Effect of transcription reinitiation in stochastic gene expression
aa r X i v : . [ q - b i o . M N ] S e p Effect of transcription reinitiation in stochastic geneexpression
Rajesh Karmakar ∗ and Amit Kumar Das September 8, 2020 ∗ Department of Physics, Ramakrishna Mission Vidyamandira, Belur Math, Howrah-711202, India.
Kharial High School, Kanaipur, Hooghly-712234, India.
Abstract
Gene expression (GE) is an inherently random or stochastic or noisy process. The randomnessin different steps of GE, e.g., transcription, translation, degradation, etc., leading to cell-to-cell variations in mRNA and protein levels. This variation appears in organisms ranging frommicrobes to metazoans. The random fluctuations in protein levels produce variability in cellularbehavior. Stochastic gene expression has important consequences for cellular function. It isbeneficial in some contexts and harmful to others. These situations include stress response,metabolism, development, cell cycle, circadian rhythms, and aging. Different model studies e.g.,constitutive, two-state, etc., reveal that the fluctuations in mRNA and protein levels arise fromdifferent steps of gene expression among which the steps in transcription have the maximumeffect. The pulsatile mRNA production through RNAP-II based reinitiation of transcription isan important part of gene expression. Though, the effect of that process on mRNA and proteinlevels is very little known. The addition of any biochemical step in the constitutive or two-stateprocess generally decreases the mean and increases the Fano factor. In this study, we show thatthe RNAP-II based reinitiation process in gene expression can increase or decrease the mean andFano factor both at the mRNA levels and therefore, can have important contributions on cellularfunctions.Keywords: stochastic gene expression, reinitiation of transcription, Fano factor
Gene expression is a fundamental cellular process consisting of several consecutive random steps liketranscription, translation, degradation, etc. The random nature of the biochemical steps of geneexpression is responsible for the stochastic or noisy production of mRNA and protein molecules.This stochasticity in gene expression gives rise to heterogeneity in an identical cell population andphenotypic variation. Phenotypic variation is generally attributed to genetic and environmentalvariation. Though it has been observed that genetically identical cells in a constant environmentshow significant phenotypic variation.The origin and consequences of noise in stochastic gene expression have been studied extensively,both theoretically and experimentally, during the last three decades [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13, 14, 15, 16, 17, 18, 19, 20]. Several studies on both prokaryotic [6, 12] and eukaryotic systems ∗ [email protected]
17, 8, 13, 17] suggest that gene transcription occurs in a discontinuous manner and that gives rise tofluctuating production of mRNAs and proteins. The random fluctuations in the number of mRNAand protein molecules in each cell constitute the noise. The cells must either exploit it, learn to copewith it, or overcome it using its internal noise suppression mechanisms. Noise in gene expressionregulates several cellular functions. It can improve fitness by generating cellular heterogeneity inclonal cell populations, thus enabling a fast response to varying environments [8]. Because of itsfunctional importance in cellular processes, it is necessary and important to identify and dissect thebiochemical processes that generate and control the noise.The transcription is an important step in stochastic gene expression. It has been observed that thetranscription process contributes maximum noise in protein level than any other biochemical steps ingene expression [4, 7, 8, 9, 11, 12, 13, 16, 17]. During the transcription process, different transcriptionfactors (TFs) bind to multiple sites on regulatory DNA in response to intracellular or extracellularsignals. On binding the regulatory systems, the TFs turn the gene into an active state from which aburst of mRNAs is produced. Transcriptional bursting has been observed across species and is oneof the primary causes of variability in gene expression in cells and tissues [7, 8, 10, 12, 13, 21, 22, 23].Many experimental observations are modeled with that burst mechanism [9, 11, 14, 16, 18, 24].The mRNA synthesis from the active gene actually takes place through interactions with RNAP-II [25, 26, 27, 28]. Experiments show that the RNAP-II based transcription, specific to eukaryotes,produces pulsatile mRNA production through reinitiation and is crucial to reproduce the experimentalobservations on noise at protein levels [7, 8]. The Reinitiation of transcription introduces the thirdstate of a gene along with the two states of the two-state model network. Recent research shows thatthe c-Fos gene in response to serum stimulation indicates that a third state along with the inactiveand active states is essential to explain the experimental data on variance [21]. Noise or Fano factorat the mRNA level is unity for constitutive gene expression. The two-state model shows the super-Poissonian Fano factor at the mRNA level because of the random nature of the gene activation anddeactivation. Only negative feedback in the two-state model network can reduce the noise in themRNA levels but not below the unity or sub-Poissonian level. Recent work on the two-state systemwith RNAP-II based transcriptional reinitiation process shows that the reinitiation process has theability to reduce the noise strength or Fano factor in mRNA level below unity [29]. There can be theother effects of transcriptional reinitiation on mRNA levels which are very little known.In this paper, we consider different gene expression models e.g., constitutive [20], two-state [11],and Suter model [17], and studied the effect of transcription reinitiation on the mean and Fano factorof mRNA levels. From our exact analytical calculations, we find that the reinitiation of transcriptionin the constitutive gene regulatory network behaves like a product independent negative feedback inthe regulatory circuit. Whereas for the two-state and Suter model, the RNAP-II based transcriptionreinitiation behaves as either positive or negative or mixed feedback circuit depending on the rateconstants of the biochemical steps. That is, the reinitiation process in gene transcription can increaseor decrease the mean and Fano factor at mRNA levels.
