Distance matters: the impact of gene proximity in bacterial gene regulation
aa r X i v : . [ q - b i o . S C ] M a y Distance matters: the impact of gene proximity in bacterial gene regulation
Otto Pulkkinen and Ralf Metzler
1, 2 Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland Institute for Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany
Following recent discoveries of colocalization of downstream-regulating genes in living cells, theimpact of the spatial distance between such genes on the kinetics of gene product formation isincreasingly recognized. We here show from analytical and numerical analysis that the distancebetween a transcription factor (TF) gene and its target gene drastically affects the speed and re-liability of transcriptional regulation in bacterial cells. For an explicit model system we develop ageneral theory for the interactions between a TF and a transcription unit. The observed variationsin regulation efficiency are linked to the magnitude of the variation of the TF concentration peaks asa function of the binding site distance from the signal source. Our results support the role of rapidbinding site search for gene colocalization and emphasize the role of local concentration differences.
PACS numbers: 87.16.-b,87.10.-e,05.40.-a
Suppose you live in a small town and start spread-ing a rumor. The time after which the rumor reaches aspecific person depends on your mutual distance, eitherthe physical distance due to word-of-mouth in the pre-telecommunications era or the topological distance inmodern social networks [1]. This distance dependenceis immediately intuitive for random propagation in largesystems. Conversely, diffusion of signaling molecules onthe microscopic scales of biological cells was observed tobe fast [2], so one might assume that spatial aspects canbe neglected. Yet recent studies strongly suggest thateven in relatively small bacterial cells distances matterwith respect to both speed and reliability of genetic reg-ulation by DNA-binding proteins, so-called transcriptionfactors (TFs) [3, 4]. Thus, the distance-dependence ofthe search time of a given TF for its target binding siteon a downstream gene was proposed to affect the order-ing of genes on the DNA, in particular, promote gene colocalization , i.e., the tendency of genes interacting viaTFs to be close together along the chromosome [6].Transcriptional regulation, the change in gene tran-scription rate caused by binding of regulatory proteinssuch as TFs, is the most prominent form of gene regula-tion in bacteria [5]. Since TFs are proteins themselves,their production consists of the inherently stochastic pro-cesses of transcription (conversion of the TF gene’s codeto RNA) and translation (conversion of the RNA code toproteins). Although a certain averaging of noise occursdue to long protein lifetimes, the noise in the TF produc-tion propagates to downstream genes regulated by thisTF [7]. The contributions of individual stochastic stepsto the total noise in protein production (magnifying glass in Fig. 1) were characterized [8], and accurate theoret-ical models for TF-regulated expression exist in the caseof known TF density at the regulatory site [9, 10].Recently, considerably effort has been invested on ex-plaining the efficiency of transcriptional regulation, espe-cially the remarkable measured speed at which TFs findtheir binding sites [2, 13–15]. This speed is due to facilit- TF gene TU gene m e s o s c o p i c r e g u l a t e d s t a t e bs u r sb su r us r bs r D D mRNAproteinNucleoid Cytosol Figure 1: Three stochastic phases in transcriptional regula-tion: Transcription factor (TF) production. TFs per-form facilitated diffusion in the nucleoid (inside the dashedline) containing the DNA. Diffusion is purely 3D in the cytosoloutside the nucleoid. TFs find the operator of the tran-scription unit (TU) gene by sliding along the DNA. The ir-regularly shaped blue objects depict other molecules whichaffect the facilitated diffusion and binding affinity of the TF. ated diffusion [16–22], in which free TF