. S3). We can choose any value from the range K G. I G wt/i G E3 ¼ 120 (here we choose K G ¼ 300). Note that the maximum fol change in gene expression is approximately K G, thus showing that the chromatin equilibrium stant can be measure by overexpression of TFs. Note that the choice of this equilibrium stant can affect the ynamic properties of the transcriptional response. In the struction of ynamical ODE type moels, the value of the chromatin equilibrium stant may also be straine by qualitative phenotypic requirements (cf. [7]). 2.5 Comparison of stochastic kinetics of gene regulation by irect tact an chromatin mechanisms So far we have focuse on the steay-state transcriptional response for combinatorial gene regulation via the chromatin mechanism an foun that the chromatin mechanism can mimic the transcriptional response of the tact mechanism when the effect of the TFs is symmetric. In such cases, it might be ifficult to istinguish the two mechanisms base on steay-state measurements of gene expression levels. However, the two mechanisms can still be istinguishe base upon the ifferences in their ynamics. ecent avances in single molecule experimental techniques offer a wealth of ata about the ynamics of transcriptional regulation in single cells [36 38]. In this section, we will use a simple example to show how single molecule experimental ata about the ynamics of gene regulation can be use to infer the mechanism of gene regulation. For simplicity, we use a toy moel with a single transcription activator to emonstrate two ifferences in the microscopic kinetics of these two moels that can be experimentally observe. Consier a gene that is regulate by a single TF that bins to a istant enhancer. TF up-regulates gene expression via irect physical tact with the transcriptional machinery or by shifting the equilibrium of local chromatin structure to an open formation in which the promoter is accessible to the transcriptional machinery. The probability of gene transcription in thermoynamic equilibrium for both mechanisms can be calculate using the framework iscusse in Section 2.1. Note that throughout this section, the superscripts an chr enote the irect tact mechanism an the chromatin mechanism, respectively. For the tact mechanism, the probability of transcription p B is calculate using (4) p B = 1 + []/K e G T (1 + v[]/k + e G T ) (1 + v[]/k ) (23) Here K is the issociation equilibrium stant of TFenhancer bining an v represents the strength of TF transcriptional machinery interaction. The probability of transcription p chr B for the chromatin mechanism is also calculate using (4) p chr B = K []/K chr e Gchr T (1 + []/K chr ) + e Gchr T (1 + []/K chr) (24) Similar to the tact moel, K chr represents the enhancer bining energy of TF an G chr T represents the free energy of transcription machinery bining. Note that there is no interaction energy between the TF an transcriptional machinery in the chromatin mechanism. It can easily be shown that the probabilities of transcription p B an p chr B are equal for all TF centrations if the parameters are chosen accoring to K chr G chr T K = v 1 = K /v = G T log(v) (25) When these three itions are satisfie, the steay-state rate of gene expression is the same for both mechanisms. However, there are still ifferences in the kinetics of the bining an issociation of the transcriptional machinery in these two mechanisms. Figs. 6a an b show the kinetic schemes for the irect tact an chromatin mechanisms, respectively. The moel for the irect tact mechanism involves four figurations of the regulatory region empty (O), TF boun (O ), transcriptional machinery boun (O ) an both TF an transcriptional machinery boun (O ). The moel for the chromatin mechanism involves five figurations: close chromatin (C), open chromatin-empty (O), TF boun (O ), transcriptional machinery boun (O ) an TF an transcriptional machinery boun (O ). We assume that only the rate stants of TF an transcriptional machinery issociation from DN are affecte by their respective affinities for the bining sites. Using this assumption an (25), the rate stants of TF/transcriptional machinery bining an issociation reactions are set to the values shown in Figs. 6a an b. itionally, in the chromatin moel k o an k c, the rate stants of spontaneous close to open chromatin (C to O) an open to close chromatin (O to C) transitions, respectively, are relate to the chromatin equilibrium stant efine in (25) as: K ¼ k c /k o. IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp oi: /iet-syb & The Institution of Engineering an Technology 2010