The essential genes in the cell always produce mRNAs and proteins. The expression from essentialgenes is modeled by the constitutive network shown in figure (1). In that model, the gene is alwaysassumed to be at the active state from which the mRNA synthesis takes place at rate constant J m .The proteins are then synthesized from the newly born mRNAs. Both the mRNAs and proteins aredegraded with rate constants k m and k p respectively. It is very easy to find out the expressions of2ean, variance and Fano factor for the constitutive gene expression at the steady-state by the Masterequation approach [30].Figure 1: Reaction scheme with rate constants for constitutive gene expression. J m ( J p ) is the tran-scription (translation) rate constant and k m ( k p ) is the mRNA (protein) degradation rate constant.Let p ( n , n , t ) be the probability density of n mRNAs and n proteins at time t . The rate ofchange of probability is given by ∂p ( n ,n ,t ) ∂t = l J m [ p ( n − , n , t ) − p ( n , n , t )]+ k m [( n + 1) p ( n + 1 , n , t ) − n p ( n , n , t )]+ J p [ n p ( n , n − , t ) − n p ( n , n , t )]+ k p [( n + 1) p ( n , n + 1 , t ) − n p ( n , n , t )] (1)where l is the copy number of the gene.The steady state solution of Eq.(1) for the constitutive gene expression process gives the mean( < m c > ) , variance ( V ar cm ) and Fano factor ( F F cm ) of mRNAs and proteins and are given by (for l = 1 ) < m c > = J m k m ; < p c > = < m c > J p k p (2) V ar cm = J m k m , F F cm = V ar cm < m c > = 1 (3) V ar cp = < p c > J p + k m + k p k m + k p , F F cp = V ar cp < p c > = J p + k m + k p k m + k p (4)The noise strength or Fano factor of mRNAs in constitutive GE is unity. That is a unique featureof the Poisson process and that can be taken as a reference to compare with other gene expressionnetwork models. In constitutive gene expression, the binding and movement of RNAP-II are ignored. But in the actualprocess, the RNAP-II molecules bind the gene to form an initiation complex [25]. In the next step,the bound RNAP-II leaves the initiation complex and starts transcription along the gene. The genethen comes again into its normal state (Fig. 2(a)). In that process, it is assumed that bound RNAP-IImust do transcription without any uncertainty. Though, that may not be possible always. Theremust be a finite probability that bound RNAP-II leaves the initiation complex without transcribingthe gene. That is considered in figure 2(b). 3igure 2: Reaction scheme with rate constants for constitutive gene expression with reinitiation (a)without reverse reaction and (b) with reverse reaction. The RNAP-II binds the gene (G) with rateconstant k and forms an initiation complex ( G c ). k is the dissociation rate constant of RNAP-IIfrom the initiation complex. J m ( J p ) is the transcription (translation) rate constant and k m ( k p ) isthe mRNA (protein) degradation rate constant.To calculate the mean and variances/Fano factors we consider p ( n , n , n , t ) be the probabilitydensity of n genes in the G c state, n mRNAs and n proteins at time t . The rate of change ofprobability density corresponding to the reaction in figure 2(b) is given by ∂p ( n ,n ,n ,t ) ∂t = k [ { l − ( n − } p ( n − , n , n , t ) − ( l − n ) p ( n , n , n , t )]+ k [ { ( n + 1) p ( n + 1 , n , n , t ) − n p ( n , n , n , t )]+ J m [ { ( n + 1) p ( n + 1 , n − , n , t ) − n p ( n , n , n , t )]+ k m [ { ( n + 1) p ( n , n + 1 , n , t ) − n p ( n , n , n , t )]+ J p [ n p ( n , n , n − , t ) − n p ( n , n , n , t )]+ k p [( n + 1) p ( n , n , n + 1 , t ) − n p ( n , n , n , t )] (5)The second term in the right-hand side will not be there for the reaction scheme in figure 2(a).The mean, variance and Fano factor corresponding to the scheme in figure 2(a) (constitutive withreinitiation) will be < m cwr > = k J m + k J m k m ; < p cwr > = < m cwr > J p k p (6) V ar cwrm = J m k m k { ( J m + k )( J m + k m ) + k } ( J m + k ) ( J m + k + k m ) = < m cwr > (1 − J m k ( J m + k )( J m + k + k m ) ) (7) F F cwrm = V ar cwrm < m cwr > = (1 − J m k ( J m + k )( J m + k + k m ) ) (8) F F cwrp = V ar cwrp < p cwr > = (1 + J p { ( J m + k + k m + k p )( J m + k + J m k ) + ( J m + k ) k m k p } ( J m + k )( J m + k + k m )( J m + k + k p )( k m + k p ) ) (9)For the reaction scheme in figure 2(b), the mean, variance, and Fano factor will be < m cwrr > = k J m + k + k J m k m ; < p cwrr > = < m cwrr > J p k p (10) V ar cwrm r = J m k m k { ( J m + k + k )( J m + k m + k ) + k + k k } ( J m + k + k ) ( J m + k + k + k m ) This expression of variance can also be expressed as