diffusion in threedimensions is interspersed by periods of one-dimensionalsliding along the DNA (Fig. 1, magnifying glass ). Fa-cilitated diffusion of lac repressor molecules has indeedbeen observed in living E. coli cells [13]. In this con-text, the colocalization hypothesis certainly makes sense:a shorter search time effects more efficient regulation [4].Concurrently, the importance of increased local proteinconcentration due to binding to DNA, occurrence of mul-tiple binding sites, formation of protein complexes, andcellular compartmentalization for prokaryotic and euka-ryotic gene regulation has been emphasized [23].Here we show that high local TF concentrations due togene proximity alone is sufficient for efficient gene regula-tion. Specifically, we extend the viewpoint of TF searchtime optimization due to colocalization to effects on theentire cascade of TF and TU gene expression, includingthe noise in TF production, facilitated diffusion of TF,and TF binding at TU by first binding non-specificallyto the DNA and then sliding to its specific binding site(magnifying glass in Fig. 1). To our knowledge, thisis the first complete, quantitative approach including allrelevant subprocesses in TF-mediated gene interaction.The time-dependent intracellular concentration of aprotein may be modeled by stationary shot-noise [10, 11] ρ ( t ) = V − R t −∞ e − γ ( t − s ) dN B ( s ), where N B is a com-pound Poisson process of protein production with com-bined transcription and translation rate a , exponentiallydistributed translational burst sizes B i , i = 0 , ± , ± , . . . of mean b [12], and a combined degradation and dilutionrate γ . V C is the (average) cell volume. The intermedi-ate translation step is excluded because of short mRNAlifetimes. Under the typical fast mixing assumption ofmolecules in the cell, the number of proteins M (Ω , t ) ina subdomain Ω of relative volume v Ω = V Ω /V C , at giventime t , is therefore a Poisson random variable of intensity R Ω ρ ( t ) d r , with Laplace transform (cid:10) e − λM (cid:11) = exp (cid:20) − a Z t −∞ b (1 − e − λ ) v Ω e − γ ( t − s ) b (1 − e − λ ) v Ω e − γ ( t − s ) ds (cid:21) = (cid:2) bv Ω (1 − e − λ ) (cid:3) − a/γ . (1)This is but the negative binomial distribution with para-meters a/γ and bv Ω / (1 + bv Ω ). In particular, the meanand the variance of the number M (Ω , t ) of proteins are h M i = abv Ω /γ, h M i − h M i = abv Ω (1 + bv Ω ) /γ. (2)Bursty protein production (large b ) clearly effects agreater variance than a simple Poissonian production ofindividual molecules. We note that the negative bino-mial distribution (1) has been previously found for thenumber of proteins in a two-stage model of stationaryexpression in the fast translation limit [24].To study the expression of a gene controlled by a con-stitutive TF, we expand the mathematical model in twoways: (i) we introduce a position dependent kernel φ ( r , t )in the shot noise ρ ( t ), to include time delays in transcrip-tion, translation, protein folding, and, notably, facilitateddiffusion of TFs to their target site. The coordinate r isthe point of observation, namely, a point in the neigh-borhood of the target site (blue operator near gene b inFig. 1, ). (ii) We allow a time-dependent transcriptionrate α ( t ), such that the mean number of protein produc-tion events in a time interval [ t , t ] equals R t t α ( s ) ds .The corresponding, time-inhomogeneous compound Pois-son process will be denoted by N α,B . In particular, therate α ( t ) may be chosen to be a random process, to modelfluctuations of the promoter [8, 25] or operator state [26], leading to transcriptional bursts [27], see below. The res-ulting process reads ρ ( r , t ) = R t −∞ φ ( r , t − s ) dN α,B ( s ).Moreover, following Berg [11] instead of a continuousexponential distribution, we will also include a discrete,geometric distribution for the burst sizes B .Even if the time evolution of the protein density ρ ( r , t )is no longer Markovian, we can write down its Laplacetransform because, for a given protocol α , protein