10 Methos section). P ON (t) represents the PDF that the first exit from the ON state lies in the interval (t 0 + t, t 0 + t + Dt), given that the system entere the ON state at time t 0. Similarly P OFF (t) represents the PDF of the first exit times from the OFF state. The fraction of time spent in the ON state is irectly relate to the rate of transcription. The ON an OFF times are relate to transcriptionally active an transcriptionally inactive states an therefore may be obtaine from the time-series ata of single-cell gene expression. In aition, several groups have alreay shown that the time spent in transcriptional machinery boun an unboun states can be tracke in vivo by aing fluorescent protein bining hairpin loops to the mn tail en [40 42] or by localisation enhancement that can etect protein bining an issociation from a specific location [37, 38]. The waiting time istributions P ON (t) an P OFF (t) can be etermine from these types of experiments an qualitative features of these istributions can be use to etermine whether the gene regulation mechanism involves irect interactions between TFs an transcriptional machinery. Figure 6 Waiting time istributions for transcriptional machinery boun an unboun states a an b Kinetic schemes of the irect tact an chromatin mechanisms for transcriptional regulation by a single activator. O, O, O an O enote empty enhancer, activator boun, transcriptional machinery boun an activator + transcriptional machinery boun figurations of the of the enhancer-gene locus in both mechanisms. In aition, these four figurations of the chromatin mechanism represent the open chromatin figurations whereas the close chromatin figuration is enote by C. The transcription machinery-boun figurations O an O together represent the ON state for both mechanisms. ll other figurations are part of the OFF state. The rate stants of the transitions between ifferent states are shown above the respective arrows (see the text for etails) c PDF for waiting times in the ON state for the tact mechanism (ashe line) shows two ifferent timescales whereas the PDF for the chromatin mechanism (soli line) shows only one timescale. The time axis is normalise by the timescale of issociation of the transcriptional machinery in the chromatin mechanism (k /v) 21 PDF for the waiting time in the OFF state for the chromatin mechanism (soli line) shows three timescales whereas the PDF for the tact mechanism (ashe line) shows only one. The time axis is normalise by the timescale of bining of the transcriptional machinery (k ) 21 e Mean waiting times in the ON state kt ONl, kt chr ONl. kt chr ONl (soli line) is not a function of TF centration whereas kt ON l (ashe line) increases with an increase in TF centration f Mean waiting times in the OFF state kt OFFl, kt chr OFFl. Waiting time kt chr OFFl ecreases with an increase in TF centration whereas kt OFFl is inepenent of TF centration We use the methos iscusse in [39] to calculate the probability ensity functions (PDFs) of waiting times in the transcriptional machinery boun (ON) states an unboun states (OFF) for the two mechanisms (see the Fig. 6c shows the PDF for time spent in the ON state for the half-saturate TF centration [] ¼ K. The waiting time PDF P chr ON(t) for the chromatin moel epens on only one rate stant of issociation of transcriptional machinery k /v P chr ON(t) = k v e k t/v (26) This happens because the rate of transcriptional machinery issociation is the same for the O an O states without irect interactions with the activator. In trast, for the tact mechanism the rate of exit from the ON state epens on whether the system is in sub-state O or O because the rate of exit from the two states is ifferent. ccoringly, the PDF P ON(t) is a sum of two exponential terms P ON(t) = w 1 r 1 e r 1t + (1 w 1 )r 2 e r 2t (27) r 1 an r 2 represent two ifferent timescales for the waiting time in the ON state an w 1 [ [0, 1] is a weighing factor that represents the probability of observing the r 1 timescale. The weighting factors an P ON(t) can be moulate with the TF centration. In the absence of TF ([] 0) the probability of being in the substate O is zero. s a result, w 1 1 an r 1 k resulting in an exponential ecay of P ON(t) with the characteristic rate k. t saturating levels of ([] 1), the substate O ominates, w 1 0 an r 2 k /v, an this results in an exponential ecay of P ON(t) with the characteristic rate k /v. The tact moel assumes a strong interaction of transcriptional machinery an the activator resulting in separation of these timescales (v. 1). For intermeiate centrations of TF two timescales are visible (see Fig. 6c). The range of centrations for which this 402 IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp & The Institution of Engineering an Technology 2010 oi: /iet-syb