V ar cwrm r = < m cwrr > (1 − J m k ( J m + k + k )( J m + k + k + k m ) ) (11)4 F cwrm r = 1 − J m k ( J m + k + k )( J m + k + k + k m ) (12) F F cwrp r = V ar cwrp < p cwr > = 1 + J p g ( J m + g )( J m + g + k m )( J m + g + k p ) g (13)where g = J m + g ( g + k m )( g + k p )+ J m (2 k +3 k + g )+ J m { k +3 k + k m k p +2 k g + k (5 k + g ) } , g = k + k , g = k m + k p It is seen that the transcriptional reinitiation in constitutive GE process (Figs. 2(a) and 2(b))decreases the mean mRNA and protein levels in comparison to that the constitutive GE process (Fig.1). The reinitiation in transcription reduces the mean mRNA level because the effective transcriptionrate is reduced due to the transcriptional reinitiation process by a factor k J m + k . The effective reductionof transcription rate causes the reduction of variance in mRNA levels. We see that the noise strengthor Fano factor (Eq.(12)) reduces below unity with the reinitiation of transcription.Figure 3: Variation of (a) mean mRNA, (b) Fano factor at mRNA levels and (c) Fano factor atprotein levels with k for different values of k with J m = 10 and k m = 1 . The solid lines are drawnfrom analytical calculations and hollow circles are generated from the simulation based on Gillespiealgorithm [31].Figures 3(a) and 3(b) show that the effect of reinitiation is strong enough at the lower values of k . We see from the figures that the mean mRNA level and F F m approaches the value observed inthe constitutive process for a given value of J m and k m for higher k . When k increases from zerovalue, the Fano factor decreases and attains a minimum value and then moves towards unity. Theminimum of the Fano factor will occur at k = q ( J m + k )( J m + k + k m ) (Fig. 3(b)). The equation(12) shows that the Fano factor has identical dependence on k and J m . As the rate constant k increases, the degree of deviation of the Fano factor below unity decreases (figure 3(b)) because thatdecreases the mean mRNA levels (figure 3(a)). If one considers pre-initiation and initiation complexesin transcriptional reinitiation process rather than only initiation complex then the Fano factor furtherreduces below unity (Appendix A).The expression for mean mRNA (Eq. 6) can be written as < m cwr > = k J m + k J m k m = < m c > β < m c > (14)This expression (Eq. 14) is identical to the gain of a linear negative feedback amplifier with thefeedback factor β = k m k [32]. The expression for the Fano factor (Eq. 8) also shows that the noise isreduced with the reinitiation process.We can also have from the equation (14) d < m cwr >< m cwr > = 11 + β < m c > ( d < m c >< m c > ) (15)5quation (15) shows that the percentage change in the mean mRNA levels with reinitiation ismuch less than that without reinitiation. That is reflected in the expression of the Fano factor ofmRNA with transcriptional reinitiation (Eq. (8)). Therefore, the equations (6), (8), (14) and (15)clearly indicate that the reinitiation of gene expression behaves as a negative feedback loop in theregulatory network.The mean mRNA with reverse reaction (Eq.(10)) can be expressed as < m cwrr > = < m c > β < m c > + α < m c > (16)where α = k m k k J m .Again from equations (10), (12) and (16), we see that the rate constant k helps to reduce themean mRNA level further but increases the Fano factor. So, the reverse transition with rate constant k behaves like negative feedback for mean mRNA level but positive feedback for the Fano factor.Thus, the successful reinitiation of transcription behaves like a negative feedback loop whereas theunsuccessful reinitiation of transcription behaves like a mixed feedback loop. It is important to notethat the negative feedback in the gene regulatory networks due to the reinitiation of transcription isproduct independent. It is completely inherent to the gene transcription regulatory network. Thenature of the variation in the Fano factor at the protein level is the same as that in mRNA levelsexcept for a change in scale (Fig. 3(c)). Therefore, the Fano factor at the protein level does notgive any new information about the effect of reinitiation. The effect of reinitiation is observed at themRNA level first. So we keep our analysis up to the mRNA level in the rest of the paper. Regulation is ubiquitous in biological processes. The regulated gene expression without feedback ina cellular system is modeled by the two-state process. Many experimental results are explained withthe help of two-state gene expression process [9, 10, 11, 12, 13, 14, 19]. In that process, the genecan be in two possible states, active ( G a ) and inactive ( G i ) (figure 4(a)) and random transitions takeplace between the states. The mRNA synthesis occurs in bursts only from the active state of thegene. The mRNAs have a specific decay rate also.Figure 4: Reaction scheme with rate constants for (a) Two-state gene expression model and (b) Two-state gene expression with reinitiation of transcription model. k a ( k d ) is the activation (deactivation)rate constant. k is the rate constant of initiation complex formation and k is the rate constant ofdissociation of RNAP-II from initiation complex. J m is the transcription rate constant and k m is themRNA degradation rate constant.Now, let us assume that there is l copy number of a particular gene exists in the cell. Let p ( n , n , t ) be the probability that at time t and there are n number of mRNAs with n number of genes in theactive state ( G a ). The number of gene in the inactive states are ( l − n ) . The time evaluation of theprobability corresponding to the chemical reactions in figure 4(a) is given by the Master equation [30]6 p ( n ,n ,t ) ∂t = k a [( l − n + 1) p ( n − , n , t ) − ( l − n ) p ( n , n , t )]+ k d [( n + 1) p ( n + 1 , n , t ) − n p ( n , n , t )]+ J m [ n p ( n , n − , t ) − n p ( n , n , t )]+ k m [( n + 1) p ( n , n + 1 , t ) − n p ( n , n , t )] (17)Solving the Eq.