pro-duction is still a time-inhomogeneous Poisson process: (cid:10) e − λρ | α (cid:11) = exp " − Z t −∞ α ( s ) b (cid:0) e λφ ( r ,t − s ) − (cid:1) b (cid:0) e λφ ( r ,t − s ) − (cid:1) + 1 ds . (3)The corresponding formula for the protein number isobtained by substituting λ → − e − λ and φ ( r , t ) → φ (Ω , t ) = R Ω φ ( r , t ) d r . In particular, the average of theprotein number M (Ω , t ) and its variance can be imme-diately calculated from the Laplace transform, yielding h M | α i = b Z t −∞ α ( s ) φ ds, (4a) h M | α i − h M | α i = b Z t −∞ α ( s )[1 + (2 b − φ ] φ ds, (4b)with φ = φ (Ω , t − s ). Eqs. (2) follow as a special caseof (4a) and (4b) with a constant transcription rate, largeburst size, and infinitely fast mixing of molecules in ahomogeneous cell volume, i.e., φ (Ω , t ) = v Ω e − γt .Let us now consider the effect of a TF (here, arepressor) to the transcription rate α TU of a transcriptionunit (TU) gene under control of the TF. We first assumea given density of unbound TF within the sliding dis-tance along the DNA from the operator site, and studythe local kinetics of the TF. We explicitly describe thelocal kinetics of the repressor molecules through facilit-ated diffusion [16, 26] near the binding site by consideringthree states of the operator (magnifying glass in Fig. 1):transcription occurs at a constant rate a whenever thereis no repressor bound to the DNA at the target. Then,the repressor is either performing a local search by slid-ing in the vicinity of the target without binding to itspecifically, or TF molecules, the mean number of whichis determined by the given density, are just hovering inthe surrounding space. The gene is considered silentwhen a repressor is bound at the operator. The linearMarkov dynamics of TF binding can be explicitly solvedby standard methods (see Supplementary Material (SM)[28]). For example, the stationary protein level is ob-tained by averaging over α in Eq. (4a), but its variancewill consist of three terms instead of the two in Eq. (4b)because of time correlations in the transcription rate α .Introducing the equilibrium constant K SP for specificTF binding to the operator and assuming fast bindingand unbinding, we integrate out the fast local searchstate in the three-state model.This leads to a simplermodel with telegraph noise at the operator, i.e., thegene is either silent or being transcribed at some effect-ive rate a eff . The transitions between these two statesoccur without intermediates at rate r on from silent toactive and vice versa with rate r off . Matching the sta-tionary mean and the variance of the protein numbersin both processes, we relate the parameters of the tele-graph model to the ones depicted in the magnifying glass of Fig. 1 [28]. This is the description of a mesoscopicrepressed state discussed in Ref. [26], where it is arguedthat this choice of retaining the completely silent statein the coarse-grained theory is justified by the separationof timescales in local search dynamics and RNA poly-merase (RNAP) binding; the rebinding of repressor isextremely fast, thus leaving hardly any time for RNAPto intervene [26]. Of course, there exists another tele-graph scenario that would leave the original transcriptionrate for the completely unbound state untouched, butwould introduce an effective, leaky transcription rate forthe combined repressed state consisting of nonspecificallyand specifically bound states. This alternative scenariois certainly plausible. For example, the leaky expressionof lac genes [30], has been associated with DNA looping[31]. We do not consider this point further here.We now address the interaction of TF and TU genesvia repression and study the transient response of theTU gene to a change in the transcription rate of the TFgene when the latter is turned on at t = 0 and thenconstitutively expressed. We study the dynamics of themoments of the TU gene transcription rate α TU as func-tions of the distance between the genes. From simulatedtrajectories (Fig. S1 [28]) of suitably normalized repressorconcentrations (see