11 timescale separation can be observe epens on the strength of the TF transcriptional machinery interaction v an k /k. (1/k ) in the waiting time istribution. P OFF (t) = k e k t (31) The mean waiting times in the ON state are as follows The mean waiting times in the OFF state are as follows kt ONl = w 1 r w 1 r 2 kt chr ONl = v k (28) kt chr OFFl = c 1 + c 2 + (1 c 1 c 2 ) r 1 r 2 r 3 kt OFFl = 1 k (32) Notably kt chr ONl is inepenent of the TF centration whereas kt ONl increases with TF centration till kt ONl ¼ kt chr ONl at saturating centrations of (see Fig. 6e). Moreover, because there are no TF transcriptional machinery interactions in the chromatin moel, kt chr ONl has only one timescale an is inepenent of TF centration. This result will hol even when multiple TFs bin to the enhancer to regulate gene expression (not shown). The situation is ifferent for the istribution of transcriptionally inactive states (Fig. 6). In this case, the waiting time istribution in the OFF state for the tact moel is exponential P OFF(t) = k e k t (29) This is a sequence of the assumption that the bining rate of the polymerase oes not change with the presence of an activator (only issociation rate oes). Therefore the ecay of PDF is etermine by the rate stant of the bining transcriptional machinery k. In trast, up to three istinct timescales can be present in the waiting time istribution in the OFF state for the chromatin mechanism (Fig. 6, soli line). The three ifferent timescales of the chromatin mechanism are reflecte in the PDF P chr OFF(t), which sists of three exponentials P chr OFF(t) = c 1 r 1 e r 1t + c 2 r 2 e r 2t + (1 c 1 c 2 )r 3 e r 3t (30) Here r 1, r 2, an r 3 represent three ifferent timescales for the waiting time in the OFF state an c 1, c 2 [ (0, 1) are weighing factors that represent the probabilities of the r 1 an r 2 timescales, respectively. P chr OFF(t) is moulate by changing the TF centration. In the absence of TF there are only two timescales. The fast timescale correspons to the irect exit from the open state (O) an the slow timescale involves switching between the open an close (C) chromatin states before exiting from the open state. s the TF centration is increase, three timescales become visible owing to the presence of an O state. t saturating centrations of the equilibrium of the open an close chromatin states shifts almost completely towars the open state. s a result, the chromatin mechanism resembles the tact mechanism an there is only a single timescale We fin that kt OFFl is inepenent of TF centrations whereas kt chr OFFl ecreases with TF centration till kt OFFl ¼ kt chr OFFl at saturating levels of TF (see Fig. 6f ). Note that while changing the TF centration affects only kt chr OFFl in the chromatin mechanism an only kt ONl in the tact mechanism, the fractional time spent in the ON state is the same for both mechanisms kt chr ONl/(kt chr ONl + kt chr OFFl) ¼ kt ONl/(kt ONl + kt OFFl). This fractional time in the ON state is proportional to the probability of transcription p B. Because we assume that the probability of transcription is the same for both mechanisms, this result shows that our analysis is self-sistent. Our results show that qualitative ifferences in the ON an OFF state waiting time istributions can be use to ientify the biophysical mechanism of gene regulation. lthough a relatively simple moel was chosen to illustrate the effect, many of the results can be generalise for combinatorial regulation by multiple TFs. more etaile investigation of the weight-time istributions will be reporte elsewhere. 3 Discussion Our results show that the chromatin mechanism an the irect tact mechanism are capable of creating functionally similar logic transcriptional gates. Notably, the NND gate is a universal gate that can be use to create any logic gate. Moreover, transcriptional logic gates are tinuous functions that can be aapte to more complicate combinatorial operations by tuning the TF bining affinities through bining site mutations as shown by the sensitivity analysis. This aaptability suggests that virtually any response function can be structe with a combination of ifferent logic gates an appropriate manipulation of TF bining affinities. lthough chromatin an irect tact mechanisms can show functionally equivalent transcriptional responses, the esigns of regulatory elements for any transcriptional input function are very ifferent between the two mechanisms. These ifferences in esign may have important implications. The chromatin mechanism is more flexible in the esign of enhancers. Specific interactions between TFs an the transcriptional machinery are unnecessary to prouce the same response as the tact mechanism. The only requirement is that the chromatin structure at the enhancer IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp oi: /iet-syb & The Institution of Engineering an Technology 2010