(17), we can easily find out the mean, variance and Fano factor of mRNAs. Theyare given by < m tswtr > = k a J m ( k a + k d ) k m (18) F F tswtrm = 1 + J m k d ( k a + k d )( k m + k a + k d ) (19)Equations (18) and (19) show that the inclusion of inactive state and the random transitionsbetween inactive and active states in the constitutive process reduces the mean and increases theFano factor.In figure 4(b), we consider the reinitiation of transcription along with the random transitionsbetween the active and inactive states of the gene. In the transcription reinitiation step, an RNAP-II binds the gene in the active state and form a initiation complex G c . Now, the bound RNAP-IIhas two choices, either it moves forward or backward. If it moves forward then again two eventsoccur: mRNA synthesis and free up of the initiation complex. That is, the initiation complex againbecomes an active state where free RNAP-II molecules can bind. The unsuccessful movement ofRNAP-II from the initiation complex of the gene brings it back to the active state by dissociating theenzyme molecules. In the two-state gene expression model, the randomness due to the transcriptionalreinitiation process is neglected assuming its insignificant contribution in mean and noise strength atthe mRNA and protein levels. Only a few works pointed out that the reinitiation of transcriptionplays important role in the phenotypic consequences of cellular systems [7, 8, 29, 33]. The above-mentioned gene expression model with reinitiation is studied in ref [29]. But for the completeness ofthis paper, we are writing down here the Master equation and the expressions of means, variances,and Fano factors.Let p ( n , n , n , t ) be the probability that at time t and there are n number of mRNAs with n number of genes in the active state ( G a ) and n number of genes in the initiation state ( G c ). Thenumber of gene in the inactive states are ( l − n − n ) with l be the copy number of the gene. Thetime evaluation of the probability corresponding to the biochemical reactions in figure 4(b) is givenby the Master equation [30] ∂p ( n ,n ,n ,t ) ∂t = k a [( l − n − n + 1) p ( n − , n , n , t ) − ( l − n − n ) p ( n , n , n , t )]+ k d [( n + 1) p ( n + 1 , n , n , t ) − n p ( n , n , n , t )]+ k [( n + 1) p ( n + 1 , n − , n , t ) − n p ( n , n , n , t )]+ k [( n + 1) p ( n − , n + 1 , n , t ) − n p ( n , n , n , t )]+ J m [( n + 1) p ( n − , n + 1 , n − , t ) − n p ( n , n , n , t )]+ k m [( n + 1) p ( n , n , n + 1 , t ) − n p ( n , n , n , t )] (20)The expressions of averages and Fano factors of mRNAs for the reaction scheme with the tran-scriptional reinitiation process in figure 4(b) are given by [29] < m tswr > = k a k a J m k m ; (21)7 F tswrm = 1 + J m k ( a − k a a ) a ( a k m + a ) (22)where a = k m + J m + k a + k d + k + k and a = k a J m + k d J m + k d k + k k a + k a k .Figure 5: Variation of mean mRNA levels with and without reinitiation as a function of (a) k a , (b) k d and (c) J m corresponding to figure (4). The solid (dashed) lines are drawn from exact analyticalexpressions (Eqs. (18) and (21)). The hollow circles are generated using the simulation based onGillespie algorithm. The rate constants are k = 50 ,k = 1 ,k m = 1 , k d = 10 and J m = 10 in figure (a), k a = 10 and J m = 10 , in figure (b) and k a = 10 and k d = 10 in figure (c).Figure 6: Plot of Fano factor at mRNA level versus (a) k a , (b) k d and (c) J m with and withoutreinitiation. The solid (dashed) lines are drawn from exact analytical expressions (Eqs. (19) and(22)). The hollow circles are obtained from stochastic simulation using the simulation based onGillespie algorithm. The rate constants are k = 50 , k = 1 , k m = 1 , k d = 10 and J m = 10 in figure(a) , k a = 10 and J m = 10 in figure (b) and k a = 10 and k d = 10 in figure (c).Figure 7: Plot of Fano factor with the variation of k with parameter (a) k a and (b) k d . The otherrate constants are J m = 10 , k m = 1 , k d = 10 in figure (a) and k a = 5 in figure (b). The solid lines aredrawn from analytical expression (Eq. 22) and hollow circles are obtained from the simulation basedon Gillespie algorithm.The variation of mean mRNA levels for both the scenarios, with and without reinitiation, areplotted with the rate constants k a , k d and J m in figure 5. We see from the figures that the reinitiationof transcription helps to keep the mean mRNA levels at higher values for lower values of k a and J m andfor almost all values of k d . Fano factor remains lower for all values of k a , k d and J m (Figs. 6(a), 6(b)and 6(c)). If one considers pre-initiation and initiation complexes in the transcriptional reinitiation8rocess [25] rather than only the initiation complex then the Fano factor further reduces below unity(Appendix B). Fano factor can also be higher due to the transcriptional reinitiation process with othersets of rate constants [7, 29]. The Fano factor can have three different phases, Poissonian ( F F = 1) ,super-Poissonian (