below) within a binding distance fromthe target and the resulting expression levels of the geneunder control, the TF shows distinct concentration peaksfor a pair of vicinal genes, and a fast decrement in ex-pression level of the TU gene due to TF binding.To analytically model the TF searching its binding site,we assume a linear dependence of the nonspecific bindingrate on the repressor concentration near the target andintroduce the equilibrium non-specific binding constant K NS . If the basal rate, in absence of repressors, of ex-pression of the TU gene is a TU , the mean and variance ofthe transcription rate α TU ( r, t ) under repression become h α TU i = h a eff r on r on + r off i = a TU p on ( r, t ) , (5a) h α i − h α TU i = a p on ( r, t ) [1 − p on ( r, t )] , (5b)where we use the probability that the TU gene is activelytranscribed at time t when the gene-gene distance is r , p on ( r, t ) = (cid:28) K NS ρ TF (Ω , t )1 + K NS (1 + K SP ) ρ TF (Ω , t ) (cid:29) . (5c)As a typical example, Ω is a tube surrounding the slidingregion around the target. Its length is 34 nm (100 base pairs), its diameter is that of DNA (2.4 nm) plus 30 nm( e.g. , the length of lac repressor is 14 nm). With Eq. (3), p on ( r, t ) = 11 + K SP (cid:18) Z ∞ e − λ − R t −∞ α TF ( s ) ℵ ℵ ds dλ (cid:19) , (6)where ℵ = b TF (exp { λ ˜ Kφ (Ω , t − s ) } −
1) and ˜ K = (1 + K SP ) K NS . The lower limit of the inner integral can beset to zero in our scenario ( α TF ( t <
0) = 0).Eq. (6) is a central result of this study. It is gen-eral and allows quantitative analyses of various transcrip-tional and translational repression scenarios in any cel-lular structure and geometry. In particular, it takes intoaccount the transciptional pulsing [27] of the TU gene in-duced by the binding of the repressor. Eq. (6) even allowsus to model RNAP binding and mRNA degradation bysetting b TF = 1 and introducing, as the TF productionrate, a new stochastic process α TF ( t ) = v TF N mRNA ( t )with a constant translation rate v TF , and the number oftranscripts N mRNA given by an immigration-death pro-cess (equivalently an M / M / ∞ queue) with mRNA pro-duction rate a TF and mRNA degradation rate γ mRNA .Since γ mRNA is of the same order as typical TF searchtimes in E. coli [2, 13], inclusion of TF mRNA dynamicsmay be necessary in some cases. The scenario can be evenfurther extended to include TF transcriptional pulsing bymodulating the immigration-death process N mRNA withtelegraph noise [25]. However, the expectation of Eq. (6)is yet to be solved for these α TF [29]. In the examples be-low, we use an approximation with a constant transcrip-tion and translation rate yielding on average 500 TF mo-lecules per cell under stationary conditions. This numberis in the ballpark of TF abundances for various levels of E. coli regulation networks [35]. Special cases with lowand high TF abundances will be studied separately.We assume the TF gene to be in the center of a spher-ical nucleoid and the TU gene at a radial distance r fromit. There is recent evidence [3] that the spatial distri-bution of TFs is highly inhomogenous. TFs bind to theDNA nonspecifically, hence under many growth condi-tions the TF concentration is higher in the nucleoid thanin the surrounding volume. Inhomogeneities were alsoobserved to affect fold repression. We thus assume thatthe diffusion constant D N within the nucleoid is muchsmaller than in the surrounding cytosol due to crowdingand nonspecific binding to the DNA (see SM [28] for com-parison with the model in Ref. [3]). The nucleoid is sur-rounded by the volume V C − V N , where V C = 4 πR / µ m and V N = 4 πR / . µ m are the cell and nucle-oid volumes. The density ρ TF ( r, t ) is subject to the radialdiffusion equation. In Eq. (6), φ (Ω , t ) ≈ V Ω φ ( r, t ) obeys ∂φ∂t = D N (cid:18) ∂ φ∂r + 2 r ∂φ∂r (cid:19) − γφ, for 0 ≤ r ≤ R N ∂φ∂t = − πR D N V C − V N ∂φ∂r − γφ, for r = R N , (7) Figure 2: Transient response to a change in TF transcriptionrate. The circles and squares are the probabilities (6) for TF-TU gene distances r = 0 .