12 an the gene unpack together. This allows a lot of flexibility in enhancer location because chromatin omains as large as several kilobases in length can open as a whole [43]. It is important to note that accoring to the chromatin mechanism, enhancers can act in a non-specific manner to activate the transcription of genes in the neighbourhoo of the target genes. In fact, this type of non-specific transcriptional activation is sistent with a number of reports regaring the effects of istant enhancers an locus trol regions in eukaryotes [44 46]. Each TF bins to the enhancer an isturbs the equilibrium between open an close chromatin states an changes the probability of bining of other TFs in a manner similar to the Mono Wyman Changeux moel for allosteric enzymes [47]. s a result, physical TF TF interactions are unnecessary for cooperativity between TFs. For example, the NND gate response functions of the two mechanisms are ientical [cf. equations (22) an (23) in Supplementary information] but the response equation for the tact mechanism has an explicit TF TF interaction term whereas the equation for the chromatin mechanism oes not. The effective cooperativity that emerges from the equilibrium between open an close chromatin in this case is equivalent to an effective free energy of interaction (see Supplementary information). The emergence of cooperativity without irect physical interaction between TFs means that any two DN bining proteins can be use as TFs uner the chromatin mechanism. In trast, the irect tact mechanism restricts the location of bining sites for transcriptional regulation. First, transcriptional repression requires bining sites in the promoter vicinity. For example, in the NND gate of the tact moel [15], bining sites for repressors an B must be in the promoter region so that they can occlue the N polymerase bining site. Se, for all tact gates the free energies of DN looping an TF transcriptional machinery interaction affect the possible enhancer location. Thir, each response function requires specific omains on TFs that are responsible for the appropriate TF transcriptional machinery interactions. Both the tact mechanism an chromatin mechanism can be utilise for combinatorial gene regulation in higher organisms an it might be necessary to investigate the particulars of the mechanism for each gene. Our moels suggest several experimental esigns to istinguish between the alternatives. lthough the two mechanisms are functionally equivalent within the operating range of TF centrations, we coul istinguish the two through force over-expression of any one of the TFs. Saturating centrations of any TF will show the same level of gene expression for the chromatin mechanism. On the other han, expression rates at saturating centrations of ifferent TFs might be ifferent for the irect tact mechanism. nother metho involves shifting the position of the enhancer relative to the promoter. egulation by tact mechanism is sensitive to such translocations because the free energy of the enhancer-boun TFs an promoter-boun transcriptional machinery epens on the istance between them. egulation by the chromatin mechanism will likely be unaffecte by translocation of the enhancer because local accessibility of DN at the enhancer can be propagate over long istances (several kbs) to establish an open chromatin state [43]. Interestingly, the sensitivities of the ND an NND logic gate esigns to free energies of TF-enhancer bining are not very ifferent for the two mechanisms. However, the chromatin mechanism O gate is more sensitive to TF-enhancer bining free energies. This increase sensitivity of the O gate response suggests that this esign is more sensitive to bining site mutations than the equivalent esign of the tact mechanism. The thermoynamic approach that we have evelope allows us to characterise gene expression input functions base on a hanful of transcriptional reporter measurements. From this perspective, gene regulation via the chromatin mechanism is easier to quantify because it oes not involve bining energies between TFs an the transcriptional machinery an therefore involves fewer parameters. In this case, the metho that we have propose can use experimental results irectly in parameter estimation. lthough the chromatin mechanism an the irect tact mechanism can prouce functionally equivalent timeaverage transcriptional responses, there are intrinsic ifferences between the two mechanisms that nevertheless lea to ifferences in the stochastic kinetics of gene expression. We have shown that the chromatin mechanism can be istinguishe from the tact mechanism base on single-molecule gene expression ata. The chromatin mechanism can easily be ientifie from such ata from the single characteristic timescale in the PDF of time spent in the transcriptionally active state. On the other han, multiple timescales are present in the PDF of time spent in transcriptionally inactive state. We have also foun that the mean waiting time in the ON state is inepenent of TF centration for the chromatin mechanism. These istinguishing ynamical properties highlight the irreucible ifferences between the two mechanisms. lthough these results were obtaine using a somewhat oversimplifie moel of transcriptional activation, we expect our observations to hol even for more complex kinetic schemes. This will be a subject of a separate investigation. We also note that transcriptional regulators using the chromatin mechanism have potentially promising applications in synthetic biology an genetic engineering. t present, synthetic biology circuits use simple promoter architectures with a single regulator to trol gene transcription. This clearly limits the transcriptional response of the gene an functional properties of the circuits. This limitation exists because combinatorial gene regulation that follows the tact mechanism requires specialise TFs with appropriately interacting omains. t the same time, cisregulatory moules of living systems, especially eukaryotes, are typically auntingly complex. The increase in complexity of gene regulation is associate with the evolutionary 404 IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp & The Institution of Engineering an Technology 2010 oi: /iet-syb

13 emergence of complex multicellular organisms [2, 48]. Moreover, the increase in proteins that trol chromatin structure an nucleosome remoelling correlates well with the increase in complexity of cis-trol elements in metazoans [2]. This aoption of the chromatin mechanism of gene regulation in higher organisms reflects the avantages of the flexibility in esign of complex combinatorial regulation. Synthetic esigns of combinatorial regulation base on the chromatin mechanism can harness this flexibility to avoi the limitations of the tact mechanism regulation. The esigns of logic gates with combinatorial regulation via the chromatin mechanism that we have iscusse in this paper are only an inication of how this mechanism can help simplify the esign of synthetic circuits for any transcriptional response function. 4 Methos 4.1 Calculation of waiting times in transcriptional machinery boun (ON) an unboun (OFF) states The methos iscusse in [39] were use to calculate the PDF of the time spent in the transcriptional machinery boun (ON) an unboun (OFF) states for the tact an chromatin mechanism kinetic schemes. The ynamics for either mechanism in the ON state can be escribe by the following system of ODEs G = (H U)G (33) t here H is a rate matrix such that each element H ij represents the rate stant of j i transition an H ii ¼ 2S j=i H ij. U ij ¼ K ij i = j, an the j i transition represents the issociation of the transcriptional machinery from the regulatory region. ll remaining elements of U are set to zero. Similarly, V ij ¼ K ij i = j an the j i transition represents the bining of transcriptional machinery to the regulatory region with the initial itions, G j (t) is the probability that the system has reache state j at time t without the issociation of transcriptional machinery given that the system was initially in the ON state. We use the equations above to calculate the PDF of time spent in ON state P ON (t) P ON (t) = (UG(t)) p in where p in = Vpss 1 Vp ss (34) Here p ss ( j), the vector of steay probability of each state, is compute using (2) an the partition function for each mechanism (see Section 2.5). The probability p in ( j)ofentering the ON substate j is a weighing factor for the calculation of the ON state PDF ( represents transpose an 1 represents a unit vector). Similarly the OFF state PDF can also be calculate as P OFF (t) = (VG(t)) p in where p in = Upss 1 Up ss (35) We use (34) to calculate the PDF for ON state waiting times for both the chromatin an tact mechanisms (see (37)) P chr ON(t) = k v e k t/v (36) where w 1 [ (0, 1) is a weighing factor. These equations show that the ON state PDF for the chromatin mechanism has only one timescale whereas the PDF for the tact mechanism has two timescales: 1/r 1 an 1/r 2. Similarly, we use (35) to calculate the PDF for OFF state waiting times for both the tact an chromatin mechanisms PON(t) = k e k t (38) POFF(t) chr = w 1 r 1 e r1t + w 2 r 2 e r2t + (1 w 1 w 2 )r 3 e r 3t (39) where w 1, w 2 [ (0, 1) are weighing factors. Equations (38) an (39) show that the waiting time istribution of the OFF state in the tact moel has only one timescale, (k ) 21 whereas the PDF of the chromatin mechanism has three timescales (see Supplementary information for etails). The moments of these waiting time istributions can be easily calculate from the PDFs. 4.2 Construction of logic gates for the chromatin mechanism We efine the response function for the logic gates as f i ([], [B]) = p B([], [B]) max(p B ) (40) where f i is the normalise rate of gene expression in the presence of TFs an B (i ¼ for the tact P ON(t) = w 1 r 1 e r1t + (1 w 1 )r 2 e r 2t r 1,2 = 1 k ( 2 v + k [] + k + k v + k ) 2 ( ) v + k [] + k + k 4 k k v v + k k [] + (k )2 (37) v v IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp oi: /iet-syb & The Institution of Engineering an Technology 2010