F F > ) and sub-Poissonian ( F F < ), when plotted against k with k a , k d and J m as parameters as shown in figure 7(a), figure 7(b) and in ref. [29] respectively. The rate constants k a = 6 . and k d = 5 in figure 7(a) and figure 7(b) respectively can be considered as the critical valueof that rate constants as that values sharply divide the super-Poissonian and sub-Poissonian Fanofactor regimes [29].The expression of mean mRNA level (Eq.(21)) can be written as < m tswr > = < m tswtr > − β < m tswtr > + β < m tswtr > (23)where β = k d k m k a J m and β = ( k a + k d ) ( J m + k ) k m k a k J m .The expression of Fano factor (Eq. (22)) can also be expressed as F F tswrm = 1 − γ < m tswtr > + γ < m tswtr > (24)where γ = k m k ( k a + k d ) a ( a k m + a ) a and γ = k m k ( k a + k d )( a k m + a ) k a .In general, for β = 0 and β = 0 ( β = 0 and β = 0 ), the expression for mean in equation (23)looks like the expression of the gain with linear negative (positive) feedback network in electroniccircuit with β ( β ) as the feedback factor. For the non-zero value of β and β , the expressionfor mean (23) can be considered as the mean mRNA from a network with mixed i.e., positive andnegative both, feedback. Therefore, β ( β ) is working here as the positive (negative) feedback factor.For β > β ( β < β ) the positive (negative) feedback nature dominates and the mean mRNA levelwith reinitiation ( < m tswr > ) becomes higher than that without reinitiation process ( < m tswtr > ).Again, as far as the Fano factor is concern, the positive (negative) feedback nature dominates for γ > γ ( γ < γ ). The expression of mean mRNA (Eq. (23)) and Fano factor (Eq. (24)) shows thatthe two-state gene regulatory network with reinitiation of transcription (Fig. 4(b)) can behave asmixed feedback network.The mean mRNA level and Fano factor can be higher or lower due to the reinitiation of transcrip-tion compared to the two-state gene expression process without reinitiation. From the equation (18)or from equation (23) we have the condition of higher average mRNA level in presence of reinitiationof transcription as β > β or ( J m + k ) < k d k k a + k d (25)From equation (22) or (24), we have the condition of sub-Poissonian Fano factor as [29] γ > γ or ( J m + k ) < k a k d ( k a + k d + k m ) (26)Two conditions in equations (25) and (26) divide the whole permissible space in ( k a , J m + k )and ( k d , J m + k ) systems into four different regions with different conditions of Fano factor andmean mRNA levels. The four regions are identified as: Region I : F F tswrm < and r > ; Region II : F F tswrm < and r < ; Region III : F F tswrm > and r < ; Region IV : F F tswrm > and r > ; where r = < m tswr > / < m tswtr > and shown in figure (8). The rate constants k a and k d are generallyfunction of transcription factors and therefore, can be modulated [7, 8]. Thus the mean and Fanofactor at the mRNA level can be changed according to the cellular requirement by changing thenumber of transcription factors in the cell, 9igure 8: The plot of ( J m + k ) versus k a (a) and versus k d (b) shows the region where both theconditions given in equations (25) and (26) are satisfied (Region I). The rate constants are taken as k d = 10 (in (a)), k = 50 and k m = 1 and k a = 10 (in (b)). The condition given in equation (25) issatisfied in Regions I and IV whereas the condition given in equation (26) is satisfied in Regions Iand II.In the two-state process (Fig. 4(a)) Fano factor is always greater than unity and there is aspecific mean mRNA level depending on the rate constants k a , k d , J m and k m . But, as reinitiation oftranscription is added in the two-state gene expression process, we get four different options on Fanofactor and mean mRNA in the ( k a , J m + k ) (Fig. 8(a)) or ( k d , J m + k ) (Fig. 8(b)) space. Among thefour regions, the first region is functionally important for cellular systems. In that region, the meanmRNA level is greater but the Fano factor is lower when compared with the two-state gene expressionprocess. With respect to the feedback features of a network, we see that the gene regulatory network((Fig. 4(b))) behaves like a network with negative (positive) feedback in the Region II (Region IV).In the Regions, I and III, the two-state network with reinitiation of transcription behaves like a generegulation with mixed feedback. The regulated gene expression is an important and essential property of a complex cellular system.Though many experimental results are modeled with the two-state process, Suter et al [17] observesomething different in the mammalian system. They observe gamma-distributed off time in generegulation rather than the exponentially distributed off time in the two-state process. Suter et almodel their experimental observation by a gene regulatory network shown in figure 9(a). In theirmodel network, the gene can be in three possible states, one active ( G ) and two inactive states ( G and G ) and random transitions take place between the states according to the reaction scheme infigure 9(a). The mRNA synthesis takes place only from the active state of the gene with rate constant J m .Figure 9: Reaction scheme with rate constants for two-state gene expression (Suter model) (a) withoutand (b) with reinitiation. k b ( k d ) is the activation (deactivation) rate constant and k a is the rateconstant for transition from G to G . k is the rate constant of initiation complex formation and k is the rate constant of dissociation of RNAP-II from initiation complex. J m is the transcription rateconstant and k m is the mRNA degradation rate constant.The Master equation corresponding to the figure 9(a) is given by10 p ( n ,n ,n ,t ) ∂t = k a [( l − n − n + 1) p ( n − , n , n , t ) − ( l − n − n ) p ( n , n , n , t )]+ k b [( n + 1) p ( n + 1 , n − , n , t ) − n p ( n , n , n , t )]+ k d [( n + 1) p ( n , n + 1 , n , t ) − n p ( n , n , n , t )]+ J m [ n p ( n , n , n − , t ) − n p ( n , n , n , t )]+ k m [( n + 1) p ( n , n , n + 1 , t ) − n p ( n , n , n , t )] (27)The expression for mean and Fano factor at mRNA level corresponding to Fig. 9(a) are given by(for l = 1 ) < m suwtr > = J m k a k b ( k a k b + k a k d + k b k d ) k m = J m k a k b C k m (28) F F suwtrm = 1+ J m k d [( k b + k m )( k a + k b ) + k a ] { k a ( k b + k d ) + k b k d }{ ( k b + k m )( k d + k m ) + k a ( k b + k d + k m ) } = 1+ J m k d [( k b + k m )( k a + k b ) + k a ] C C (29)where C = k a ( k b + k d ) + k b k d and C = ( k b + k m )( k d + k m ) + k a ( k b + k d + k m ) .Now let us consider the gene transcriptional regulatory network with the reinitiation of tran-scription by RNAP-II (figure 9(b)). We have the Master equation corresponding to the figure 9(b)as ∂p ( n ,n ,n ,n ,,t ) ∂t = k a [ { l − ( n − n + n } p ( n − , n , n , n , t ) −{ l − ( n + n + n ) } p ( n , n , n , n , t )]+ k b [( n + 1) p ( n + 1 , n − , n , n , t ) − n p ( n , n , n , n , t )]+ k d [( n + 1) p ( n , n + 1 , n , n , t ) − n p ( n , n , n , n , t )]+ k [( n + 1) p ( n , n + 1 , n − , n , t ) − n p ( n , n , n , n , t )]+ k [( n + 1) p ( n , n − , n + 1 , n − , t ) − n p ( n , n , n , n , t )]+ J m [( n + 1) p ( n , n − , n + 1 , n − , t ) − n p ( n , n , n , n , t )]+ k m [( n + 1) p ( n , n , n , n + 1 , t ) − n p ( n , n , n , n , t )] (30)The mean mRNA and Fano factor corresponding to figure 9(b) are given by (for l = 1 ) < m suwr > = J m k k a k b [ k a k b ( J m + k + k ) + ( J m + k )( k a k d + k b k d )] k m (31) F F suwrm = 1 − J m k k a k b [ k k a k b + ( J m + k ) C ] k m + J m C [ C + ( J m + k + k m ) C ] k m (32)where C = k ( k a + k m )( k b + k m ) , 11igure 10: Variation of mean mRNA with (a) k a , (b) k b , (c) k d , and (d) J m . The dashed (solid)lines are from analytical calculations corresponding to Eq. (31) (Eq. (28)). The hollow circles aregenerated from the simulation based on Gillespie algorithm. The rate constants are k = 50 , k = 1 ,k m = 1 . In figure (a) k d = 10 k b = 20 and J m = 10 . In figure (b) k a = 10 , k d = 10 and J m = 10 . Infigure (c) k a = 10 , k b = 20 and J m = 10 . In figure (d) k a = 10 , k d = 10 and k b = 20 .Figures 10(a), 10(b), 10(c) and 10(d) show the variation of mean mRNA number with the rateconstants k a , k b , k d and J m respectively. All figures show that the reinitiation of transcription favoursthe higher mean mRNA levels with reinitiation of transcription.Figure 11: Variation of Fano factor in mRNA level ( F F m ) with different conditions. Dashed (solid)lines are drawn from analytical calculation for Fig. 9(b) (Fig. 9(a)). The hollow circles are generatedfrom the simulation based on Gillespie algorithm. The rate constants are k = 50 , k = 1 , k m = 1 .In figure (a) k d = 10 , k b = 20 and J m = 10 . In figure (b) k a = 10 , k d = 10 and J m = 10 . In figure(c) k a = 10 , k b = 20 and J m = 10 . In figure (d) k a = 10 , k d = 10 and k b = 20 .12igure 12: Variation of Fano factor with the rate constant k with (a) k a (b) k b (c) k d and (d) J m as parameter. The solid lines are drawn from analytical expression (Eq. 32) and hollow circles areobtained from the simulation based on Gillespie algorithm. The rate constants are taken as k d = 10 ,k b = 20 , k = 50 , k = 1 , J m = 10 and k m = 1 in (a). The rate constants are k a = 10 , k d = 10 , k = 50 , k = 1 , J m = 10 and k m = 1 . in (b). In (c), the rate constants are k a = 10 , k b = 20 , k = 50 ,k = 1 , J m = 10 and k m = 1 . In (d), the rate constants are k a = 10 , k d = 10 , k b = 20 , k = 50 ,k = 1 , and k m = 1 .The variation of Fano factors are plotted with the rate constants k a , k b , k d and J m in figures11(a) - 11(d). The figures show that the Fano factor is always lower with the reinitiation of genetranscription. The variation of the Fano factor with the rate constant k is shown in figures 12.The figures show that the three different Fano factor regimes, Poissonian, sub-Poissonian and super-Poissonian, are likely to occur with the reinitiation of transcription. That is a unique feature of theFano factor for a gene regulatory network with reinitiation of transcription. In the Suter model, k b is an extra parameter by which the mean and Fano factor can be controlled. It can be shown thatfor higher k b ( k b > ) the Suter model merges in the two-state model.The expression of mean mRNA (Eq. (31)) can also be expressed as < m suwr > = < m suwtr > − δ < m suwtr > + δ < m suwtr > (33)where δ = k d k m ( k a + k b ) k a k b J m and δ = C ( J m + k ) k m k a k J m k b . Here δ ( δ ) is working as the positive (negative)feedback factor. For δ > δ ( δ < δ ) the positive (negative) feedback nature dominates and themean mRNA level with reinitiation ( < m suwr > ) becomes higher than that without reinitiationprocess ( < m suwtr > ).The expression of Fano factor (Eq. (22)) can also be expressed as F F suwrm = 1 − B < m suwtr > + B < m suwtr > (34)where B = ( J m + k ) k + k a k b C and B = C C { C +( J m + k + k m ) C } k a k b .Again, as far as Fano factor is concern, the positive (negative) feedback nature dominates for B > B ( B < B ). The expressions of mean mRNA (Eq. (33)) and Fano factor (Eq. (34)) showthat the gene regulatory network following Suter model with reinitiation of transcription (Fig. 9(b))can also behave as mixed feedback network.We see that the