05 and 0 . µ m, and the solid anddashed lines show the corresponding mean field approxima-tions (see main text). The inset shows the variation of TF con-centration around the target site at various TF-TU distances.The equilibrium constants are K NS = 10, K SP = 1000, andthe rest of the parameters as described in the main text. with a dilution rate γ = 1 /
20 min − due to cell growthand with the condition that the TFs are initially uni-formly distributed in the close vicinity (say, within a ra-dius R I = 20 nm) of the TF gene. This is justified fromthe observed localization of transcripts near their tran-scription site in bacteria [33]. The explicit solution ofEqs. (7) for our spherical geometry is [32] φ ( r, t ) = e − γt V C + 32 π ∞ X n =1 e − ( D N q n + γ ) t sin( q n r ) R N r ×× k ψ n + 3(2 k + 3) ψ n + 9 k ψ n + 9( k + 1) ψ n · sin( θ n ) − θ n cos( θ n ) R I θ n , (8)where ψ n = q n R N , θ n = q n R I and k = ( V C − V N ) /V N , andthe q n are the positive solutions of (3+ kR q ) tan( qR I ) =3 qR I . Eq. (8) is our other central result.Fig. 2 shows the probabilities (6) as function of timefor short and long distances between the TF and TUgenes. Accordingly, the distance impacts vastly the reg-ulation efficiency: the response is significantly strongerand faster for short distances, this difference persistingfor minutes. Fig. 2 also demonstrates that it is neces-sary to consider this exact expression instead of a meanfield approximation obtained by taking expectations ofthe density separately in the numerator and denominatorin Eq. (5c). The mean field approximation would over-estimate the spatial differences in regulation. Therefore,it is of importance to use the exact formula (6) instead.The inset of Fig. 2 shows the reason for the differencebetween exact and mean field approaches: as already sug-gested by the simulated trajectories in Fig. S1, the amp-litude variation of the TF concentration contributing tononspecific binding at the target depends heavily on theseparation of TF and TU genes. The TU genes far away from the TF gene receive a more diluted signal than thoseclose-by. Specifically, both Fig. S1 and the inset of Fig. 2show ˜ Kφ (Ω , t ), characterizing both the availability of TFand its binding affinity to the target. Its values shouldbe compared to 1, the scale set by the first term in theexponential of Eq. (6). The truncation of the peak ob-served at short distances causes the mean field theory tofail. Note that smaller TF copy numbers than used herelead to a similar spatial effect in p on ; e.g., the same setof parameters but with a stationary mean number of 100TFs leads to a roughly constant difference of the order0 . p on with r = 0 .
33 and 0 . µ m in a windowof 1 min. The magnitude of the effect depends naturallyon the TF binding affinity at the target. Both the expres-sion levels and binding specificity are known to dependon whether the TF is a local or global regulator [34, 35].With Eq. (5b), we assess the noise propagation in theTF-TU system, in particular, the variance of the tran-scription rate of the TU gene. Since the variances areproportional to the product p on ( r, t ) [1 − p on ( r, t )], we seefrom Fig. 2 that they peak at a few seconds and at tenseconds for r = 0 .
05 and 0 . µ m, respectively. The prob-ability p on grows with distance to the TU gene, and thesame hence applies to the variance after the initial tran-sient peak. The total time-integrated variance is greaterfor the distant gene, and its transcription is thereforemore susceptible to stochastic variation in TF produc-tion. However, the effect in Eq. (5b) is small for small α TU , and the situation may be different under stationaryconditions. Fig. 2 shows that the distance variation in ex-pression levels in the long time limit can be small, even ifthe transient response shows considerable variation. Thesame applies to expression fluctuations. Experimentalobservations [36] show that the protein level fluctuationsare, in general, determined by the mean expression level,and are independent of system details. The dependenceof protein number fluctuations on the TF-TU distanceunder stationary conditions needs to be explored further.Concluding, we established a quantitative model forthe distance dependence of gene regulation efficiency andstochasticity in bacteria. Intracellular structure and non-specific binding to the DNA are taken into account interms of an inhomogeneous diffusion rate. The bindingat the target is facilitated by a local search process, whichwas modeled by an intermediate fast degree of freedom.Significant spatial effects in the regulation efficiency wasdemonstrated, strongly supporting the regulation hypo-thesis for gene colocalization. We note that more precisemodels, for instance, with multiple TFs sliding simul-taneously near the target can be solved, as well. Theexpressions are more elaborate (except for infinite num-bers) but the binding probabilities show roughly the samebehavior as above. 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