14 mechanism; i ¼ chr for the chromatin mechanism) relative to the maximum rate of expression. Parameters an equations for the tact mechanism logic gates were taken from for each of the logic gates were erive using (4) an the appropriate expressions for Z ON an Z OFF from Section 2.2. [15]. The expressions for p chr B f an f chr were use to struct an objective function S that represents the mean square ifference between the response functions of the two mechanisms. S = ( f ([], [B]) f chr ([], [B])) 2 log[] log[b] (41) The parameters for the ND, O an XO gates of the chromatin mechanism were estimate numerically by minimising S. The fminimax library routine of MTLB was use to solve the non-linear optimisation. Parameters of the NND gate of the chromatin mechanism were erive analytically from the parameters of the NND gate of the tact mechanism. See Supplementary information for the etails. 5 cknowlegments The authors woul like to thank Dr. Bertie Gottgens an Dr. ileen Smith for useful iscussions regaring the regulation of Gata2. We also wish to thank J. Christian J. ay an bhinav Tiwari for their comments on the theoretical aspects of this stuy. J.N. an O..I. are supporte by ice University startup funs an NSF awar MCB eferences [1] DVIDSON E.H., ST J.P., OLIVIEI P., ET L.: genomic regulatory network for evelopment, Science, 2002, 295, (5560), pp [2] LEVINE M., TJIN.: Transcription regulation an animal iversity, Nature, 2003, 424, (6945), pp [3] BLCKWOOD E.M., KDONG J.T.: Going the istance: a current view of enhancer action, Science, 1998, 281, (5373), pp [4] PTSHNE M., GNN.: Transcriptional activation by recruitment, Nature, 1997, 386, (6625), pp [5] POLCH K.J., WIDOM J.: Mechanism of protein access to specific DN sequences in chromatin: a ynamic equilibrium moel for gene regulation, J. Mol. Biol., 1995, 254, (2), pp [6] VEH-SDK T., LEVO M., SEGL E.: Incorporating nucleosomes into thermoynamic moels of transcription regulation, Genome es., 2009, 19, (8), pp [7] NUL J., SMITH.M., GOTTGENS B., IGOSHIN O..: Moeling reveals bistability an low-pass filtering in the network moule etermining stem cell fate, PLoS Comput. Biol., 2009, 6, (5), pp. e [8] PIMND J.E., DONLDSON I.J., DE BUJIN M.F., ET L.: The SCL transcriptional network an BMP signaling pathway interact to regulate UNX1 activity, Proc. Natl. ca. Sci. US, 2007, 104, (3), pp [9] WLTES M.C., FIEING S., EIDEMILLE J., ET L.: Enhancers increase the probability but not the level of gene expression, Proc. Natl. ca. Sci. US, 1995, 92, (15), pp [10] KHOUY G., GUSS P.: Enhancer elements, Cell, 1983, 33, (2), pp [11] CHVES M., LBET., SONTG E.D.: obustness an fragility of Boolean moels for genetic regulatory networks, J. Theoret. Biol., 2005, 235, (3), pp [12] KUFFMN S., PETESON C., SMUELSSON B., TOEIN C.: anom Boolean network moels an the yeast transcriptional network, Proc. Natl. ca. Sci. US, 2003, 100, (25), pp [13] LSLO P., SPOONE C.J., WMFLSH., ET L.: Multilineage transcriptional priming an etermination of alternate hematopoietic cell fates, Cell, 2006, 126, (4), pp [14] BINTU L., BUCHLE N.E., GCI H.G., ET L.: Transcriptional regulation by the numbers: moels, Curr. Opin. Genet. Dev., 2005, 15, (2), pp [15] BUCHLE N.E., GELND U., HW T.: On schemes of combinatorial transcription logic, Proc. Natl. ca. Sci. US, 2003, 100, (9), pp [16] SHE M.., CKES G.K.: The O trol system of bacteriophage lamba. physical chemical moel for gene regulation, J. Mol. Biol., 1985, 181, (2), pp [17] HN M., GUNSTEIN M.: Nucleosome loss activates yeast ownstream promoters in vivo, Cell, 1988, 55, (6), pp [18] LOCH Y., LPOINTE J.W., KONBEG.D.: Nucleosomes inhibit the initiation of transcription but allow chain elongation with the isplacement of histones, Cell, 1987, 49, (2), pp IET Syst. Biol., 2010, Vol. 4, Iss. 6, pp & The Institution of Engineering an Technology 2010 oi: /iet-syb