Fano factor is reduced by the transcriptional reinitiation process as observed intwo-state also. From the equations (33) and (34) we find that the average mRNA level can be greaterwith the reinitiation in gene transcription provided13 J m + k ) < k k d ( k a + k b )( k a k b + k a k d + k b k d ) (35)From equation (22), we have the condition of sub-Poissonian Fano factor as [29] ( J m + k ) < k a k b k d ( k b + k m )( k d + k m ) + k a ( k b + k d + k m )( k a + k b + k a k b + k a k m + k b k m ) (36)Here also, two conditions in equations (35) and (36) divide the whole permissible space in ( k a , J m + k ) and ( k d , J m + k ) into four different regions with different conditions of Fano factor and meanmRNA levels. The four regions are identified as: Region I : F F suwrm < and s > ; Region II : F F suwrm < and s < ; Region III : F F suwrm > and s < ; Region IV : F F suwrm > and s > ; where s = < m suwr > / < m suwtr > and shown in figure (13).Figure 13: The plot of ( J m + k ) versus k a (a) and versus k d (b) shows the region where both theconditions given in equations (35) and (36) are satisfied (Region I). The rate constants are taken as k b = 20 , k = 50 , k m = 1 , k d = 10 (in (a)), and k a = 10 (in (b)). The condition given in equation(35) is satisfied in regions I and IV whereas the condition given in equation (36) is satisfied in regionsII and III.Figure 13 shows the four different regions in the ( k a , J m + k ) (Fig. 13(a)) or ( k d , J m + k ) (Fig.13(b)) space in Suter model. With the gradual decrease in the rate constant k b , the area of the RegionII decreases gradually and the the two curves intersect at higher (lower) value of k a ( k d ) in figure13(a) (Fig. 13(b)). By modulating the rate constants k a , k d or k b the cellular system can change thefunctional region according to its requirement. In this article, we study the effect of transcriptional reinitiation by RNAP-II in gene expression.Transcriptional reinitiation is an important step in gene expression though it is ignored in most ofthe model networks assuming it has insignificant role in mRNA and protein levels. But, Blake et. el.identify that reinitiation of transcription can be crucial in eukaryotic systems [7, 8]. To find out theeffect of transcriptional reinitiation on phenotypic variability, we consider different gene regulatorynetworks, with and without the reinitiation step. When the constitutive gene network is analyzed inpresence of reinitiation, we observe that the mean mRNA level and Fano factor both are reduced.The behaviour is similar to a negative feedback amplifier which reduces the gain and noise. Though,there is a fundamental difference between a negative feedback amplifier in electronic circuits andthe observed negative feedback like behavior in the constitutive gene network with transcriptionalreinitiation. In the electronic negative feedback circuit, a fraction of the output voltage is fed back tothe input but in the present reinitiation based circuit the gene product/mRNA levels are not involvedat all. Thus the reinitiation based negative feedback in constitutive gene transcription is completelyinherent in nature. 14hen we study the effect of transcriptional reinitiation in the two-state gene expression process.Here, in absence of reinitiation, the Fano factor at mRNA level is higher than unity due to ran-dom transitions between the active and inactive states of the gene. Now, with the reinitiation oftranscription in the two-state model, we observe four different phenotypic outcomes (
F F tswrm < and r > ; F F tswrm < and r < ; F F tswrm > and r < ; F F tswrm > and r > ; where r = < m tswr > / < m tswtr > ) depending on the rate constants of the reactions. Similar behaviour isobserved for the Suter model also. Though the mean mRNA level is higher and the Fano factor islower over a wide region of parameter variation in the Suter model. We find that the gene regulatorynetwork like the two-state and Suter model with RNAP-II based transcriptional reinitiation processcan behave as the mixed feedback process. In this work, the rate constants are chosen from differentworks [11, 29].Noise in gene expression plays an important role in cellular behaviour and disease control [36, 37,38, 39]. For the appropriate functioning of the cellular system, a specific average level of mRNA andprotein is crucial [34, 35]. Recent work shows that reinitiation of transcription has the capability toreduce the Fano factor below unity i.e., to the sub-Poissonian regime in the two-state process[29].We now observe that transcriptional reinitiation has an important role not only controlling the Fanofactor but also the average in the mRNA levels. In the two-state gene expression process, the cellularsystem can regulate its mean and Fano factor by controlling the rate constants responsible for randomtransitions between the gene states. In that process, the Fano factor can be reduced up to unity forlarge k a and small k d . But, in presence of reinitiation of gene transcription, the cellular system candecrease the Fano factor below unity at a lower value of k a . At the same time, the average mRNAlevel compared to the two-state process can be increased. This is very much important to controldiseases like haploinsufficiency [34, 35, 38]. The noise in gene expression can also be the survivalstrategy for cells in adverse environmental conditions [36, 37, 39]. The reinitiation of gene expressioncan also be helpful in such situations by selecting the higher Fano factor and appropriate mean mRNAvalues. Thus, the reinitiation of gene transcription can play a crucial role to determine the phenotypicoutcome of the cellular systems. Appendix
A. Constitutive gene expression with pre-initiation and initia-tion complexes
The biochemical reactions for constitutive gene expression with pre-initiation and initiation complexesare shown in figure (14). The RNAP-II molecule binds the active gene and forms a pre-initiationcomplex G t . Then further modifications in that produce the initiation complex from which mRNAsynthesis takes place [25]. 15igure 14: Reaction scheme with the rate constants for constitutive gene expression with pre-initiationand initiation complexes. G is the open active state and k is the rate constant for the open activeto the pre-initiation complex formation. k is the rate constant for the pre-initiation to the initiationcomplex formation. J m is the transcription rate constant and k m is the mRNA degradation rateconstant.Let, there are l copy number of a particular gene exist in the cell. The Master equation describingthe rate of change of probability P ( n , n , n , t ) with n number of mRNAs and n number of genesin the pre-initiation state ( G t ) and n number of genes in the initiation complex ( G c ) is given by ∂p ( n ,n ,n ,t ) ∂t = k [( l − n − n + 1) p ( n − , n , n , t ) − ( l − n − n ) p ( n , n , n , t )]+ k [( n + 1) p ( n + 1 , n − , n , t ) − n p ( n , n , n , t )]+ J m [ { ( n + 1) p ( n , n + 1 , n − , t ) − n p ( n , n , n , t )]+ k m [( n + 1) p ( n , n , n + 1 , t ) − n p ( n , n , n , t )] (37)If the reinitiation process happens in two states (as shown in figure (14)) then the expression ofmean mRNA and Fano factor at mRNA level are given by < m cwrtsr > = k k J m k + J m k + k k J m k m ; (38) F F cwrtsm = 1 − J m k k ( J m + k + k + k m )( J m k + J m k + k k ) { J m ( k + k + k m ) + ( k + k m )( k + k m ) } (39)Figure 15: Variation of mean mRNA and Fano factor with J m for the rate constants k = k = 4 and k m = 1 . Figure (15) shows that consideration of the pre-initiation complex in the transcription initiationprocess results the decrease in mean and Fano factor further.
B. Two-state gene expression with pre-initiation and initiation complexes
The biochemical reactions for the two-state gene activation process with pre-initiation and initiationsteps are shown in figure (16). The gene state G t is the pre-initiation complex and G c is the initiation16omplex. From the initiation complex, the RNAP-II starts transcription for mRNA synthesis andthe gene turns into an open active state.Figure 16: Reaction scheme for the two-state gene expression with pre-initiation and initiation com-plexes. G i ( G a ) is the inactive (active) state. k a ( k d ) is the activation (deactivation) rate constant. k ( k ) is the rate constant of pre-initiation (initiation) complex formation and k ( k ) is the rateconstant of dissociation of RNAP-II from pre-initiation (initiation) complex. J m is the transcriptionrate constant and k m is the mRNA degradation rate constant.Let p ( n , n , n , n , t ) be the probability that at time t , there are n number of mRNAs with n number of genes in the active state ( G a ) , n number of genes in the pre-initiation state ( G t ) and n number of genes in the transcription initiation complex ( G c ). The number of gene in the inactivestate ( G i ) are ( l − n − n − n ) with l being the copy number of a particular gene. The time evaluationof the probability is given by the Master equation ∂p ( n ,n ,n ,n ,t ) ∂t = k a [ { l − ( n − n + n ) } p ( n − , n , n , n , t ) −{ l − ( n + n + n ) } p ( n , n , n , n , t )]+ k d [( n + 1) p ( n + 1 , n , n , n , t ) − n p ( n , n , n , n , t )]+ k [( n + 1) p ( n + 1 , n − , n , n , t ) − n p ( n , n , n , n , t )]+ k [( n + 1) p ( n − , n + 1 , n , n , t ) − n p ( n , n , n , n , t )]+ k [( n + 1) p ( n , n + 1 , n − , n , t ) − n p ( n , n , n , n , t )]+ k [( n + 1) p ( n , n − , n + 1 , n , t ) − n p ( n , n , n , n , t )]+ J m [( n + 1) p ( n − , n , n + 1 , n − , t ) − n p ( n , n , n , n , t )]+ k m [( n + 1) p ( n , n , n , n + 1 , t ) − n p ( n , n , n , n , t )] (40)The expressions for mean mRNA and Fano factor for mRNA ( F F m ) are given by < m tswrth > = k a k m b ( b + b ) (41) F F tswrthm = 1 − < m tswrth > + ( k a + k m ) b ( k m ( b + J m ( k d ( b − k ) + ( k a + k m ) b ) + k a ( k ( k + k m ) + b ( k + k m ))) (42)where b = J m k k , b = J m ( k a k + k a k + k d k + k a k + k d k ) , b = k m ( k d k + k d k + k k + k d k + k k + k k ) , b = k a ( k k + k k + k k ) + k d k k , b = k m ( k d + k + k + k + k + k m ) , b = ( k + k + k + k m ) , b = ( k + k + k m ) , b = k d k k + b + b .17igure 17: Plot of mean mRNA and Fano factor corresponding to the figure (16) with J m for therate constants k a = k d = 10 , k = k = 1 , k m = 1 and three different sets of k and k . The solid linecorresponds to the two-state process without reinitiation (Fig. 4(a)).Figure 17 shows that the mean and the Fano factor can be controlled in better ways by controllingthe rate constants k and k . References [1] Ko M S 1991 A stochastic model for gene induction
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