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17 Detaile Methos 1. Calculation of waiting times in transcriptional machinery boun (ON) an unboun (OFF) states The methos iscusse in [1] were use to calculate the probability istribution function (PDF) of the time spent in the transcriptional machinery boun (ON) an unboun (OFF) states for the tact an chromatin mechanism kinetic schemes. The ynamics for either mechanism can be escribe by a system of orinary ifferential equations. Let H be a rate matrix such that each element H ij represents the rate stant of j itransition an Hii = H ij. j i To obtain the PDF of the time spent in the ON state we first require the probability that the system oes not release the boun transcriptional machinery for time t given that the transcriptional machinery bins to the regulatory region at t = 0. Given that the system starts in the ON sub-state j, then G () t, the probability that the system is in the ON sub-state i at time i t without releasing the transcriptional machinery is given by the rate equation : G = ( HU - ) G (1) t with the initial itions G j (0) = 1, G k (0) = 0 k j. Here U is the matrix of rate stants such that Uij = Kij i jan the j itransition represents the issociation of the transcriptional machinery from the regulatory region. ll remaining elements of U are set to zero. Similarly we can efine a matrix V such that Vij = Kij i jan the j itransition represents the bining of transcriptional machinery to the regulatory region. The solution of

18 equation (1) is G( t) = exp(( K U)) t G (0). The PDF of spening time τ in the ON state epens on three factors: 1) the probability of entering the ON substate j at time t 0, 2) the probability of going from substate j substate i at time t 0 i in time τ an 3) the probability of exiting the ON state from ON + τ. This PDF can be calculate as: P ON ( τ) = ( UG( τ) ) p where, p ss Vp = 1 Vp i n ss in (2) Here p ss ( j), the vector of steay probability of each state is compute using equation (2) an the partition function for each mechanism (see section 2.5). The probability p ( j) of entering the ON substate j is a weighing factor for the calculation of the ON state PDF ( represents transpose an 1 represents a unit vector). Similarly the OFF state PDF can also be calculate: in P OFF where, ( τ) = ( VG( τ)) p p in ss Up = 1 Up ss in (3) For the tact mechanism we use the following numbering of sub-states: O (1), O (2), O (3) an O (4). For Figure 6(a), the matrices H, U an V for the tact mechanism are: ( k + k [ ]) k k 0 k [ ] ( k + k ) 0 k / ω H = (4) k 0 ( k + k [ ]) k / ω 0 k k [ ] ( k / ω + k / ω)

19 0 0 k k / ω U = (5) V = (6) k k 0 0 To fin P ON () τ, the PDF of waiting time in the ON state for the tact moel equations, first the equations (1), (2) an (4)-(6) were use with the Laplace transform to solve for P% ( s) : ON P% ON kk k kk kk [ ] k k [ ] + ( k [ ] + k + s) + k + + k + s ω ω ω ω ω () s = k k ( k + k [ ] ) ( k + s) + ( k [ ] + k + s) + s (7) ω ω The inverse Laplace transform of terms: P% ( s) shows that the PDF ON P ON () τ is a sum of two exponential P r ON 1,2 ( τ ) = wre + (1 w) re rt 1 r2t k k k k kk kk [ ] = k [ ] k k [ ] k ± ω ω ω ω ω ω ω ( k ) 2 (8) where w 1 (0,1) is a weighing factor. Equation (8) shows that the PDF has two timescales: 1/ r1 an 1/ r 2. The point of separation of these two timescales τ s is the point at which the two exponential terms are equal.

20 log w1r 1 ( 1 w1 ) r 2 τ s = r -r 1 2 (9) Using equations (1), (3) an (4)-(6) we can calculate P%, the Laplace transform of the OFF () s OFF state waiting time istribution: k P% OFF ( s) = (10) k + s The inverse Laplace transform of term: P% shows that the PDF OFF () s P OFF () τ has a single exponential k t P ( τ ) = k e (11) OFF Thus the waiting time istribution of the OFF state in the tact moel has only one timescale: ( k ) 1. We use the same metho to calculate the waiting time istributions for the chromatin mechanism. We use the following numbering of sub-states: C(1), O (2), O (3), O (4) an O (5). The matrices H, U an V for the chromatin mechanism are: ko kc ko ( kc k k [ ]) k / ω k / ω H = 0 k [ ] ( k + k / ω) 0 k / ω 0 k 0 ( k / ω+ k [ ]) k / ω 0 0 k k [ ] ( k / ω + k / ω) (12)

21 k / ω 0 U = k / ω (13) V = (14) 0 k k 0 0 Using equations (1), (2) an (12)-(14) we solve for the Laplace transform of the ON state waiting time istribution P% ( s) : chr ON P% chr ON k / ω () s = k / ω+ s (15) The inverse Laplace transform of term: P% ( s) shows that the PDF chr ON P chr ON () τ has a single exponential k chr P k t/ ON ( ) e ω τ = (16) ω gain using equations (1), (3) an (12)-(14) we can calculate P% the Laplace transform of chr OFF () s the OFF state waiting time istribution:

22 P% chr OFF N() s () s = Ds () k 2 Ns () = k ω 2 k ko + s + ko + s ( k + k + s) + ω ( ) ( ) [ ] ( ) [ ] [ ]( o + ) + o( + ) + ( c + + ) k k k s k k s s k k s ( ko( k s) s( kc k s) ) k k Ds () = + k [ ] ω ω ( k + s) ( k [ ]( ko + s) + ko( k + s) + s( kc + k + s) ) (17) The inverse Laplace transform of exponential terms: P% shows that the PDF chr OFF () s P chr OFF () τ is a sum of three P r r chr OFF rt 1 r2 t rt 3 ( τ ) = w e + w e + (1 w w ) re (18) where w1, w 2 (0,1) are weighing factors that represent the probability of timescales r 1 an r 2 respectively. The moments of these waiting time istributions can be easily calculate from the PDFs. For the calculations shown in the main text the rate of bining of TF was assume to be near the iffusion limit -1-1 k = 0.001nM s [2] an a typical value was chosen for the TF issociation stant K = k / = 1nM [3, 4]. The bining rate stant for the transcriptional k machinery was also assume to be iffusion limite, -1 k = 0.001s (note that this is a first orer rate stant unlike k ) [2, 5, 6]. Bining of the transcriptional machinery to core promoters is typically weak [4, 7], so we assume that -1 k = 0. 1s. The chromatin equilibrium stant K an the strength of TF-transcriptional machinery interactions ω are both known to be in the range 10-

23 1000 [8, 9] so we assume ω = K+1= 20. The centration of TF was chosen to ensure that the response is not saturate [ ] = 1nM. 2. Construction of logic gates for the chromatin mechanism Parameters for the tact mechanism logic gates were taken from ef. [10]. The parameters for each logic gate of the chromatin mechanism were chosen to ensure that the response of this esign was as close as possible to the response of the corresponing tact mechanism logic gate. The response function for the logic gates is given by i pb ([ ],[ B]) f ([ ],[ B]) = (19) max( p ) B where i f is the normalize rate of gene expression in the presence of TFs an B ( i = for the tact mechanism; expression. i = chr for the chromatin mechanism) relative to the maximum rate of For the ND, O an XO gates the parameters were estimate numerically by minimizing the square of the ifference between the response functions of each logic gate of the two mechanisms. The normalize rate of gene expression for the ND logic of the tact mechanism ( f ) was calculate by using equation (4) for the probability of transcription p. ND B In a similar fashion, the normalize rate of gene expression for the ND logic of the chromatin mechanism ( (4) to fin p chr B. chr f ND ) was calculate by substituting ZON an ZOFF from equation (11) in equation

24 f ND an chr f ND were use to struct the objective function S. c on chr 2 ( fnd ([ ],[ ]) fnd ([ ],[ ])) log[ ] log[ ] (20) S = B B B 3 To approximate this integral, the range of centrations ( 110, ) was iscretize into 40 loguniform intervals for each TF. The square of the ifference between the normalize transcription rates of the two mechanisms was calculate at each combination of TF centrations ( ],[ B ] ; i, j = 1, ) an summe to struct the following iscrete version of the [ i j objective function: S = ( f B B (21) i, j chr 2 ND ([ ],[ i ] j) f ND([ ],[ i ] j) ) T The parameters K, e G, G, G B an GB for the chromatin mechanism were chosen to minimize S. The fminimax library routine of MTLB was use to solve the nonlinear optimization. The parameters for the O gate an XO gate were estimate using similar objective functions. Normalize transcription rates for these gates can be easily calculate with the equations liste in sections 2.3. Parameters of the NND gate of the chromatin mechanism were erive analytically from the parameters of the NND gate of the tact mechanism. The probability of transcription is highest when [ ] = [ B] = 0in the case of NND logic. Numerical methos were not necessary for estimating parameters of the NND gate because there are no TF-transcriptional machinery interactions in the tact moel of the NND gate (see ef [10]). The probabilities p B an chr p B for the NND gate of the tact mechanism an the NND gate of chromatin mechanism are given by:

25 p G e T B GT G GB G GB GB = (22) e + 1+ [ ] e + [ B] e + [ ][ B] e p chr GT chr e B = chr chr chr chr chr GT G GB G GB e + K + 1+ [ ] e + [ B] e + [ ][ B] e (23) Using the substitutions G G G e T B GB = G = G = G chr T chr chr B G G G = K + 1 B B B,, an (24) the response function for the tact mechanism in equation (22) can be rearrange to give the response function for the chromatin mechanism shown in equation (23). Clearly the two expressions are analytically ientical an the parameters of the chromatin mechanism can be erive from the substitutions use above. Note that in the response function for the chromatin mechanism the TFs an B o not interact. This implies that the cooperativity between the two TFs emerges from the equilibrium between open an close chromatin states. The strength of this emergent cooperativity matches the free energy of the TF-TF interaction in equation (22). G = log( K+ 1) (25) B

26 Supplementary Figures Figure S1. Gene expression response functions for tact moel logic gates. Parameters an equations have been aapte from ef. [10]. (a) Gene expression rates relative to the maximum 3 level of expression ([ ] = [ B] = 10 ) for the ND logic with typical cellular centrations of G TFs an B ( G = GB = 12. 5, GB = 2. 99, GP = GBP = 299., GBP = -668., e T = ). (b) Gene expression rates for the O logic ( single promoter moel) normalize relative to the 3 maximum expression level at [ ] = [ B] = 10 ( G = GB = 2. 31, GB = 0, G GP = GBP = 299., GBP = -369., e T = 005. ). (c) For the NND logic transcription rates were normalize relative to the maximum expression level at [ ] = [ B] = 0 GT ( G = GB = 2. 31, GB = 299., e = 100 ). () For the XO logic the maximum expression 3 level at [ ] = 10,[ B] = 0was use to normalize the transcription rates ( G = GB = 1. 62, GB = 0, 1 1., GP = GBP = 299 GBP = -369., G = GB = 0. 11, GB = 299., G e T = 01. ).

27 Figure S2. Comparison of the sensitivities to free energy values of the ND, NND an XO logic gate responses. (a)-() The ND gate response is most sensitive to the free energies G an G B at high centrations of TFs an B, respectively. The sensitivity of the ND gate response to these free energies is similar for the chromatin mechanism an the irect tact mechanism. (The sensitivity to GB is symmetric to the sensitivity to G ). (e)-(h) Similar to the ND gate response, the NND gate response is most sensitive to G an GB at high TF centrations. The sensitivities of the tact an chromatin mechanisms are ientical. (i) In the 1 tact moel the XO gate response is sensitive to G only at intermeiate centrations of the TFs. (j) The XO gate response for the chromatin mechanism is less sensitive than the tact mechanism at intermeiate centrations of TFs an more sensitive at high 1 centrations of TFs. (k),(l) The tact mechanism s response is more sensitive to GB at high TF centrations than the chromatin mechanism s response. ather than parameters sensitivities these ifferences are largely a result of the ifference in XO response of the two

28 mechanisms (see section 2.3). Parameters an equations for the tact moel were abstracte from [10]. Figure S3. Sensitivity of the parameter estimation metho to the choice of chromatin equilibrium stant K. (a) an (c) show the Gata2 response function for K = 250an K = 500 respectively, as etermine using the expression ata given in Figure 5(a) (the colorbars represent fol-change increase in gene expression relative to the expression rate at = = ). Note that the response function within the range of wil-type TF [ Gata2] [ Fli1] 0 centrations ( 0< [ Gata2 ],[ Fli1] 1; emarcate by white lines) oes not change over this twofol change in K. However the response functions are rastically ifferent when the TFs are over-expresse. In fact the maximum fol-change (at saturating TF centrations) in each case are approximately equal to K. Therefore the value of K can be etermine by over-expressing the enhancer bining TFs. (b) an () show the sensitivity of the Gata2 response to the value of K for K = 250an K 500 = respectively. In the range of wil-type centrations ( 0< [ Gata2 ],[ Fli1] 1; area boune by white lines) the response functions are not sensitive to the value of K, but outsie this region the response function epens strongly on the chosen value of K. Therefore the response function insie the region of wil-type centrations can be reliably preicte even without the knowlege of K.